2つの形状パラメータαとβを持つベータ分布は、範囲[0,1]または(0,1)でサポートされています。分布の位置とスケールを変更するには、分布の最小値aと最大値c ( c > a )を表す2つのパラメータを追加し、[1] 無次元変数xを新しい変数y (サポート[ a , c ]または( a , c ))とパラメータaとcで置き換える線形変換を使用します。
4 パラメータ ベータ分布の確率密度関数は、範囲 ( c − a ) でスケーリングされた 2 パラメータ分布に等しく (密度曲線の下の合計面積が確率 1 に等しくなるように)、"y" 変数は次のようにシフトおよびスケーリングされます。
ベータ分布の（平均−幾何平均）とαとβの関係を0から2まで示したもので、幾何平均のαとβの非対称性を示している。ベータ分布の幾何平均 紫 = G ( x )、黄色 = G (1 − x )、小さい方の値はαとβベータ分布の幾何平均。紫 = G ( x )、黄色 = G (1 − x )、前方に大きい値αとβ
ランダム変数Xを持つ分布の幾何平均G Xの対数はln( X )の算術平均、または同等にその期待値です。
このため、正規化された結果、つまり基準値に対する比率として提示される結果を平均化する場合、幾何平均が唯一の正しい平均となります。 [12] これが関連しているのは、ベータ分布がパーセンテージのランダムな動作に適したモデルであり、特に比率の統計的モデリングに適しているためです。幾何平均は、最尤推定において中心的な役割を果たします。「パラメータ推定、最尤」のセクションを参照してください。実際、最尤推定を実行すると、ランダム変数 X に基づく幾何平均G X の他に、もう 1 つの幾何平均も自然に現れます。それは、線形変換(1 − X )に基づく幾何平均で、 Xの鏡像であり、G (1− X )と表されます。
G Xと G (1− X ) は両方とも非対称ですが、両方の形状パラメータが等しい場合( α = β )、幾何平均は等しくなります: G X = G (1− X )。この等式は、両方の幾何平均の間に示される次の対称性から生じます:
調和平均
0 < α < 5 および 0 < β < 5のベータ分布の調和平均 0から2までのベータ分布とαおよびβの調和平均ベータ分布の調和平均 紫 = H ( X )、黄色 = H (1 − X )、前方に小さい値αとβベータ分布の調和平均: 紫 = H ( X )、黄色 = H (1 − X )、前方に大きい値αとβ
ランダム変数Xを持つ分布の調和平均( H X )の逆数は、 1/ Xの算術平均、または同等にその期待値です。したがって、形状パラメータαとβを持つベータ分布の調和平均( H X ) は次のようになります。
α < 1のベータ分布の調和平均( H X )は定義されていません。これは、定義式が形状パラメータα が1 未満の場合に [0, 1] で制限されないためです。
ジョンソン[24]は、ロジット変換された変数ln( X /1− X ) の分布を、そのモーメント生成関数と形状パラメータの大きな値に対する近似を含めて検討した。この変換は、元の変数Xに基づく有限サポート[0, 1]を、実数直線(−∞, +∞)の両方向の無限サポートに拡張する。ベータ変量のロジットはロジスティックベータ分布に従う。
(for further discussion see section on Fisher information matrix). Thus, it is not possible to strictly carry on the maximum likelihood estimation for some well known distributions belonging to the four-parameter beta distribution family, like the uniform distribution (Beta(1, 1, a, c)), and the arcsine distribution (Beta(1/2, 1/2, a, c)). N.L.Johnson and S.Kotz[1] ignore the equations for the harmonic means and instead suggest "If a and c are unknown, and maximum likelihood estimators of a, c, α and β are required, the above procedure (for the two unknown parameter case, with X transformed as X = (Y − a)/(c − a)) can be repeated using a succession of trial values of a and c, until the pair (a, c) for which maximum likelihood (given a and c) is as great as possible, is attained" (where, for the purpose of clarity, their notation for the parameters has been translated into the present notation).
Fisher information matrix
Let a random variable X have a probability density f(x;α). The partial derivative with respect to the (unknown, and to be estimated) parameter α of the log likelihood function is called the score. The second moment of the score is called the Fisher information:
The expectation of the score is zero, therefore the Fisher information is also the second moment centered on the mean of the score: the variance of the score.
If the log likelihood function is twice differentiable with respect to the parameter α, and under certain regularity conditions,[50] then the Fisher information may also be written as follows (which is often a more convenient form for calculation purposes):
Thus, the Fisher information is the negative of the expectation of the second derivative with respect to the parameter α of the log likelihood function. Therefore, Fisher information is a measure of the curvature of the log likelihood function of α. A low curvature (and therefore high radius of curvature), flatter log likelihood function curve has low Fisher information; while a log likelihood function curve with large curvature (and therefore low radius of curvature) has high Fisher information. When the Fisher information matrix is computed at the evaluates of the parameters ("the observed Fisher information matrix") it is equivalent to the replacement of the true log likelihood surface by a Taylor's series approximation, taken as far as the quadratic terms.[51] The word information, in the context of Fisher information, refers to information about the parameters. Information such as: estimation, sufficiency and properties of variances of estimators. The Cramér–Rao bound states that the inverse of the Fisher information is a lower bound on the variance of any estimator of a parameter α:
The precision to which one can estimate the estimator of a parameter α is limited by the Fisher Information of the log likelihood function. The Fisher information is a measure of the minimum error involved in estimating a parameter of a distribution and it can be viewed as a measure of the resolving power of an experiment needed to discriminate between two alternative hypothesis of a parameter.[52]
Under certain regularity conditions,[50] the Fisher Information Matrix may also be written in the following form, which is often more convenient for computation:
With X1, ..., XNiid random variables, an N-dimensional "box" can be constructed with sides X1, ..., XN. Costa and Cover[53] show that the (Shannon) differential entropy h(X) is related to the volume of the typical set (having the sample entropy close to the true entropy), while the Fisher information is related to the surface of this typical set.
Two parameters
For X1, ..., XN independent random variables each having a beta distribution parametrized with shape parameters α and β, the joint log likelihood function for Niid observations is:
therefore the joint log likelihood function per Niid observations is
For the two parameter case, the Fisher information has 4 components: 2 diagonal and 2 off-diagonal. Since the Fisher information matrix is symmetric, one of these off diagonal components is independent. Therefore, the Fisher information matrix has 3 independent components (2 diagonal and 1 off diagonal).
Aryal and Nadarajah[54] calculated Fisher's information matrix for the four-parameter case, from which the two parameter case can be obtained as follows:
Since the Fisher information matrix is symmetric
The Fisher information components are equal to the log geometric variances and log geometric covariance. Therefore, they can be expressed as trigamma functions, denoted ψ1(α), the second of the polygamma functions, defined as the derivative of the digamma function:
These derivatives are also derived in the § Two unknown parameters and plots of the log likelihood function are also shown in that section. § Geometric variance and covariance contains plots and further discussion of the Fisher information matrix components: the log geometric variances and log geometric covariance as a function of the shape parameters α and β. § Moments of logarithmically transformed random variables contains formulas for moments of logarithmically transformed random variables. Images for the Fisher information components and are shown in § Geometric variance.
The determinant of Fisher's information matrix is of interest (for example for the calculation of Jeffreys prior probability). From the expressions for the individual components of the Fisher information matrix, it follows that the determinant of Fisher's (symmetric) information matrix for the beta distribution is:
From Sylvester's criterion (checking whether the diagonal elements are all positive), it follows that the Fisher information matrix for the two parameter case is positive-definite (under the standard condition that the shape parameters are positive α > 0 and β > 0).
Four parameters
Fisher Information I(a,a) for α = β vs range (c − a) and exponent α = βFisher Information I(α,a) for α = β, vs. range (c − a) and exponent α = β
If Y1, ..., YN are independent random variables each having a beta distribution with four parameters: the exponents α and β, and also a (the minimum of the distribution range), and c (the maximum of the distribution range) (section titled "Alternative parametrizations", "Four parameters"), with probability density function:
the joint log likelihood function per Niid observations is:
For the four parameter case, the Fisher information has 4*4=16 components. It has 12 off-diagonal components = (4×4 total − 4 diagonal). Since the Fisher information matrix is symmetric, half of these components (12/2=6) are independent. Therefore, the Fisher information matrix has 6 independent off-diagonal + 4 diagonal = 10 independent components. Aryal and Nadarajah[54] calculated Fisher's information matrix for the four parameter case as follows:
In the above expressions, the use of X instead of Y in the expressions var[ln(X)] = ln(varGX) is not an error. The expressions in terms of the log geometric variances and log geometric covariance occur as functions of the two parameter X ~ Beta(α, β) parametrization because when taking the partial derivatives with respect to the exponents (α, β) in the four parameter case, one obtains the identical expressions as for the two parameter case: these terms of the four parameter Fisher information matrix are independent of the minimum a and maximum c of the distribution's range. The only non-zero term upon double differentiation of the log likelihood function with respect to the exponents α and β is the second derivative of the log of the beta function: ln(B(α, β)). This term is independent of the minimum a and maximum c of the distribution's range. Double differentiation of this term results in trigamma functions. The sections titled "Maximum likelihood", "Two unknown parameters" and "Four unknown parameters" also show this fact.
The Fisher information for Ni.i.d. samples is N times the individual Fisher information (eq. 11.279, page 394 of Cover and Thomas[27]). (Aryal and Nadarajah[54] take a single observation, N = 1, to calculate the following components of the Fisher information, which leads to the same result as considering the derivatives of the log likelihood per N observations. Moreover, below the erroneous expression for in Aryal and Nadarajah has been corrected.)
The lower two diagonal entries of the Fisher information matrix, with respect to the parameter a (the minimum of the distribution's range): , and with respect to the parameter c (the maximum of the distribution's range): are only defined for exponents α > 2 and β > 2 respectively. The Fisher information matrix component for the minimum a approaches infinity for exponent α approaching 2 from above, and the Fisher information matrix component for the maximum c approaches infinity for exponent β approaching 2 from above.
The Fisher information matrix for the four parameter case does not depend on the individual values of the minimum a and the maximum c, but only on the total range (c − a). Moreover, the components of the Fisher information matrix that depend on the range (c − a), depend only through its inverse (or the square of the inverse), such that the Fisher information decreases for increasing range (c − a).
The accompanying images show the Fisher information components and . Images for the Fisher information components and are shown in § Geometric variance. All these Fisher information components look like a basin, with the "walls" of the basin being located at low values of the parameters.
The following four-parameter-beta-distribution Fisher information components can be expressed in terms of the two-parameter: X ~ Beta(α, β) expectations of the transformed ratio ((1 − X)/X) and of its mirror image (X/(1 − X)), scaled by the range (c − a), which may be helpful for interpretation:
These are also the expected values of the "inverted beta distribution" or beta prime distribution (also known as beta distribution of the second kind or Pearson's Type VI) [1] and its mirror image, scaled by the range (c − a).
Also, the following Fisher information components can be expressed in terms of the harmonic (1/X) variances or of variances based on the ratio transformed variables ((1-X)/X) as follows:
See section "Moments of linearly transformed, product and inverted random variables" for these expectations.
The determinant of Fisher's information matrix is of interest (for example for the calculation of Jeffreys prior probability). From the expressions for the individual components, it follows that the determinant of Fisher's (symmetric) information matrix for the beta distribution with four parameters is:
Using Sylvester's criterion (checking whether the diagonal elements are all positive), and since diagonal components and have singularities at α=2 and β=2 it follows that the Fisher information matrix for the four parameter case is positive-definite for α>2 and β>2. Since for α > 2 and β > 2 the beta distribution is (symmetric or unsymmetric) bell shaped, it follows that the Fisher information matrix is positive-definite only for bell-shaped (symmetric or unsymmetric) beta distributions, with inflection points located to either side of the mode. Thus, important well known distributions belonging to the four-parameter beta distribution family, like the parabolic distribution (Beta(2,2,a,c)) and the uniform distribution (Beta(1,1,a,c)) have Fisher information components () that blow up (approach infinity) in the four-parameter case (although their Fisher information components are all defined for the two parameter case). The four-parameter Wigner semicircle distribution (Beta(3/2,3/2,a,c)) and arcsine distribution (Beta(1/2,1/2,a,c)) have negative Fisher information determinants for the four-parameter case.
Examples of beta distributions used as prior probabilities to represent ignorance of prior parameter values in Bayesian inference are Beta(1,1), Beta(0,0) and Beta(1/2,1/2).
Rule of succession
A classic application of the beta distribution is the rule of succession, introduced in the 18th century by Pierre-Simon Laplace[55] in the course of treating the sunrise problem. It states that, given s successes in nconditionally independentBernoulli trials with probability p, that the estimate of the expected value in the next trial is . This estimate is the expected value of the posterior distribution over p, namely Beta(s+1, n−s+1), which is given by Bayes' rule if one assumes a uniform prior probability over p (i.e., Beta(1, 1)) and then observes that p generated s successes in n trials. Laplace's rule of succession has been criticized by prominent scientists. R. T. Cox described Laplace's application of the rule of succession to the sunrise problem ([56] p. 89) as "a travesty of the proper use of the principle". Keynes remarks ([57] Ch.XXX, p. 382) "indeed this is so foolish a theorem that to entertain it is discreditable". Karl Pearson[58] showed that the probability that the next (n + 1) trials will be successes, after n successes in n trials, is only 50%, which has been considered too low by scientists like Jeffreys and unacceptable as a representation of the scientific process of experimentation to test a proposed scientific law. As pointed out by Jeffreys ([59] p. 128) (crediting C. D. Broad[60] ) Laplace's rule of succession establishes a high probability of success ((n+1)/(n+2)) in the next trial, but only a moderate probability (50%) that a further sample (n+1) comparable in size will be equally successful. As pointed out by Perks,[61] "The rule of succession itself is hard to accept. It assigns a probability to the next trial which implies the assumption that the actual run observed is an average run and that we are always at the end of an average run. It would, one would think, be more reasonable to assume that we were in the middle of an average run. Clearly a higher value for both probabilities is necessary if they are to accord with reasonable belief." These problems with Laplace's rule of succession motivated Haldane, Perks, Jeffreys and others to search for other forms of prior probability (see the next § Bayesian inference). According to Jaynes,[52] the main problem with the rule of succession is that it is not valid when s=0 or s=n (see rule of succession, for an analysis of its validity).
Bayes–Laplace prior probability (Beta(1,1))
The beta distribution achieves maximum differential entropy for Beta(1,1): the uniform probability density, for which all values in the domain of the distribution have equal density. This uniform distribution Beta(1,1) was suggested ("with a great deal of doubt") by Thomas Bayes[62] as the prior probability distribution to express ignorance about the correct prior distribution. This prior distribution was adopted (apparently, from his writings, with little sign of doubt[55]) by Pierre-Simon Laplace, and hence it was also known as the "Bayes–Laplace rule" or the "Laplace rule" of "inverse probability" in publications of the first half of the 20th century. In the later part of the 19th century and early part of the 20th century, scientists realized that the assumption of uniform "equal" probability density depended on the actual functions (for example whether a linear or a logarithmic scale was most appropriate) and parametrizations used. In particular, the behavior near the ends of distributions with finite support (for example near x = 0, for a distribution with initial support at x = 0) required particular attention. Keynes ([57] Ch.XXX, p. 381) criticized the use of Bayes's uniform prior probability (Beta(1,1)) that all values between zero and one are equiprobable, as follows: "Thus experience, if it shows anything, shows that there is a very marked clustering of statistical ratios in the neighborhoods of zero and unity, of those for positive theories and for correlations between positive qualities in the neighborhood of zero, and of those for negative theories and for correlations between negative qualities in the neighborhood of unity. "
Haldane's prior probability (Beta(0,0))
: The Haldane prior probability expressing total ignorance about prior information, where we are not even sure whether it is physically possible for an experiment to yield either a success or a failure. As α, β → 0, the beta distribution approaches a two-point Bernoulli distribution with all probability density concentrated at each end, at 0 and 1, and nothing in between. A coin-toss: one face of the coin being at 0 and the other face being at 1.
The Beta(0,0) distribution was proposed by J.B.S. Haldane,[63] who suggested that the prior probability representing complete uncertainty should be proportional to p−1(1−p)−1. The function p−1(1−p)−1 can be viewed as the limit of the numerator of the beta distribution as both shape parameters approach zero: α, β → 0. The Beta function (in the denominator of the beta distribution) approaches infinity, for both parameters approaching zero, α, β → 0. Therefore, p−1(1−p)−1 divided by the Beta function approaches a 2-point Bernoulli distribution with equal probability 1/2 at each end, at 0 and 1, and nothing in between, as α, β → 0. A coin-toss: one face of the coin being at 0 and the other face being at 1. The Haldane prior probability distribution Beta(0,0) is an "improper prior" because its integration (from 0 to 1) fails to strictly converge to 1 due to the singularities at each end. However, this is not an issue for computing posterior probabilities unless the sample size is very small. Furthermore, Zellner[64] points out that on the log-odds scale, (the logit transformation ln(p/1 − p)), the Haldane prior is the uniformly flat prior. The fact that a uniform prior probability on the logit transformed variable ln(p/1 − p) (with domain (−∞, ∞)) is equivalent to the Haldane prior on the domain [0, 1] was pointed out by Harold Jeffreys in the first edition (1939) of his book Theory of Probability ([59] p. 123). Jeffreys writes "Certainly if we take the Bayes–Laplace rule right up to the extremes we are led to results that do not correspond to anybody's way of thinking. The (Haldane) rule dx/(x(1 − x)) goes too far the other way. It would lead to the conclusion that if a sample is of one type with respect to some property there is a probability 1 that the whole population is of that type." The fact that "uniform" depends on the parametrization, led Jeffreys to seek a form of prior that would be invariant under different parametrizations.
Jeffreys' prior probability (Beta(1/2,1/2) for a Bernoulli or for a binomial distribution)
Jeffreys prior probability for the beta distribution: the square root of the determinant of Fisher's information matrix: is a function of the trigamma function ψ1 of shape parameters α, βPosterior Beta densities with samples having success = "s", failure = "f" of s/(s + f) = 1/2, and s + f = {3,10,50}, based on 3 different prior probability functions: Haldane (Beta(0,0), Jeffreys (Beta(1/2,1/2)) and Bayes (Beta(1,1)). The image shows that there is little difference between the priors for the posterior with sample size of 50 (with more pronounced peak near p = 1/2). Significant differences appear for very small sample sizes (the flatter distribution for sample size of 3)Posterior Beta densities with samples having success = "s", failure = "f" of s/(s + f) = 1/4, and s + f ∈ {3,10,50}, based on three different prior probability functions: Haldane (Beta(0,0), Jeffreys (Beta(1/2,1/2)) and Bayes (Beta(1,1)). The image shows that there is little difference between the priors for the posterior with sample size of 50 (with more pronounced peak near p = 1/4). Significant differences appear for very small sample sizes (the very skewed distribution for the degenerate case of sample size = 3, in this degenerate and unlikely case the Haldane prior results in a reverse "J" shape with mode at p = 0 instead of p = 1/4. If there is sufficient sampling data, the three priors of Bayes (Beta(1,1)), Jeffreys (Beta(1/2,1/2)) and Haldane (Beta(0,0)) should yield similar posterior probability densities.Posterior Beta densities with samples having success = s, failure = f of s/(s + f) = 1/4, and s + f ∈ {4,12,40}, based on three different prior probability functions: Haldane (Beta(0,0), Jeffreys (Beta(1/2,1/2)) and Bayes (Beta(1,1)). The image shows that there is little difference between the priors for the posterior with sample size of 40 (with more pronounced peak near p = 1/4). Significant differences appear for very small sample sizes
Harold Jeffreys[59][65] proposed to use an uninformative prior probability measure that should be invariant under reparameterization: proportional to the square root of the determinant of Fisher's information matrix. For the Bernoulli distribution, this can be shown as follows: for a coin that is "heads" with probability p ∈ [0, 1] and is "tails" with probability 1 − p, for a given (H,T) ∈ {(0,1), (1,0)} the probability is pH(1 − p)T. Since T = 1 − H, the Bernoulli distribution is pH(1 − p)1 − H. Considering p as the only parameter, it follows that the log likelihood for the Bernoulli distribution is
The Fisher information matrix has only one component (it is a scalar, because there is only one parameter: p), therefore:
It will be shown in the next section that the normalizing constant for Jeffreys prior is immaterial to the final result because the normalizing constant cancels out in Bayes theorem for the posterior probability. Hence Beta(1/2,1/2) is used as the Jeffreys prior for both Bernoulli and binomial distributions. As shown in the next section, when using this expression as a prior probability times the likelihood in Bayes theorem, the posterior probability turns out to be a beta distribution. It is important to realize, however, that Jeffreys prior is proportional to for the Bernoulli and binomial distribution, but not for the beta distribution. Jeffreys prior for the beta distribution is given by the determinant of Fisher's information for the beta distribution, which, as shown in the § Fisher information matrix is a function of the trigamma function ψ1 of shape parameters α and β as follows:
As previously discussed, Jeffreys prior for the Bernoulli and binomial distributions is proportional to the arcsine distribution Beta(1/2,1/2), a one-dimensional curve that looks like a basin as a function of the parameter p of the Bernoulli and binomial distributions. The walls of the basin are formed by p approaching the singularities at the ends p → 0 and p → 1, where Beta(1/2,1/2) approaches infinity. Jeffreys prior for the beta distribution is a 2-dimensional surface (embedded in a three-dimensional space) that looks like a basin with only two of its walls meeting at the corner α = β = 0 (and missing the other two walls) as a function of the shape parameters α and β of the beta distribution. The two adjoining walls of this 2-dimensional surface are formed by the shape parameters α and β approaching the singularities (of the trigamma function) at α, β → 0. It has no walls for α, β → ∞ because in this case the determinant of Fisher's information matrix for the beta distribution approaches zero.
It will be shown in the next section that Jeffreys prior probability results in posterior probabilities (when multiplied by the binomial likelihood function) that are intermediate between the posterior probability results of the Haldane and Bayes prior probabilities.
Jeffreys prior may be difficult to obtain analytically, and for some cases it just doesn't exist (even for simple distribution functions like the asymmetric triangular distribution). Berger, Bernardo and Sun, in a 2009 paper[66] defined a reference prior probability distribution that (unlike Jeffreys prior) exists for the asymmetric triangular distribution. They cannot obtain a closed-form expression for their reference prior, but numerical calculations show it to be nearly perfectly fitted by the (proper) prior
where θ is the vertex variable for the asymmetric triangular distribution with support [0, 1] (corresponding to the following parameter values in Wikipedia's article on the triangular distribution: vertex c = θ, left end a = 0,and right end b = 1). Berger et al. also give a heuristic argument that Beta(1/2,1/2) could indeed be the exact Berger–Bernardo–Sun reference prior for the asymmetric triangular distribution. Therefore, Beta(1/2,1/2) not only is Jeffreys prior for the Bernoulli and binomial distributions, but also seems to be the Berger–Bernardo–Sun reference prior for the asymmetric triangular distribution (for which the Jeffreys prior does not exist), a distribution used in project management and PERT analysis to describe the cost and duration of project tasks.
Clarke and Barron[67] prove that, among continuous positive priors, Jeffreys prior (when it exists) asymptotically maximizes Shannon's mutual information between a sample of size n and the parameter, and therefore Jeffreys prior is the most uninformative prior (measuring information as Shannon information). The proof rests on an examination of the Kullback–Leibler divergence between probability density functions for iid random variables.
Effect of different prior probability choices on the posterior beta distribution
If samples are drawn from the population of a random variable X that result in s successes and f failures in nBernoulli trialsn = s + f, then the likelihood function for parameters s and f given x = p (the notation x = p in the expressions below will emphasize that the domain x stands for the value of the parameter p in the binomial distribution), is the following binomial distribution:
If beliefs about prior probability information are reasonably well approximated by a beta distribution with parameters α Prior and β Prior, then:
According to Bayes' theorem for a continuous event space, the posterior probability density is given by the product of the prior probability and the likelihood function (given the evidence s and f = n − s), normalized so that the area under the curve equals one, as follows:
appears both in the numerator and the denominator of the posterior probability, and it does not depend on the integration variable x, hence it cancels out, and it is irrelevant to the final result. Similarly the normalizing factor for the prior probability, the beta function B(αPrior,βPrior) cancels out and it is immaterial to the final result. The same posterior probability result can be obtained if one uses an un-normalized prior
because the normalizing factors all cancel out. Several authors (including Jeffreys himself) thus use an un-normalized prior formula since the normalization constant cancels out. The numerator of the posterior probability ends up being just the (un-normalized) product of the prior probability and the likelihood function, and the denominator is its integral from zero to one. The beta function in the denominator, B(s + α Prior, n − s + β Prior), appears as a normalization constant to ensure that the total posterior probability integrates to unity.
The ratio s/n of the number of successes to the total number of trials is a sufficient statistic in the binomial case, which is relevant for the following results.
For the Bayes' prior probability (Beta(1,1)), the posterior probability is:
For the Jeffreys' prior probability (Beta(1/2,1/2)), the posterior probability is:
and for the Haldane prior probability (Beta(0,0)), the posterior probability is:
From the above expressions it follows that for s/n = 1/2) all the above three prior probabilities result in the identical location for the posterior probability mean = mode = 1/2. For s/n < 1/2, the mean of the posterior probabilities, using the following priors, are such that: mean for Bayes prior > mean for Jeffreys prior > mean for Haldane prior. For s/n > 1/2 the order of these inequalities is reversed such that the Haldane prior probability results in the largest posterior mean. The Haldane prior probability Beta(0,0) results in a posterior probability density with mean (the expected value for the probability of success in the "next" trial) identical to the ratio s/n of the number of successes to the total number of trials. Therefore, the Haldane prior results in a posterior probability with expected value in the next trial equal to the maximum likelihood. The Bayes prior probability Beta(1,1) results in a posterior probability density with mode identical to the ratio s/n (the maximum likelihood).
In the case that 100% of the trials have been successful s = n, the Bayes prior probability Beta(1,1) results in a posterior expected value equal to the rule of succession (n + 1)/(n + 2), while the Haldane prior Beta(0,0) results in a posterior expected value of 1 (absolute certainty of success in the next trial). Jeffreys prior probability results in a posterior expected value equal to (n + 1/2)/(n + 1). Perks[61] (p. 303) points out: "This provides a new rule of succession and expresses a 'reasonable' position to take up, namely, that after an unbroken run of n successes we assume a probability for the next trial equivalent to the assumption that we are about half-way through an average run, i.e. that we expect a failure once in (2n + 2) trials. The Bayes–Laplace rule implies that we are about at the end of an average run or that we expect a failure once in (n + 2) trials. The comparison clearly favours the new result (what is now called Jeffreys prior) from the point of view of 'reasonableness'."
Conversely, in the case that 100% of the trials have resulted in failure (s = 0), the Bayes prior probability Beta(1,1) results in a posterior expected value for success in the next trial equal to 1/(n + 2), while the Haldane prior Beta(0,0) results in a posterior expected value of success in the next trial of 0 (absolute certainty of failure in the next trial). Jeffreys prior probability results in a posterior expected value for success in the next trial equal to (1/2)/(n + 1), which Perks[61] (p. 303) points out: "is a much more reasonably remote result than the Bayes–Laplace result 1/(n + 2)".
Jaynes[52] questions (for the uniform prior Beta(1,1)) the use of these formulas for the cases s = 0 or s = n because the integrals do not converge (Beta(1,1) is an improper prior for s = 0 or s = n). In practice, the conditions 0<s<n necessary for a mode to exist between both ends for the Bayes prior are usually met, and therefore the Bayes prior (as long as 0 < s < n) results in a posterior mode located between both ends of the domain.
As remarked in the section on the rule of succession, K. Pearson showed that after n successes in n trials the posterior probability (based on the Bayes Beta(1,1) distribution as the prior probability) that the next (n + 1) trials will all be successes is exactly 1/2, whatever the value of n. Based on the Haldane Beta(0,0) distribution as the prior probability, this posterior probability is 1 (absolute certainty that after n successes in n trials the next (n + 1) trials will all be successes). Perks[61] (p. 303) shows that, for what is now known as the Jeffreys prior, this probability is ((n + 1/2)/(n + 1))((n + 3/2)/(n + 2))...(2n + 1/2)/(2n + 1), which for n = 1, 2, 3 gives 15/24, 315/480, 9009/13440; rapidly approaching a limiting value of as n tends to infinity. Perks remarks that what is now known as the Jeffreys prior: "is clearly more 'reasonable' than either the Bayes–Laplace result or the result on the (Haldane) alternative rule rejected by Jeffreys which gives certainty as the probability. It clearly provides a very much better correspondence with the process of induction. Whether it is 'absolutely' reasonable for the purpose, i.e. whether it is yet large enough, without the absurdity of reaching unity, is a matter for others to decide. But it must be realized that the result depends on the assumption of complete indifference and absence of knowledge prior to the sampling experiment."
Following are the variances of the posterior distribution obtained with these three prior probability distributions:
for the Bayes' prior probability (Beta(1,1)), the posterior variance is:
for the Jeffreys' prior probability (Beta(1/2,1/2)), the posterior variance is:
and for the Haldane prior probability (Beta(0,0)), the posterior variance is:
So, as remarked by Silvey,[50] for large n, the variance is small and hence the posterior distribution is highly concentrated, whereas the assumed prior distribution was very diffuse. This is in accord with what one would hope for, as vague prior knowledge is transformed (through Bayes theorem) into a more precise posterior knowledge by an informative experiment. For small n the Haldane Beta(0,0) prior results in the largest posterior variance while the Bayes Beta(1,1) prior results in the more concentrated posterior. Jeffreys prior Beta(1/2,1/2) results in a posterior variance in between the other two. As n increases, the variance rapidly decreases so that the posterior variance for all three priors converges to approximately the same value (approaching zero variance as n → ∞). Recalling the previous result that the Haldane prior probability Beta(0,0) results in a posterior probability density with mean (the expected value for the probability of success in the "next" trial) identical to the ratio s/n of the number of successes to the total number of trials, it follows from the above expression that also the Haldane prior Beta(0,0) results in a posterior with variance identical to the variance expressed in terms of the max. likelihood estimate s/n and sample size (in § Variance):
with the mean μ = s/n and the sample size ν = n.
In Bayesian inference, using a prior distribution Beta(αPrior,βPrior) prior to a binomial distribution is equivalent to adding (αPrior − 1) pseudo-observations of "success" and (βPrior − 1) pseudo-observations of "failure" to the actual number of successes and failures observed, then estimating the parameter p of the binomial distribution by the proportion of successes over both real- and pseudo-observations. A uniform prior Beta(1,1) does not add (or subtract) any pseudo-observations since for Beta(1,1) it follows that (αPrior − 1) = 0 and (βPrior − 1) = 0. The Haldane prior Beta(0,0) subtracts one pseudo observation from each and Jeffreys prior Beta(1/2,1/2) subtracts 1/2 pseudo-observation of success and an equal number of failure. This subtraction has the effect of smoothing out the posterior distribution. If the proportion of successes is not 50% (s/n ≠ 1/2) values of αPrior and βPrior less than 1 (and therefore negative (αPrior − 1) and (βPrior − 1)) favor sparsity, i.e. distributions where the parameter p is closer to either 0 or 1. In effect, values of αPrior and βPrior between 0 and 1, when operating together, function as a concentration parameter.
The accompanying plots show the posterior probability density functions for sample sizes n ∈ {3,10,50}, successes s ∈ {n/2,n/4} and Beta(αPrior,βPrior) ∈ {Beta(0,0),Beta(1/2,1/2),Beta(1,1)}. Also shown are the cases for n = {4,12,40}, success s = {n/4} and Beta(αPrior,βPrior) ∈ {Beta(0,0),Beta(1/2,1/2),Beta(1,1)}. The first plot shows the symmetric cases, for successes s ∈ {n/2}, with mean = mode = 1/2 and the second plot shows the skewed cases s ∈ {n/4}. The images show that there is little difference between the priors for the posterior with sample size of 50 (characterized by a more pronounced peak near p = 1/2). Significant differences appear for very small sample sizes (in particular for the flatter distribution for the degenerate case of sample size = 3). Therefore, the skewed cases, with successes s = {n/4}, show a larger effect from the choice of prior, at small sample size, than the symmetric cases. For symmetric distributions, the Bayes prior Beta(1,1) results in the most "peaky" and highest posterior distributions and the Haldane prior Beta(0,0) results in the flattest and lowest peak distribution. The Jeffreys prior Beta(1/2,1/2) lies in between them. For nearly symmetric, not too skewed distributions the effect of the priors is similar. For very small sample size (in this case for a sample size of 3) and skewed distribution (in this example for s ∈ {n/4}) the Haldane prior can result in a reverse-J-shaped distribution with a singularity at the left end. However, this happens only in degenerate cases (in this example n = 3 and hence s = 3/4 < 1, a degenerate value because s should be greater than unity in order for the posterior of the Haldane prior to have a mode located between the ends, and because s = 3/4 is not an integer number, hence it violates the initial assumption of a binomial distribution for the likelihood) and it is not an issue in generic cases of reasonable sample size (such that the condition 1 < s < n − 1, necessary for a mode to exist between both ends, is fulfilled).
In Chapter 12 (p. 385) of his book, Jaynes[52] asserts that the Haldane prior Beta(0,0) describes a prior state of knowledge of complete ignorance, where we are not even sure whether it is physically possible for an experiment to yield either a success or a failure, while the Bayes (uniform) prior Beta(1,1) applies if one knows that both binary outcomes are possible. Jaynes states: "interpret the Bayes–Laplace (Beta(1,1)) prior as describing not a state of complete ignorance, but the state of knowledge in which we have observed one success and one failure...once we have seen at least one success and one failure, then we know that the experiment is a true binary one, in the sense of physical possibility." Jaynes [52] does not specifically discuss Jeffreys prior Beta(1/2,1/2) (Jaynes discussion of "Jeffreys prior" on pp. 181, 423 and on chapter 12 of Jaynes book[52] refers instead to the improper, un-normalized, prior "1/pdp" introduced by Jeffreys in the 1939 edition of his book,[59] seven years before he introduced what is now known as Jeffreys' invariant prior: the square root of the determinant of Fisher's information matrix. "1/p" is Jeffreys' (1946) invariant prior for the exponential distribution, not for the Bernoulli or binomial distributions). However, it follows from the above discussion that Jeffreys Beta(1/2,1/2) prior represents a state of knowledge in between the Haldane Beta(0,0) and Bayes Beta (1,1) prior.
Similarly, Karl Pearson in his 1892 book The Grammar of Science[68][69] (p. 144 of 1900 edition) maintained that the Bayes (Beta(1,1) uniform prior was not a complete ignorance prior, and that it should be used when prior information justified to "distribute our ignorance equally"". K. Pearson wrote: "Yet the only supposition that we appear to have made is this: that, knowing nothing of nature, routine and anomy (from the Greek ανομία, namely: a- "without", and nomos "law") are to be considered as equally likely to occur. Now we were not really justified in making even this assumption, for it involves a knowledge that we do not possess regarding nature. We use our experience of the constitution and action of coins in general to assert that heads and tails are equally probable, but we have no right to assert before experience that, as we know nothing of nature, routine and breach are equally probable. In our ignorance we ought to consider before experience that nature may consist of all routines, all anomies (normlessness), or a mixture of the two in any proportion whatever, and that all such are equally probable. Which of these constitutions after experience is the most probable must clearly depend on what that experience has been like."
If there is sufficient sampling data, and the posterior probability mode is not located at one of the extremes of the domain (x = 0 or x = 1), the three priors of Bayes (Beta(1,1)), Jeffreys (Beta(1/2,1/2)) and Haldane (Beta(0,0)) should yield similar posterior probability densities. Otherwise, as Gelman et al.[70] (p. 65) point out, "if so few data are available that the choice of noninformative prior distribution makes a difference, one should put relevant information into the prior distribution", or as Berger[4] (p. 125) points out "when different reasonable priors yield substantially different answers, can it be right to state that there is a single answer? Would it not be better to admit that there is scientific uncertainty, with the conclusion depending on prior beliefs?."
Occurrence and applications
Order statistics
The beta distribution has an important application in the theory of order statistics. A basic result is that the distribution of the kth smallest of a sample of size n from a continuous uniform distribution has a beta distribution.[39] This result is summarized as:
In standard logic, propositions are considered to be either true or false. In contradistinction, subjective logic assumes that humans cannot determine with absolute certainty whether a proposition about the real world is absolutely true or false. In subjective logic the posteriori probability estimates of binary events can be represented by beta distributions.[71]
Wavelet analysis
A wavelet is a wave-like oscillation with an amplitude that starts out at zero, increases, and then decreases back to zero. It can typically be visualized as a "brief oscillation" that promptly decays. Wavelets can be used to extract information from many different kinds of data, including – but certainly not limited to – audio signals and images. Thus, wavelets are purposefully crafted to have specific properties that make them useful for signal processing. Wavelets are localized in both time and frequency whereas the standard Fourier transform is only localized in frequency. Therefore, standard Fourier Transforms are only applicable to stationary processes, while wavelets are applicable to non-stationary processes. Continuous wavelets can be constructed based on the beta distribution. Beta wavelets[72] can be viewed as a soft variety of Haar wavelets whose shape is fine-tuned by two shape parameters α and β.
where and ; here F is (Wright's) genetic distance between two populations.
Project management: task cost and schedule modeling
The beta distribution can be used to model events which are constrained to take place within an interval defined by a minimum and maximum value. For this reason, the beta distribution — along with the triangular distribution — is used extensively in PERT, critical path method (CPM), Joint Cost Schedule Modeling (JCSM) and other project management/control systems to describe the time to completion and the cost of a task. In project management, shorthand computations are widely used to estimate the mean and standard deviation of the beta distribution:[38]
where a is the minimum, c is the maximum, and b is the most likely value (the mode for α > 1 and β > 1).
The above estimate for the mean is known as the PERTthree-point estimation and it is exact for either of the following values of β (for arbitrary α within these ranges):
Otherwise, these can be poor approximations for beta distributions with other values of α and β, exhibiting average errors of 40% in the mean and 549% in the variance.[74][75][76]
Random variate generation
If X and Y are independent, with and then
So one algorithm for generating beta variates is to generate , where X is a gamma variate with parameters (α, 1) and Y is an independent gamma variate with parameters (β, 1).[77] In fact, here and are independent, and . If and is independent of and , then and is independent of . This shows that the product of independent and random variables is a random variable.
Also, the kth order statistic of nuniformly distributed variates is , so an alternative if α and β are small integers is to generate α + β − 1 uniform variates and choose the α-th smallest.[39]
Another way to generate the Beta distribution is by Pólya urn model. According to this method, one start with an "urn" with α "black" balls and β "white" balls and draw uniformly with replacement. Every trial an additional ball is added according to the color of the last ball which was drawn. Asymptotically, the proportion of black and white balls will be distributed according to the Beta distribution, where each repetition of the experiment will produce a different value.
A beta distribution with α ~ β and α and β >> 1 is approximately normal with mean 1/2 and variance 1/(4(2α + 1)). If α ≥ β the normal approximation can be improved by taking the cube-root of the logarithm of the reciprocal of [78]
History
Thomas Bayes, in a posthumous paper [62] published in 1763 by Richard Price, obtained a beta distribution as the density of the probability of success in Bernoulli trials (see § Applications, Bayesian inference), but the paper does not analyze any of the moments of the beta distribution or discuss any of its properties.
Karl Pearson analyzed the beta distribution as the solution Type I of Pearson distributions
The first systematic modern discussion of the beta distribution is probably due to Karl Pearson.[79][80] In Pearson's papers[20][32] the beta distribution is couched as a solution of a differential equation: Pearson's Type I distribution which it is essentially identical to except for arbitrary shifting and re-scaling (the beta and Pearson Type I distributions can always be equalized by proper choice of parameters). In fact, in several English books and journal articles in the few decades prior to World War II, it was common to refer to the beta distribution as Pearson's Type I distribution. William P. Elderton in his 1906 monograph "Frequency curves and correlation"[42] further analyzes the beta distribution as Pearson's Type I distribution, including a full discussion of the method of moments for the four parameter case, and diagrams of (what Elderton describes as) U-shaped, J-shaped, twisted J-shaped, "cocked-hat" shapes, horizontal and angled straight-line cases. Elderton wrote "I am chiefly indebted to Professor Pearson, but the indebtedness is of a kind for which it is impossible to offer formal thanks." Elderton in his 1906 monograph [42] provides an impressive amount of information on the beta distribution, including equations for the origin of the distribution chosen to be the mode, as well as for other Pearson distributions: types I through VII. Elderton also included a number of appendixes, including one appendix ("II") on the beta and gamma functions. In later editions, Elderton added equations for the origin of the distribution chosen to be the mean, and analysis of Pearson distributions VIII through XII.
As remarked by Bowman and Shenton[44] "Fisher and Pearson had a difference of opinion in the approach to (parameter) estimation, in particular relating to (Pearson's method of) moments and (Fisher's method of) maximum likelihood in the case of the Beta distribution." Also according to Bowman and Shenton, "the case of a Type I (beta distribution) model being the center of the controversy was pure serendipity. A more difficult model of 4 parameters would have been hard to find." The long running public conflict of Fisher with Karl Pearson can be followed in a number of articles in prestigious journals. For example, concerning the estimation of the four parameters for the beta distribution, and Fisher's criticism of Pearson's method of moments as being arbitrary, see Pearson's article "Method of moments and method of maximum likelihood" [45] (published three years after his retirement from University College, London, where his position had been divided between Fisher and Pearson's son Egon) in which Pearson writes "I read (Koshai's paper in the Journal of the Royal Statistical Society, 1933) which as far as I am aware is the only case at present published of the application of Professor Fisher's method. To my astonishment that method depends on first working out the constants of the frequency curve by the (Pearson) Method of Moments and then superposing on it, by what Fisher terms "the Method of Maximum Likelihood" a further approximation to obtain, what he holds, he will thus get, 'more efficient values' of the curve constants".
David and Edwards's treatise on the history of statistics[81] cites the first modern treatment of the beta distribution, in 1911,[82] using the beta designation that has become standard, due to Corrado Gini, an Italian statistician, demographer, and sociologist, who developed the Gini coefficient. N.L.Johnson and S.Kotz, in their comprehensive and very informative monograph[83] on leading historical personalities in statistical sciences credit Corrado Gini[84] as "an early Bayesian...who dealt with the problem of eliciting the parameters of an initial Beta distribution, by singling out techniques which anticipated the advent of the so-called empirical Bayes approach."
References
^ a b c d e f g h i j k l m n o p q r s t u v w x yJohnson, Norman L.; Kotz, Samuel; Balakrishnan, N. (1995). "Chapter 25: Beta Distributions". Continuous Univariate Distributions Vol. 2 (2nd ed.). Wiley. ISBN 978-0-471-58494-0.
^ a bRose, Colin; Smith, Murray D. (2002). Mathematical Statistics with MATHEMATICA. Springer. ISBN 978-0387952345.
^ a b cKruschke, John K. (2011). Doing Bayesian data analysis: A tutorial with R and BUGS. Academic Press / Elsevier. p. 83. ISBN 978-0123814852.
^ a bBerger, James O. (2010). Statistical Decision Theory and Bayesian Analysis (2nd ed.). Springer. ISBN 978-1441930743.
^ a b c dFeller, William (1971). An Introduction to Probability Theory and Its Applications, Vol. 2. Wiley. ISBN 978-0471257097.
^Kruschke, John K. (2015). Doing Bayesian Data Analysis: A Tutorial with R, JAGS and Stan. Academic Press / Elsevier. ISBN 978-0-12-405888-0.
^ a bWadsworth, George P. and Joseph Bryan (1960). Introduction to Probability and Random Variables. McGraw-Hill.
^ a b c d e f gGupta, Arjun K., ed. (2004). Handbook of Beta Distribution and Its Applications. CRC Press. ISBN 978-0824753962.
^ a bKerman, Jouni (2011). "A closed-form approximation for the median of the beta distribution". arXiv:1111.0433 [math.ST].
^Mosteller, Frederick and John Tukey (1977). Data Analysis and Regression: A Second Course in Statistics. Addison-Wesley Pub. Co. Bibcode:1977dars.book.....M. ISBN 978-0201048544.
^ a bFeller, William (1968). An Introduction to Probability Theory and Its Applications. Vol. 1 (3rd ed.). ISBN 978-0471257080.
^Philip J. Fleming and John J. Wallace. How not to lie with statistics: the correct way to summarize benchmark results. Communications of the ACM, 29(3):218–221, March 1986.
^"NIST/SEMATECH e-Handbook of Statistical Methods 1.3.6.6.17. Beta Distribution". National Institute of Standards and Technology Information Technology Laboratory. April 2012. Retrieved May 31, 2016.
^Oguamanam, D.C.D.; Martin, H. R.; Huissoon, J. P. (1995). "On the application of the beta distribution to gear damage analysis". Applied Acoustics. 45 (3): 247–261. doi:10.1016/0003-682X(95)00001-P.
^Zhiqiang Liang; Jianming Wei; Junyu Zhao; Haitao Liu; Baoqing Li; Jie Shen; Chunlei Zheng (27 August 2008). "The Statistical Meaning of Kurtosis and Its New Application to Identification of Persons Based on Seismic Signals". Sensors. 8 (8): 5106–5119. Bibcode:2008Senso...8.5106L. doi:10.3390/s8085106. PMC3705491. PMID 27873804.
^Kenney, J. F., and E. S. Keeping (1951). Mathematics of Statistics Part Two, 2nd edition. D. Van Nostrand Company Inc.{{cite book}}: CS1 maint: multiple names: authors list (link)
^ a b c dAbramowitz, Milton and Irene A. Stegun (1965). Handbook Of Mathematical Functions With Formulas, Graphs, And Mathematical Tables. Dover. ISBN 978-0-486-61272-0.
^Weisstein., Eric W. "Kurtosis". MathWorld--A Wolfram Web Resource. Retrieved 13 August 2012.
^ a bPanik, Michael J (2005). Advanced Statistics from an Elementary Point of View. Academic Press. ISBN 978-0120884940.
^ a b c d e fPearson, Karl (1916). "Mathematical contributions to the theory of evolution, XIX: Second supplement to a memoir on skew variation". Philosophical Transactions of the Royal Society A. 216 (538–548): 429–457. Bibcode:1916RSPTA.216..429P. doi:10.1098/rsta.1916.0009. JSTOR 91092.
^Billingsley, Patrick (1995). "30". Probability and measure (3rd ed.). Wiley-Interscience. ISBN 978-0-471-00710-4.
^ a bMacKay, David (2003). Information Theory, Inference and Learning Algorithms. Cambridge University Press; First Edition. Bibcode:2003itil.book.....M. ISBN 978-0521642989.
^ a bJohnson, N.L. (1949). "Systems of frequency curves generated by methods of translation" (PDF). Biometrika. 36 (1–2): 149–176. doi:10.1093/biomet/36.1-2.149. hdl:10338.dmlcz/135506. PMID 18132090.
^Verdugo Lazo, A. C. G.; Rathie, P. N. (1978). "On the entropy of continuous probability distributions". IEEE Trans. Inf. Theory. 24 (1): 120–122. doi:10.1109/TIT.1978.1055832.
^Shannon, Claude E. (1948). "A Mathematical Theory of Communication". Bell System Technical Journal. 27 (4): 623–656. doi:10.1002/j.1538-7305.1948.tb01338.x.
^ a b cCover, Thomas M. and Joy A. Thomas (2006). Elements of Information Theory 2nd Edition (Wiley Series in Telecommunications and Signal Processing). Wiley-Interscience; 2 edition. ISBN 978-0471241959.
^Plunkett, Kim, and Jeffrey Elman (1997). Exercises in Rethinking Innateness: A Handbook for Connectionist Simulations (Neural Network Modeling and Connectionism). A Bradford Book. p. 166. ISBN 978-0262661058.{{cite book}}: CS1 maint: multiple names: authors list (link)
^Nallapati, Ramesh (2006). The smoothed dirichlet distribution: understanding cross-entropy ranking in information retrieval (Thesis). Computer Science Dept., University of Massachusetts Amherst.
^ a bPearson, Egon S. (July 1969). "Some historical reflections traced through the development of the use of frequency curves". THEMIS Statistical Analysis Research Program, Technical Report 38. Office of Naval Research, Contract N000014-68-A-0515 (Project NR 042–260).
^Hahn, Gerald J.; Shapiro, S. (1994). Statistical Models in Engineering (Wiley Classics Library). Wiley-Interscience. ISBN 978-0471040651.
^ a bPearson, Karl (1895). "Contributions to the mathematical theory of evolution, II: Skew variation in homogeneous material". Philosophical Transactions of the Royal Society. 186: 343–414. Bibcode:1895RSPTA.186..343P. doi:10.1098/rsta.1895.0010. JSTOR 90649.
^Buchanan, K.; Rockway, J.; Sternberg, O.; Mai, N. N. (May 2016). "Sum-difference beamforming for radar applications using circularly tapered random arrays". 2016 IEEE Radar Conference (RadarConf). pp. 1–5. doi:10.1109/RADAR.2016.7485289. ISBN 978-1-5090-0863-6. S2CID 32525626.
^Buchanan, K.; Flores, C.; Wheeland, S.; Jensen, J.; Grayson, D.; Huff, G. (May 2017). "Transmit beamforming for radar applications using circularly tapered random arrays". 2017 IEEE Radar Conference (RadarConf). pp. 0112–0117. doi:10.1109/RADAR.2017.7944181. ISBN 978-1-4673-8823-8. S2CID 38429370.
^Ryan, Buchanan, Kristopher (2014-05-29). "Theory and Applications of Aperiodic (Random) Phased Arrays". {{cite journal}}: Cite journal requires |journal= (help)CS1 maint: multiple names: authors list (link)
^Pham-Gia, T. (January 2000). "Distributions of the ratios of independent beta variables and applications". Communications in Statistics - Theory and Methods. 29 (12): 2693–2715. doi:10.1080/03610920008832632. ISSN 0361-0926. Retrieved 13 November 2024.
^Herrerías-Velasco, José Manuel and Herrerías-Pleguezuelo, Rafael and René van Dorp, Johan. (2011). Revisiting the PERT mean and Variance. European Journal of Operational Research (210), p. 448–451.
^ a bMalcolm, D. G.; Roseboom, J. H.; Clark, C. E.; Fazar, W. (September–October 1958). "Application of a Technique for Research and Development Program Evaluation". Operations Research. 7 (5): 646–669. doi:10.1287/opre.7.5.646. ISSN 0030-364X.
^ a b c dDavid, H. A., Nagaraja, H. N. (2003) Order Statistics (3rd Edition). Wiley, New Jersey pp 458. ISBN 0-471-38926-9
^ a b c d e f g hElderton, William Palin (1906). Frequency-Curves and Correlation. Charles and Edwin Layton (London).
^Elderton, William Palin and Norman Lloyd Johnson (2009). Systems of Frequency Curves. Cambridge University Press. ISBN 978-0521093361.
^ a b cBowman, K. O.; Shenton, L. R. (2007). "The beta distribution, moment method, Karl Pearson and R.A. Fisher" (PDF). Far East J. Theo. Stat. 23 (2): 133–164.
^ a bPearson, Karl (June 1936). "Method of moments and method of maximum likelihood". Biometrika. 28 (1/2): 34–59. doi:10.2307/2334123. JSTOR 2334123.
^ a b cJoanes, D. N.; C. A. Gill (1998). "Comparing measures of sample skewness and kurtosis". The Statistician. 47 (Part 1): 183–189. doi:10.1111/1467-9884.00122.
^Beckman, R. J.; G. L. Tietjen (1978). "Maximum likelihood estimation for the beta distribution". Journal of Statistical Computation and Simulation. 7 (3–4): 253–258. doi:10.1080/00949657808810232.
^Gnanadesikan, R., Pinkham and Hughes (1967). "Maximum likelihood estimation of the parameters of the beta distribution from smallest order statistics". Technometrics. 9 (4): 607–620. doi:10.2307/1266199. JSTOR 1266199.{{cite journal}}: CS1 maint: multiple names: authors list (link)
^Fackler, Paul. "Inverse Digamma Function (Matlab)". Harvard University School of Engineering and Applied Sciences. Retrieved 2012-08-18.
^ a b cSilvey, S.D. (1975). Statistical Inference. Chapman and Hal. p. 40. ISBN 978-0412138201.
^Edwards, A. W. F. (1992). Likelihood. The Johns Hopkins University Press. ISBN 978-0801844430.
^ a b c d e fJaynes, E.T. (2003). Probability theory, the logic of science. Cambridge University Press. ISBN 978-0521592710.
^Costa, Max, and Cover, Thomas (September 1983). On the similarity of the entropy power inequality and the Brunn Minkowski inequality (PDF). Tech.Report 48, Dept. Statistics, Stanford University.{{cite book}}: CS1 maint: multiple names: authors list (link)
^ a b cAryal, Gokarna; Saralees Nadarajah (2004). "Information matrix for beta distributions" (PDF). Serdica Mathematical Journal (Bulgarian Academy of Science). 30: 513–526.
^ a bLaplace, Pierre Simon, marquis de (1902). A philosophical essay on probabilities. New York : J. Wiley; London : Chapman & Hall. ISBN 978-1-60206-328-0.{{cite book}}: CS1 maint: multiple names: authors list (link)
^Cox, Richard T. (1961). Algebra of Probable Inference. The Johns Hopkins University Press. ISBN 978-0801869822.
^ a bKeynes, John Maynard (2010) [1921]. A Treatise on Probability: The Connection Between Philosophy and the History of Science. Wildside Press. ISBN 978-1434406965.
^Pearson, Karl (1907). "On the Influence of Past Experience on Future Expectation". Philosophical Magazine. 6 (13): 365–378.
^ a b c dJeffreys, Harold (1998). Theory of Probability. Oxford University Press, 3rd edition. ISBN 978-0198503682.
^Broad, C. D. (October 1918). "On the relation between induction and probability". MIND, A Quarterly Review of Psychology and Philosophy. 27 (New Series) (108): 389–404. doi:10.1093/mind/XXVII.4.389. JSTOR 2249035.
^ a b c dPerks, Wilfred (January 1947). "Some observations on inverse probability including a new indifference rule". Journal of the Institute of Actuaries. 73 (2): 285–334. doi:10.1017/S0020268100012270. Archived from the original on 2014-01-12. Retrieved 2012-09-19.
^ a bBayes, Thomas; communicated by Richard Price (1763). "An Essay towards solving a Problem in the Doctrine of Chances". Philosophical Transactions of the Royal Society. 53: 370–418. doi:10.1098/rstl.1763.0053. JSTOR 105741.
^Haldane, J.B.S. (1932). "A note on inverse probability". Mathematical Proceedings of the Cambridge Philosophical Society. 28 (1): 55–61. Bibcode:1932PCPS...28...55H. doi:10.1017/s0305004100010495. S2CID 122773707.
^Zellner, Arnold (1971). An Introduction to Bayesian Inference in Econometrics. Wiley-Interscience. ISBN 978-0471169376.
^Jeffreys, Harold (September 1946). "An Invariant Form for the Prior Probability in Estimation Problems". Proceedings of the Royal Society. A 24. 186 (1007): 453–461. Bibcode:1946RSPSA.186..453J. doi:10.1098/rspa.1946.0056. PMID 20998741.
^Berger, James; Bernardo, Jose; Sun, Dongchu (2009). "The formal definition of reference priors". The Annals of Statistics. 37 (2): 905–938. arXiv:0904.0156. Bibcode:2009arXiv0904.0156B. doi:10.1214/07-AOS587. S2CID 3221355.
^Clarke, Bertrand S.; Andrew R. Barron (1994). "Jeffreys' prior is asymptotically least favorable under entropy risk" (PDF). Journal of Statistical Planning and Inference. 41: 37–60. doi:10.1016/0378-3758(94)90153-8.
^Pearson, Karl (1892). The Grammar of Science. Walter Scott, London.
^Pearson, Karl (2009). The Grammar of Science. BiblioLife. ISBN 978-1110356119.
^Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B. (2003). Bayesian Data Analysis. Chapman and Hall/CRC. ISBN 978-1584883883.{{cite book}}: CS1 maint: multiple names: authors list (link)
^H.M. de Oliveira and G.A.A. Araújo,. Compactly Supported One-cyclic Wavelets Derived from Beta Distributions. Journal of Communication and Information Systems. vol.20, n.3, pp.27-33, 2005.
^Balding, David J.; Nichols, Richard A. (1995). "A method for quantifying differentiation between populations at multi-allelic loci and its implications for investigating identity and paternity". Genetica. 96 (1–2). Springer: 3–12. doi:10.1007/BF01441146. PMID 7607457. S2CID 30680826.
^Keefer, Donald L. and Verdini, William A. (1993). Better Estimation of PERT Activity Time Parameters. Management Science 39(9), p. 1086–1091.
^Keefer, Donald L. and Bodily, Samuel E. (1983). Three-point Approximations for Continuous Random variables. Management Science 29(5), p. 595–609.
^"Defense Resource Management Institute - Naval Postgraduate School". www.nps.edu.
^van der Waerden, B. L., "Mathematical Statistics", Springer, ISBN 978-3-540-04507-6.
^On normalizing the incomplete beta-function for fitting to dose-response curves M.E. Wise Biometrika vol 47, No. 1/2, June 1960, pp. 173–175
^"Library and Archive catalogue". Sackler Digital Archive. Royal Society. Archived from the original on 2011-10-25. Retrieved 2011-07-01.
^David, H. A. and A.W.F. Edwards (2001). Annotated Readings in the History of Statistics. Springer; 1 edition. ISBN 978-0387988443.
^Gini, Corrado (1911). "Considerazioni Sulle Probabilità Posteriori e Applicazioni al Rapporto dei Sessi Nelle Nascite Umane". Studi Economico-Giuridici della Università de Cagliari. Anno III (reproduced in Metron 15, 133, 171, 1949): 5–41.
^Johnson, Norman L. and Samuel Kotz, ed. (1997). Leading Personalities in Statistical Sciences: From the Seventeenth Century to the Present (Wiley Series in Probability and Statistics. Wiley. ISBN 978-0471163817.
^Metron journal. "Biography of Corrado Gini". Metron Journal. Archived from the original on 2012-07-16. Retrieved 2012-08-18.
External links
Wikimedia Commons has media related to Beta distribution.
![{\displaystyle \operatorname {E} [X]={\frac {\alpha }{\alpha +\beta }}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3905662ceed484cba5580951e29eda96f4d2605e)
![{\displaystyle \operatorname {E} [\ln X]=\psi (\alpha )-\psi (\alpha +\beta )\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de67df996fa33237ab7f415e7edc9fa8e71997a0)
![{\displaystyle \operatorname {E} [X\,\ln X]={\frac {\alpha }{\alpha +\beta }}\,\left[\psi (\alpha +1)-\psi (\alpha +\beta +1)\right]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50106a787db7d72ce3066a5a3238813cffebcc2e)
![{\displaystyle {\begin{matrix}I_{\frac {1}{2}}^{[-1]}(\alpha ,\beta ){\text{ (一般に) }}\\[0.5em]\approx {\frac {\alpha -{\tfrac {1}{3}}}{\alpha +\beta -{\tfrac {2}{3}}}}{\text{ }}\alpha ,\beta >1\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af887ef0331cde970dad14ad670cf3592334f845)
![{\displaystyle \operatorname {var} [X]={\frac {\alpha \beta }{(\alpha +\beta )^{2}(\alpha +\beta +1)}}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f90a6ad61b4b436749ca37a6c2a1aa077b032ce3)
![{\displaystyle \operatorname {var} [\ln X]=\psi _{1}(\alpha )-\psi _{1}(\alpha +\beta )\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4941f45412823abd34d3befea7f8fbf544135e4)
![{\displaystyle {\frac {6[(\alpha -\beta )^{2}(\alpha +\beta +1)-\alpha \beta (\alpha +\beta +2)]}{\alpha \beta (\alpha +\beta +2)(\alpha +\beta +3)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eea65a8d7c9e00ba6299b727eab679117776f41e)
![{\displaystyle {\begin{matrix}\ln \mathrm {B} (\alpha ,\beta )-(\alpha -1)\psi (\alpha )-(\beta -1)\psi (\beta )\\[0.5em]{}+(\alpha +\beta -2)\psi (\alpha +\beta )\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff4b6cc1848fe96318adb734393b701cb816f88a)
![{\displaystyle {\begin{bmatrix}\operatorname {var} [\ln X]&\operatorname {cov} [\ln X,\ln(1-X)]\\\operatorname {cov} [\ln X, \ln(1-X)]&\演算子名 {var} [\ln(1-X)]\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/881f91af0ab1d6bf3809a4ed6ca9e6384544292f)
}{V[X]}}-1\right)E[X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2b596a180ef813a0baa1d6f2063950e20da1f62)
}{V[X]}}-1\right)(1-E[X])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05ace15e23f6ac9be43eea861f44c018fd3d00de)
![{\displaystyle {\begin{aligned}f(x;\alpha ,\beta )&=\mathrm {constant} \cdot x^{\alpha -1}(1-x)^{\beta -1}\\ [3pt]&={\frac {x^{\alpha -1}(1-x)^{\beta -1}}{\displaystyle \int _{0}^{1}u^{\alpha -1}(1-u)^{\beta -1}\,du}}\\[6pt]&={\frac {\Gamma (\alpha + \beta )}{\Gamma (\alpha )\Gamma (\beta )}}\,x^{\alpha -1}(1-x)^{\beta -1}\\[6pt]&={\frac {1}{\mathrm {B} (\alpha ,\beta )}}x^{\alpha -1}(1-x)^{\beta -1}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5fc18388353b219c482e8e35ca4aae808ab1be81)
![{\displaystyle {\begin{aligned}\mu _{Y}&=\mu _{X}(ca)+a\\&=\left({\frac {\alpha }{\alpha +\beta }}\right)(ca)+a={\frac {\alpha c+\beta a}{\alpha +\beta }}\\[8pt]{\text{mode}}(Y)&={\text{mode}}(X)(ca)+a\\&=\left({\frac {\alpha -1}{\alpha +\beta -2}}\right)(ca)+a={\frac {(\alpha -1)c+(\beta -1)a}{\alpha +\beta -2}}\ ,\qquad {\text{ if }}\alpha ,\beta >1\\[8pt]{\text{median}}(Y)&={\text{median}}(X)(ca)+a\\&=\left(I_{\frac {1}{2}}^{[-1]}(\alpha ,\beta )\right)(ca)+a\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f65af6c528ca47e8838d3d5722a94f9e3f128100)
![{\displaystyle {\text{尖度過剰}}(Y)={\text{尖度過剰}}(X)={\frac {6[(\alpha -\beta )^{2}(\alpha +\beta +1)-\alpha \beta (\alpha +\beta +2)]}{\alpha \beta (\alpha +\beta +2)(\alpha +\beta +3)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99ddb3577b02ee0b10163123af23c6d7f728946b)
_for_Beta_distribution_versus_alpha_and_beta_from_0_to_2_-_J._Rodal.jpg/500px-(Mean_-_Median)_for_Beta_distribution_versus_alpha_and_beta_from_0_to_2_-_J._Rodal.jpg)
![{\displaystyle x=I_{1/2}^{[-1]}(\alpha ,\beta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7510f94efa49f254eb3924678b527a6fd22d0fc)
![1 ≤ α ≤ 5 および 1 ≤ β ≤ 5 の場合のベータ分布の Abs[(中央値-近似値)/中央値]](https://img-server.japedia.wiki/wikipedia/commons/thumb/a/af/Relative_Error_for_Approximation_to_Median_of_Beta_Distribution_for_alpha_and_beta_from_1_to_5_-_J._Rodal.jpg/500px-Relative_Error_for_Approximation_to_Median_of_Beta_Distribution_for_alpha_and_beta_from_1_to_5_-_J._Rodal.jpg)
![1≤α≤5および1≤β≤5のベータ分布の絶対値[(中央値-近似値)/(平均-最頻値)]](https://img-server.japedia.wiki/wikipedia/commons/thumb/e/e8/Error_in_Median_Apprx._relative_to_Mean-Mode_distance_for_Beta_Distribution_with_alpha_and_beta_from_1_to_5_-_J._Rodal.jpg/500px-Error_in_Median_Apprx._relative_to_Mean-Mode_distance_for_Beta_Distribution_with_alpha_and_beta_from_1_to_5_-_J._Rodal.jpg)
![{\displaystyle {\begin{aligned}\mu =\operatorname {E} [X]&=\int _{0}^{1}xf(x;\alpha ,\beta )\,dx\\&=\int _{0}^{1}x\,{\frac {x^{\alpha -1}(1-x)^{\beta -1}}{\mathrm {B} (\alpha ,\beta )}}\,dx\\&={\frac {\alpha }{\alpha +\beta }}\\&={\frac {1}{1+{\frac {\beta }{\alpha }}}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9137834d9d47360ed6c23550c6236fed5fd35f7)
_for_Beta_Distribution_versus_alpha_and_beta_from_0_to_2_-_J._Rodal.jpg/500px-(Mean_-_GeometricMean)_for_Beta_Distribution_versus_alpha_and_beta_from_0_to_2_-_J._Rodal.jpg)
,_Yellow=G(1-X),_smaller_values_alpha_and_beta_in_front_-_J._Rodal.jpg/500px-Geometric_Means_for_Beta_distribution_Purple=G(X),_Yellow=G(1-X),_smaller_values_alpha_and_beta_in_front_-_J._Rodal.jpg)
,_Yellow=G(1-X),_larger_values_alpha_and_beta_in_front_-_J._Rodal.jpg/500px-Geometric_Means_for_Beta_distribution_Purple=G(X),_Yellow=G(1-X),_larger_values_alpha_and_beta_in_front_-_J._Rodal.jpg)
![{\displaystyle \ln G_{X}=\operatorname {E} [\ln X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64b67cb73b90bc0e09ba41003b44f84b6e1d3feb)
![{\displaystyle {\begin{aligned}\operatorname {E} [\ln X]&=\int _{0}^{1}\ln x\,f(x;\alpha ,\beta )\,dx\\[4pt]&=\int _{0}^{1}\ln x\,{\frac {x^{\alpha -1}(1-x)^{\beta -1}}{\mathrm {B} (\alpha ,\beta )}}\,dx\\[4pt]&={\frac {1}{\mathrm {B} (\alpha ,\beta )}}\,\int _{0}^{1}{\frac {\partial x^{\alpha -1}(1-x)^{\beta -1}}{\partial \alpha }}\,dx\\[4pt]&={\frac {1}{\mathrm {B} (\alpha 、\beta )}}{\frac {\partial }{\partial \alpha }}\int _{0}^{1}x^{\alpha -1}(1-x)^{\beta -1}\,dx\\[4pt]&={\frac {1}{\mathrm {B} (\alpha 、\beta )}}{\frac {\partial \mathrm {B} (\alpha 、\beta )}{\partial \alpha }}\\[4pt]&={\frac {\partial \ln \mathrm {B} (\alpha 、\beta )}{\partial \alpha }}\\[4pt]&={\frac {\partial \ln \Gamma (\alpha )}{\partial \alpha }}-{\frac {\partial \ln \Gamma (\alpha +\beta )}{\partial \alpha }}\\[4pt]&=\psi (\alpha )-\psi (\alpha +\beta )\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd9db519e08e3c72cd6f9e2f0c90a7c57bdba035)
![{\displaystyle G_{X}=e^{\オペレータ名 {E} [\ln X]}=e^{\psi (\alpha )-\psi (\alpha +\beta )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c93ffa7f0155fa3816fcb151c3eb677700aabca2)
![{\displaystyle G_{(1-X)}=e^{\operatorname {E} [\ln(1-X)]}=e^{\psi (\beta )-\psi (\alpha +\beta ) }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8013e6a62bd5140a4e7919686761225dd54847e)
_for_Beta_distribution_versus_alpha_and_beta_from_0_to_2_-_J._Rodal.jpg/500px-(Mean_-_HarmonicMean)_for_Beta_distribution_versus_alpha_and_beta_from_0_to_2_-_J._Rodal.jpg)
,_Yellow=H(1-X),_smaller_values_alpha_and_beta_in_front_-_J._Rodal.jpg/500px-Harmonic_Means_for_Beta_distribution_Purple=H(X),_Yellow=H(1-X),_smaller_values_alpha_and_beta_in_front_-_J._Rodal.jpg)
,_Yellow=H(1-X),_larger_values_alpha_and_beta_in_front_-_J._Rodal.jpg/500px-Harmonic_Means_for_Beta_distribution_Purple=H(X),_Yellow=H(1-X),_larger_values_alpha_and_beta_in_front_-_J._Rodal.jpg)
![{\displaystyle {\begin{aligned}H_{X}&={\frac {1}{\operatorname {E} \left[{\frac {1}{X}}\right]}}\\&={\frac {1}{\int _{0}^{1}{\frac {f(x;\alpha ,\beta )}{x}}\,dx}}\\&={\frac {1}{\int _{0}^{1}{\frac {x^{\alpha -1}(1-x)^{\beta -1}}{x\mathrm {B} (\alpha ,\beta )}}\,dx}}\\&={\frac {\alpha -1}{\alpha +\beta -1}}{\text{ }}\alpha >1{\text{ かつ }}\beta >0 の場合\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed7d99dd7493b9c085cd5d407861730e2a2abf6c)
![{\displaystyle H_{1-X}={\frac {1}{\operatorname {E} \left[{\frac {1}{1-X}}\right]}}={\frac {\beta -1}{\alpha +\beta -1}}{\text{ }}\beta >1、{\text{ かつ }}\alpha >0 の場合。}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48f4fd69f20c4259cb8a50e754df8dfed5a1ddca)
![{\displaystyle \operatorname {var} (X)=\operatorname {E} [(X-\mu )^{2}]={\frac {\alpha \beta }{(\alpha +\beta )^{2}(\alpha +\beta +1)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d96555f71897dc80e2f31ec71b3cbfbcf39950bc)
![{\displaystyle {\begin{aligned}\ln \operatorname {var} _{GX}&=\operatorname {E} \left[(\ln X-\ln G_{X})^{2}\right]\\&=\operatorname {E} [(\ln X-\operatorname {E} \left[\ln X])^{2}\right]\\&=\operatorname {E} \left[(\ln X)^{2}\right]-(\operatorname {E} [\ln X])^{2}\\&=\operatorname {var} [\ln X]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5737429860855b238c6ac72ae064e4bb6d8cb772)
![{\displaystyle \operatorname {var} _{GX}=e^{\operatorname {var} [\ln X]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/524cf664ccfd5eb381fd1987926209f1c401a200)
![{\displaystyle {\begin{aligned}\ln \operatorname {var_{G(1-X)}} &=\operatorname {E} [(\ln(1-X)-\ln G_{1-X})^{2}]\\&=\operatorname {E} [(\ln(1-X)-\operatorname {E} [\ln(1-X)])^{2}]\\&=\operatorname {E} [(\ln(1-X))^{2}]-(\operatorname {E} [\ln(1-X)])^{2}\\&=\operatorname {var} [\ln(1-X)]\\&\\\operatorname {var_{G(1-X)}} &=e^{\operatorname {var} [\ln(1-X)]}\\&\\\ln \operatorname {cov_{G{X,1-X}}} &=\オペレーター名 {E} [(\ln X-\ln G_{X})(\ln(1-X)-\ln G_{1-X})]\\&=\オペレーター名 {E} [(\ln X-\オペレーター名 {E} [\ln X])(\ln(1-X)-\オペレーター名 {E} [\ln(1-X)])]\\&=\オペレーター名 {E} \left[\ln X\ln(1-X)\right]-\オペレーター名 {E} [\ln X]\オペレーター名 {E} [\ln(1-X)]\\&=\オペレーター名 {cov} [\ln X,\ln(1-X)]\\&\\\オペレーター名 {cov} _{G{X,(1-X)}}&=e^{\オペレーター名 {cov} [\ln X,\ln(1-X)]}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9951a17fba87115b493918bfd9271c8e2193d0a8)
![{\displaystyle \operatorname {var} [\ln X]=\psi _{1}(\alpha )-\psi _{1}(\alpha +\beta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e396e8700267735eb741f73e8906445579c43bc6)
![{\displaystyle \operatorname {var} [\ln(1-X)]=\psi _{1}(\beta )-\psi _{1}(\alpha +\beta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70eefadef46c7d56cc13c8221aa3df1d71596b7f)
![{\displaystyle \operatorname {cov} [\ln X,\ln(1-X)]=-\psi _{1}(\alpha +\beta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7a515ada0b9d62c5a3a7b35662b03256d66e3b9)
![{\displaystyle \ln \operatorname {var} _{GX}=\operatorname {var} [\ln X]=\psi _{1}(\alpha )-\psi _{1}(\alpha +\beta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/194b00552edda5d8d026a24872cdb27b604516c9)
![{\displaystyle \ln \operatorname {var} _{G(1-X)}=\operatorname {var} [\ln(1-X)]=\psi _{1}(\beta )-\psi _{1}(\alpha +\beta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96dd82553307c025c84da68a3c373aad7467abd2)
![{\displaystyle \ln \operatorname {cov} _{GX,1-X}=\operatorname {cov} [\ln X,\ln(1-X)]=-\psi _{1}(\alpha +\ベータ版）}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40a793c0271e457f671edb0668edc15bbae8740f)
![{\displaystyle \operatorname {E} [|XE[X]|]={\frac {2\alpha ^{\alpha }\beta ^{\beta }}{\mathrm {B} (\alpha ,\beta ) (\アルファ +\ベータ )^{\アルファ +\ベータ +1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d1c6330a91df22b40cedc7903dbc70120d66cf9)
![{\displaystyle {\begin{aligned}{\frac {\text{平均値からの絶対偏差}}{\text{標準偏差}}}&={\frac {\operatorname {E} [|XE[X]|]}{\sqrt {\operatorname {var} (X)}}}\\&\approx {\sqrt {\frac {2}{\pi }}}\left(1+{\frac {7}{12(\alpha +\beta )}}{}-{\frac {1}{12\alpha }}-{\frac {1}{12\beta }}\right),{\text{ if }}\alpha ,\beta >1.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c196a5a2eb110b71471a3dc019241c6cb8c3f927)
![{\displaystyle \operatorname {E} [|XE[X]|]={\frac {2\mu ^{\mu \nu }(1-\mu )^{(1-\mu )\nu }}{ \nu \mathrm {B} (\mu \nu ,(1-\mu )\nu )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/027efecf8aaefea8c805194e47a1374ffcb63cb8)
![{\displaystyle {\begin{aligned}\operatorname {E} [|XE[X]|]={\frac {2^{1-\nu }}{\nu \mathrm {B} ({\tfrac {\ nu }{2}},{\tfrac {\nu }{2}})}}&={\frac {2^{1-\nu }\Gamma (\nu )}{\nu (\Gamma ({\tfrac {\nu }{2}}))^{2}}}\\\lim _{\nu \to 0}\left(\lim _{\mu \to {\frac { 1}{2}}}\オペレーター名 {E} [|XE[X]|]\right)&={\tfrac {1}{2}}\\\lim _{\nu \to \infty }\left (\lim _{\mu \to {\frac {1}{2}}}\operatorname {E} [|XE[X]|]\right)&=0\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87aa0eff7a4da0f5abe2d211e2b7dda8c8fff801)
![{\displaystyle {\begin{aligned}\lim _{\beta \to 0}\operatorname {E} [|XE[X]|]&=\lim _{\alpha \to 0}\operatorname {E} [ |XE[X]|]=0\\\lim _{\beta \to \infty }\operatorname {E} [|XE[X]|]&=\lim _{\alpha \to \infty }\operatorname {E} [|XE[X]|]=0\\\lim _{\mu \to 0}\operatorname {E} [|XE[X]|]& =\lim _{\mu \to 1}\operatorname {E} [|XE[X]|]=0\\\lim _{\nu \to 0}\operatorname {E} [|XE[X]|]&={\sqrt {\mu (1-\mu )}}\\\lim _{\nu \to \infty }\operatorname {E} [|XE[X]|] &=0\end{整列}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87c43b4a05f8ea3acf3f15b0a16f6ee07811ac6b)
![{\displaystyle \gamma _{1}={\frac {\operatorname {E} [(X-\mu )^{3}]}{(\operatorname {var} (X))^{3/2}}}={\frac {2(\beta -\alpha ){\sqrt {\alpha +\beta +1}}}{(\alpha +\beta +2){\sqrt {\alpha \beta }}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2334c6fa1e6326760870b716521460bba115a92)
![{\displaystyle \gamma _{1}={\frac {\operatorname {E} [(X-\mu )^{3}]}{(\operatorname {var} (X))^{3/2}} }={\frac {2(1-2\mu ){\sqrt {1+\nu }}}{(2+\nu ){\sqrt {\mu (1-\mu )}}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d71042628a42bfedc972e01f8ba5c27f63c3fae4)
![{\displaystyle \gamma _{1}={\frac {\operatorname {E} [(X-\mu )^{3}]}{(\operatorname {var} (X))^{3/2}}}={\frac {2(1-2\mu ){\sqrt {\text{ var }}}}{\mu (1-\mu )+\operatorname {var} }}{\text{ if }}\operatorname {var} <\mu (1-\mu )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e30c818c1af07028336494d35562333ffd903f1)
![{\displaystyle (\gamma _{1})^{2}={\frac {(\operatorname {E} [(X-\mu )^{3}])^{2}}{(\operatorname {var} (X))^{3}}}={\frac {4}{(2+\nu )^{2}}}{\bigg (}{\frac {1}{\text{var}}}-4(1+\nu ){\bigg )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9d9343b5e45ab6ac483a8f0fb8efd1699e0158d)
![{\displaystyle {\begin{aligned}{\text{過剰尖度}}&={\text{尖度}}-3\\&={\frac {\operatorname {E} [(X-\mu )^{4}]}{(\operatorname {var} (X))^{2}}}-3\\&={\frac {6[\alpha ^{3}-\alpha ^{2}(2\beta -1)+\beta ^{2}(\beta +1)-2\alpha \beta (\beta +2)]}{\alpha \beta (\alpha +\beta +2)(\alpha +\beta +3)}}\\&={\frac {6[(\alpha -\beta )^{2}(\alpha +\beta +1)-\alpha \beta (\alpha +\beta +2)]}{\alpha \beta (\alpha +\beta +2)(\alpha +\beta +3)}}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed8320d4f38ba9260f8ad91c30238abc08306dc8)
_Beta_Distr_alpha=beta_from_0_to_25_Back_-_J._Rodal.jpg/500px-Re(CharacteristicFunction)_Beta_Distr_alpha=beta_from_0_to_25_Back_-_J._Rodal.jpg)
_Beta_Distr_alpha=beta_from_0_to_25_Front-_J._Rodal.jpg/500px-Re(CharacteristicFunc)_Beta_Distr_alpha=beta_from_0_to_25_Front-_J._Rodal.jpg)
_Beta_Distr_alpha_from_0_to_25_and_beta=alpha+0.5_Back_-_J._Rodal.jpg/500px-Re(CharacteristFunc)_Beta_Distr_alpha_from_0_to_25_and_beta=alpha+0.5_Back_-_J._Rodal.jpg)
_Beta_Distrib._beta_from_0_to_25,_alpha=beta+0.5_Back_-_J._Rodal.jpg/500px-Re(CharacterFunc)_Beta_Distrib._beta_from_0_to_25,_alpha=beta+0.5_Back_-_J._Rodal.jpg)
_Beta_Distr._beta_from_0_to_25,_alpha=beta+0.5_Front_-_J._Rodal.jpg/500px-Re(CharacterFunc)_Beta_Distr._beta_from_0_to_25,_alpha=beta+0.5_Front_-_J._Rodal.jpg)
![{\displaystyle {\begin{aligned}\varphi _{X}(\alpha ;\beta ;t)&=\operatorname {E} \left[e^{itX}\right]\\&=\int _{0}^{1}e^{itx}f(x;\alpha ,\beta )\,dx\\&={}_{1}F_{1}(\alpha ;\alpha +\beta ;it)\!\\&=\sum _{n=0}^{\infty }{\frac {\alpha ^{(n)}(it)^{n}}{(\alpha +\beta )^{(n)}n!}}\\&=1+\sum _{k=1}^{\infty }\left(\prod _{r=0}^{k-1}{\frac {\alpha +r}{\alpha +\beta +r}}\right){\frac {(it)^{k}}{k!}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5dac7a59df675d0b8758523c87e85888609a213c)
![{\displaystyle \operatorname {Re} \left[{}_{1}F_{1}(\alpha ;\alpha +\beta ;it)\right]=\operatorname {Re} \left[{}_{1}F_{1}(\alpha ;\alpha +\beta ;-it)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/468f2135d76bd1b522c84092679209ff6abd5845)
![{\displaystyle \operatorname {Im} \left[{}_{1}F_{1}(\alpha ;\alpha +\beta ;it)\right]=-\operatorname {Im} \left[{}_{1}F_{1}(\alpha ;\alpha +\beta ;-it)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0fca8984292cc85cb8a37ecb5a2b7c04b5596282)
![{\displaystyle {\begin{aligned}M_{X}(\alpha ;\beta ;t)&=\operatorname {E} \left[e^{tX}\right]\\[4pt]&=\int _{0}^{1}e^{tx}f(x;\alpha ,\beta )\,dx\\[4pt]&={}_{1}F_{1}(\alpha ;\alpha +\beta ;t)\\[4pt]&=\sum _{n=0}^{\infty }{\frac {\alpha ^{(n)}}{(\alpha +\beta )^{(n)}}}{\frac {t^{n}}{n!}}\\[4pt]&=1+\sum _{k=1}^{\infty }\left(\prod _{r=0}^{k-1}{\frac {\alpha +r}{\alpha +\beta +r}}\right){\frac {t^{k}}{k!}}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e664dd90aeae487c19dc89c87c7d1abe47bfcf3d)
![{\displaystyle \operatorname {E} [X^{k}]={\frac {\alpha ^{(k)}}{(\alpha +\beta )^{(k)}}}=\prod _{r=0}^{k-1}{\frac {\alpha +r}{\alpha +\beta +r}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e03c03f31b903a1bc73ea8b637e3134b110a85a2)
![{\displaystyle \operatorname {E} [X^{k}]={\frac {\alpha +k-1}{\alpha +\beta +k-1}}\operatorname {E} [X^{k-1}].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/069cb373a905b1e8a5a82a0e3b028e88f63672e2)
![{\displaystyle {\begin{aligned}\operatorname {E} [1-X]&={\frac {\beta }{\alpha +\beta }}\\\operatorname {E} [X(1-X)]&=\operatorname {E} [(1-X)X]={\frac {\alpha \beta }{(\alpha +\beta )(\alpha +\beta +1)}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43fc49b9eafccd56d39c236b26d222dde51638ce)
![{\displaystyle \operatorname {var} [(1-X)]=\operatorname {var} [X]=-\operatorname {cov} [X,(1-X)]={\frac {\alpha \beta }{(\alpha +\beta )^{2}(\alpha +\beta +1)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7273cc84a6c789724b985c34059fa75a62bce631)
![{\displaystyle {\begin{aligned}\operatorname {E} \left[{\frac {1}{X}}\right]&={\frac {\alpha +\beta -1}{\alpha -1}}&&{\text{ if }}\alpha >1\\\operatorname {E} \left[{\frac {1}{1-X}}\right]&={\frac {\alpha +\beta -1}{\beta -1}}&&{\text{ if }}\beta >1\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/972a9c1853e6991aac6666b06c0a633a0caad5b7)
![{\displaystyle {\begin{aligned}\operatorname {E} \left[{\frac {X}{1-X}}\right]&={\frac {\alpha }{\beta -1}}&&{\text{ if }}\beta >1\\\operatorname {E} \left[{\frac {1-X}{X}}\right]&={\frac {\beta }{\alpha -1}}&&{\text{ if }}\alpha >1\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db2a0928a7f906bcbe5cfd9d0c57713c9ab5cfb7)
![{\displaystyle \operatorname {var} \left[{\frac {1}{X}}\right]=\operatorname {E} \left[\left({\frac {1}{X}}-\operatorname {E} \left[{\frac {1}{X}}\right]\right)^{2}\right]=\operatorname {var} \left[{\frac {1-X}{X}}\right]=\operatorname {E} \left[\left({\frac {1-X}{X}}-\operatorname {E} \left[{\frac {1-X}{X}}\right]\right)^{2}\right]={\frac {\beta (\alpha +\beta -1)}{(\alpha -2)(\alpha -1)^{2}}}{\text{ if }}\alpha >2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af7a7cb782c6c42658374ac45365cde388c2326a)
![{\displaystyle \operatorname {var} \left[{\frac {1}{1-X}}\right]=\operatorname {E} \left[\left({\frac {1}{1-X}}-\operatorname {E} \left[{\frac {1}{1-X}}\right]\right)^{2}\right]=\operatorname {var} \left[{\frac {X}{1-X}}\right]=\operatorname {E} \left[\left({\frac {X}{1-X}}-\operatorname {E} \left[{\frac {X}{1-X}}\right]\right)^{2}\right]={\frac {\alpha (\alpha +\beta -1)}{(\beta -2)(\beta -1)^{2}}}{\text{ if }}\beta >2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22dd8c7aad68227962947ad69019b01a79c41b6d)
![{\displaystyle \operatorname {cov} \left[{\frac {1}{X}},{\frac {1}{1-X}}\right]=\operatorname {cov} \left[{\frac {1-X}{X}},{\frac {X}{1-X}}\right]=\operatorname {cov} \left[{\frac {1}{X}},{\frac {X}{1-X}}\right]=\operatorname {cov} \left[{\frac {1-X}{X}},{\frac {1}{1-X}}\right]={\frac {\alpha +\beta -1}{(\alpha -1)(\beta -1)}}{\text{ if }}\alpha ,\beta >1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa2342c3fe24a7ed5b840fd9583c055aa5791486)
![{\displaystyle {\begin{aligned}\operatorname {E} [\ln(X)]&=\psi (\alpha )-\psi (\alpha +\beta )=-\operatorname {E} \left[\ln \left({\frac {1}{X}}\right)\right],\\\operatorname {E} [\ln(1-X)]&=\psi (\beta )-\psi (\alpha +\beta )=-\operatorname {E} \left[\ln \left({\frac {1}{1-X}}\right)\right].\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6125bcdc7551c82f7451b444b941c97b83abdf20)
![{\displaystyle {\begin{aligned}\operatorname {E} \left[\ln \left({\frac {X}{1-X}}\right)\right]&=\psi (\alpha )-\psi (\beta )=\operatorname {E} [\ln(X)]+\operatorname {E} \left[\ln \left({\frac {1}{1-X}}\right)\right],\\\operatorname {E} \left[\ln \left({\frac {1-X}{X}}\right)\right]&=\psi (\beta )-\psi (\alpha )=-\operatorname {E} \left[\ln \left({\frac {X}{1-X}}\right)\right].\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/519d77a6f894b5ebc9240b813a675477194f8858)
![{\displaystyle {\begin{aligned}\operatorname {E} \left[\ln ^{2}(X)\right]&=(\psi (\alpha )-\psi (\alpha +\beta ))^{2}+\psi _{1}(\alpha )-\psi _{1}(\alpha +\beta ),\\\operatorname {E} \left[\ln ^{2}(1-X)\right]&=(\psi (\beta )-\psi (\alpha +\beta ))^{2}+\psi _{1}(\beta )-\psi _{1}(\alpha +\beta ),\\\operatorname {E} \left[\ln(X)\ln(1-X)\right]&=(\psi (\alpha )-\psi (\alpha +\beta ))(\psi (\beta )-\psi (\alpha +\beta ))-\psi _{1}(\alpha +\beta ).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b42eb1276e349df39df3051df11e0e16afe88e2e)
![{\displaystyle {\begin{aligned}\operatorname {cov} [\ln(X),\ln(1-X)]&=\operatorname {E} \left[\ln(X)\ln(1-X)\right]-\operatorname {E} [\ln(X)]\operatorname {E} [\ln(1-X)]=-\psi _{1}(\alpha +\beta )\\&\\\operatorname {var} [\ln X]&=\operatorname {E} [\ln ^{2}(X)]-(\operatorname {E} [\ln(X)])^{2}\\&=\psi _{1}(\alpha )-\psi _{1}(\alpha +\beta )\\&=\psi _{1}(\alpha )+\operatorname {cov} [\ln(X),\ln(1-X)]\\&\\\operatorname {var} [\ln(1-X)]&=\operatorname {E} [\ln ^{2}(1-X)]-(\operatorname {E} [\ln(1-X)])^{2}\\&=\psi _{1}(\beta )-\psi _{1}(\alpha +\beta )\\&=\psi _{1}(\beta )+\operatorname {cov} [\ln(X),\ln(1-X)]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e76df52180954247e1d1aa62646a20f7e3d68a36)
![{\displaystyle {\begin{aligned}\operatorname {var} \left[\ln \left({\frac {1}{X}}\right)\right]&=\operatorname {var} [\ln(X)]=\psi _{1}(\alpha )-\psi _{1}(\alpha +\beta ),\\\operatorname {var} \left[\ln \left({\frac {1}{1-X}}\right)\right]&=\operatorname {var} [\ln(1-X)]=\psi _{1}(\beta )-\psi _{1}(\alpha +\beta ),\\\operatorname {cov} \left[\ln \left({\frac {1}{X}}\right),\ln \left({\frac {1}{1-X}}\right)\right]&=\operatorname {cov} [\ln(X),\ln(1-X)]=-\psi _{1}(\alpha +\beta ).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d335167cf1713083b3fb1c4d0cfadb04c1d40d43)
![{\displaystyle \operatorname {var} \left[\ln \left({\frac {X}{1-X}}\right)\right]=\operatorname {var} \left[\ln \left({\frac {1-X}{X}}\right)\right]=-\operatorname {cov} \left[\ln \left({\frac {X}{1-X}}\right),\ln \left({\frac {1-X}{X}}\right)\right]=\psi _{1}(\alpha )+\psi _{1}(\beta ).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52585d88405196366b14e2e7183f48be9b5de874)
![{\displaystyle {\begin{aligned}h(X)&=\operatorname {E} [-\ln(f(x;\alpha ,\beta ))]\\[4pt]&=\int _{0}^{1}-f(x;\alpha ,\beta )\ln(f(x;\alpha ,\beta ))\,dx\\[4pt]&=\ln(\mathrm {B} (\alpha ,\beta ))-(\alpha -1)\psi (\alpha )-(\beta -1)\psi (\beta )+(\alpha +\beta -2)\psi (\alpha +\beta )\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc995dbbf41c91536ce4d0efaff554d8b0a77fe)
![{\displaystyle {\begin{aligned}H(X_{1},X_{2})&=\int _{0}^{1}-f(x;\alpha ,\beta )\ln(f(x;\alpha ',\beta '))\,dx\\[4pt]&=\ln \left(\mathrm {B} (\alpha ',\beta ')\right)-(\alpha '-1)\psi (\alpha )-(\beta '-1)\psi (\beta )+(\alpha '+\beta '-2)\psi (\alpha +\beta ).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/711e6ef61e56ccd8ba515b8db5013d1847939ec8)
![{\displaystyle {\begin{aligned}D_{\mathrm {KL} }(X_{1}\parallel X_{2})&=\int _{0}^{1}f(x;\alpha ,\beta )\ln \left({\frac {f(x;\alpha ,\beta )}{f(x;\alpha ',\beta ')}}\right)\,dx\\[4pt]&=\left(\int _{0}^{1}f(x;\alpha ,\beta )\ln(f(x;\alpha ,\beta ))\,dx\right)-\left(\int _{0}^{1}f(x;\alpha ,\beta )\ln(f(x;\alpha ',\beta '))\,dx\right)\\[4pt]&=-h(X_{1})+H(X_{1},X_{2})\\[4pt]&=\ln \left({\frac {\mathrm {B} (\alpha ',\beta ')}{\mathrm {B} (\alpha ,\beta )}}\right)+(\alpha -\alpha ')\psi (\alpha )+(\beta -\beta ')\psi (\beta )+(\alpha '-\alpha +\beta '-\beta )\psi (\alpha +\beta ).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d62cb13ae214eaa48e12d453fbae8bdd58b301a)
_Parameter_estimates_vs._excess_Kurtosis_and_(squared)_Skewness_Beta_distribution_-_J._Rodal.png/500px-(alpha_and_beta)_Parameter_estimates_vs._excess_Kurtosis_and_(squared)_Skewness_Beta_distribution_-_J._Rodal.png)
![{\displaystyle \operatorname {E} [|XE[X]|](\mathrm {B} (\alpha ,\beta ))=\operatorname {E} [|XE[X]|](\mathrm {B} (\ベータ、\アルファ))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83468b7365d9095b07e04bfbb5c9cff50c64ea2d)
![{\displaystyle {\text{Re}}[{}_{1}F_{1}(\alpha ;\alpha +\beta ;it)]={\text{Re}}[{}_{1}F_{1}(\alpha ;\alpha +\beta ;-it)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55bbddff6ec53eb39eccc603618459ba36e10ad2)
![{\displaystyle {\text{Im}}[{}_{1}F_{1}(\alpha ;\alpha +\beta ;it)]=-{\text{Im}}[{}_{1}F_{1}(\alpha ;\alpha +\beta ;-it)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebb682c9915a60ef8d982340de042127ab262a0f)
![{\displaystyle {\text{Abs}}[{}_{1}F_{1}(\alpha ;\alpha +\beta ;it)]={\text{Abs}}[{}_{1}F_{1}(\alpha ;\alpha +\beta ;-it)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08c1c15ff4c63020b66d2404d439514d965946d0)
_for_Beta_distribution_Maxima_at_alpha=beta=2_-_J._Rodal.png/500px-Max_(Joint_Log_Likelihood_per_N)_for_Beta_distribution_Maxima_at_alpha=beta=2_-_J._Rodal.png)
_for_Beta_distribution_Maxima_at_alpha=beta=_0.25,0.5,1,2,4,6,8_-_J._Rodal.png/500px-Max_(Joint_Log_Likelihood_per_N)_for_Beta_distribution_Maxima_at_alpha=beta=_0.25,0.5,1,2,4,6,8_-_J._Rodal.png)
![{\displaystyle \operatorname {var} [\ln(X)]=\operatorname {E} [\ln ^{2}(X)]-(\operatorname {E} [\ln(X)])^{2}=\psi _{1}(\alpha )-\psi _{1}(\alpha +\beta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7737d681fea7490e27f8760c6bcc8fccb154904)
![{\displaystyle \operatorname {var} [\ln(1-X)]=\operatorname {E} [\ln ^{2}(1-X)]-(\operatorname {E} [\ln(1-X)])^{2}=\psi _{1}(\beta )-\psi _{1}(\alpha +\beta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f84c3747955206cf2190c61bd7875a6cd739ac04)
![{\displaystyle \operatorname {var} [\ln(X)]>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5fb5a5d0db057469fb9dad8df2902fe93e3f3b0d)
![{\displaystyle \operatorname {var} [\ln(1-X)]>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c66a5aaa8362f578beb9b141b4108138b9d21e89)
![{\displaystyle {\begin{aligned}{\hat {\operatorname {E} }}[\ln(X)]&=\psi ({\hat {\alpha }})-\psi ({\hat {\alpha }}+{\hat {\beta }})={\frac {1}{N}}\sum _{i=1}^{N}\ln X_{i}=\ln {\hat {G}}_{X}\\{\hat {\operatorname {E} }}[\ln(1-X)]&=\psi ({\hat {\beta }})-\psi ({\hat {\alpha }}+{\hat {\beta }})={\frac {1}{N}}\sum _{i=1}^{N}\ln(1-X_{i})=\ln {\hat {G}}_{(1-X)}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e43bfd770a92437c9b0bbef4c7c6ead3534eba3)
![{\displaystyle {\hat {\operatorname {E} }}\left[\ln \left({\frac {X}{1-X}}\right)\right]=\psi ({\hat {\alpha }})-\psi ({\hat {\beta }})={\frac {1}{N}}\sum _{i=1}^{N}\ln {\frac {X_{i}}{1-X_{i}}}=\ln {\hat {G}}_{X}-\ln \left({\hat {G}}_{(1-X)}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d5d9a3105854f3fe65f5a72aa4418da4a50ba11)
![{\displaystyle {\frac {\partial ^{2}\ln {\mathcal {L}}(\alpha ,\beta \mid X)}{\partial \alpha ^{2}}}=-\operatorname {var} [\ln X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/517a09a3b13d22689a3e1e400cbcab3af08e130c)
![{\displaystyle {\frac {\partial ^{2}\ln {\mathcal {L}}(\alpha ,\beta \mid X)}{\partial \beta ^{2}}}=-\operatorname {var} [\ln(1-X)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1715f50fc408ceba307005a3f9e404520edd5a20)
![{\displaystyle \mathrm {var} ({\hat {\alpha }})\geq {\frac {1}{\operatorname {var} [\ln X]}}\geq {\frac {1}{\psi _{1}({\hat {\alpha }})-\psi _{1}({\hat {\alpha }}+{\hat {\beta }})}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/744f1e8421337ed7a2e6cae00fccec1eaf68e3dc)
![{\displaystyle \mathrm {var} ({\hat {\beta }})\geq {\frac {1}{\operatorname {var} [\ln(1-X)]}}\geq {\frac {1 }{\psi _{1}({\hat {\beta }})-\psi _{1}({\hat {\alpha }}+{\hat {\beta }})}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f01dc5c3eb614cfa71a500fd34f3fa430c183c76)
![{\displaystyle \alpha =2:\quad \operatorname {E} \left[-{\frac {1}{N}}{\frac {\partial ^{2}\ln {\mathcal {L}}(\alpha ,\beta ,a,c\mid Y)}{\partial a^{2}}}\right]={\mathcal {I}}_{a,a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53538160a2404a5d7b74ae2033fbbb2dbc1045eb)
![{\displaystyle \beta =2:\quad \operatorname {E} \left[-{\frac {1}{N}}{\frac {\partial ^{2}\ln {\mathcal {L}}(\alpha ,\beta ,a,c\mid Y)}{\partial c^{2}}}\right]={\mathcal {I}}_{c,c}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee2c6ecfcafe60e54799ab4a16451cf478d65f7d)
![{\displaystyle \alpha =2:\quad \operatorname {E} \left[-{\frac {1}{N}}{\frac {\partial ^{2}\ln {\mathcal {L}}(\alpha ,\beta ,a,c\mid Y)}{\partial \alpha \partial a}}\right]={\mathcal {I}}_{\alpha ,a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b8af544d7c5e0cc278aa725daba7f3de2f70d31)
![{\displaystyle \beta =1:\quad \operatorname {E} \left[-{\frac {1}{N}}{\frac {\partial ^{2}\ln {\mathcal {L}}(\alpha ,\beta ,a,c\mid Y)}{\partial \beta \partial c}}\right]={\mathcal {I}}_{\beta ,c}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f87b32b39997964dcbc95bc64c1364e6832db1d)
![{\displaystyle {\mathcal {I}}(\alpha )=\operatorname {E} \left[\left({\frac {\partial }{\partial \alpha }}\ln {\mathcal {L}}(\alpha \mid X)\right)^{2}\right],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/daec13972d17a073bcd447abfde55a6b0e168720)
![{\displaystyle {\mathcal {I}}(\alpha )=-\operatorname {E} \left[{\frac {\partial ^{2}}{\partial \alpha ^{2}}}\ln({\mathcal {L}}(\alpha \mid X))\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92244569d1fab7e801097aa25a9c883ff75be5aa)
![{\displaystyle \operatorname {var} [{\hat {\alpha }}]\geq {\frac {1}{{\mathcal {I}}(\alpha )}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d93d4983a2717258c52eb47d1562e849a3a66c5c)
![{\displaystyle ({\mathcal {I}}(\theta ))_{i,j}=\operatorname {E} \left[\left({\frac {\partial }{\partial \theta _{i}}}\ln {\mathcal {L}}\right)\left({\frac {\partial }{\partial \theta _{j}}}\ln {\mathcal {L}}\right)\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e4390c2c908fe7a3e4f06ee9e3526df1ff20f18)
![{\displaystyle ({\mathcal {I}}(\theta ))_{i,j}=-\operatorname {E} \left[{\frac {\partial ^{2}}{\partial \theta _{i}\,\partial \theta _{j}}}\ln({\mathcal {L}})\right]\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b7d9fd3ec7fa358ec6450f166aa89592bcc55ab)
![{\displaystyle -{\frac {\partial ^{2}\ln {\mathcal {L}}(\alpha ,\beta \mid X)}{N\partial \alpha ^{2}}}=\operatorname {var} [\ln(X)]=\psi _{1}(\alpha )-\psi _{1}(\alpha +\beta )={\mathcal {I}}_{\alpha ,\alpha }=\operatorname {E} \left[-{\frac {\partial ^{2}\ln {\mathcal {L}}(\alpha ,\beta \mid X)}{N\partial \alpha ^{2}}}\right]=\ln \operatorname {var} _{GX}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c90003d5bd2f6d2bcfe2c788585689726b4b7e36)
![{\displaystyle -{\frac {\partial ^{2}\ln {\mathcal {L}}(\alpha ,\beta \mid X)}{N\,\partial \beta ^{2}}}=\operatorname {var} [\ln(1-X)]=\psi _{1}(\beta )-\psi _{1}(\alpha +\beta )={\mathcal {I}}_{\beta ,\beta }=\operatorname {E} \left[-{\frac {\partial ^{2}\ln {\mathcal {L}}(\alpha ,\beta \mid X)}{N\partial \beta ^{2}}}\right]=\ln \operatorname {var} _{G(1-X)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b59bcc31f14f1f4b3b07bd92a66426bb6ac126b1)
![{\displaystyle -{\frac {\partial ^{2}\ln {\mathcal {L}}(\alpha ,\beta \mid X)}{N\,\partial \alpha \,\partial \beta }}=\operatorname {cov} [\ln X,\ln(1-X)]=-\psi _{1}(\alpha +\beta )={\mathcal {I}}_{\alpha ,\beta }=\operatorname {E} \left[-{\frac {\partial ^{2}\ln {\mathcal {L}}(\alpha ,\beta \mid X)}{N\,\partial \alpha \,\partial \beta }}\right]=\ln \operatorname {cov} _{G{X,(1-X)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6c3268da3b23c062ce981ab02d4c454a0267365)
![{\displaystyle {\begin{aligned}\det({\mathcal {I}}(\alpha ,\beta ))&={\mathcal {I}}_{\alpha ,\alpha }{\mathcal {I}}_{\beta ,\beta }-{\mathcal {I}}_{\alpha ,\beta }{\mathcal {I}}_{\alpha ,\beta }\\[4pt]&=(\psi _{1}(\alpha )-\psi _{1}(\alpha +\beta ))(\psi _{1}(\beta )-\psi _{1}(\alpha +\beta ))-(-\psi _{1}(\alpha +\beta ))(-\psi _{1}(\alpha +\beta ))\\[4pt]&=\psi _{1}(\alpha )\psi _{1}(\beta )-(\psi _{1}(\alpha )+\psi _{1}(\beta ))\psi _{1}(\alpha +\beta )\\[4pt]\lim _{\alpha \to 0}\det({\mathcal {I}}(\alpha ,\beta ))&=\lim _{\beta \to 0}\det({\mathcal {I}}(\alpha ,\beta ))=\infty \\[4pt]\lim _{\alpha \to \infty }\det({\mathcal {I}}(\alpha ,\beta ))&=\lim _{\beta \to \infty }\det({\mathcal {I}}(\alpha ,\beta ))=0\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2c5ccf59b05ea730fc108360c07e9ac9634e829)
_for_alpha=beta_vs_range_(c-a)_and_exponent_alpha=beta_-_J._Rodal.png/500px-Fisher_Information_I(a,a)_for_alpha=beta_vs_range_(c-a)_and_exponent_alpha=beta_-_J._Rodal.png)
_for_alpha=beta,_vs._range_(c_-_a)_and_exponent_alpha=beta_-_J._Rodal.png/500px-Fisher_Information_I(alpha,a)_for_alpha=beta,_vs._range_(c_-_a)_and_exponent_alpha=beta_-_J._Rodal.png)
![{\displaystyle -{\frac {1}{N}}{\frac {\partial ^{2}\ln {\mathcal {L}}(\alpha ,\beta ,a,c\mid Y)}{\partial \alpha ^{2}}}=\operatorname {var} [\ln(X)]=\psi _{1}(\alpha )-\psi _{1}(\alpha +\beta )={\mathcal {I}}_{\alpha ,\alpha }=\operatorname {E} \left[-{\frac {1}{N}}{\frac {\partial ^{2}\ln {\mathcal {L}}(\alpha ,\beta ,a,c\mid Y)}{\partial \alpha ^{2}}}\right]=\ln(\operatorname {var_{GX}} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31be86f2c53663c6d3975bc2676806ba3e538423)
![{\displaystyle -{\frac {1}{N}}{\frac {\partial ^{2}\ln {\mathcal {L}}(\alpha ,\beta ,a,c\mid Y)}{\partial \beta ^{2}}}=\operatorname {var} [\ln(1-X)]=\psi _{1}(\beta )-\psi _{1}(\alpha +\beta )={\mathcal {I}}_{\beta ,\beta }=\operatorname {E} \left[-{\frac {1}{N}}{\frac {\partial ^{2}\ln {\mathcal {L}}(\alpha ,\beta ,a,c\mid Y)}{\partial \beta ^{2}}}\right]=\ln(\operatorname {var_{G(1-X)}} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25ab885119f25fae0b9919326db96395d13e3bc3)
![{\displaystyle -{\frac {1}{N}}{\frac {\partial ^{2}\ln {\mathcal {L}}(\alpha ,\beta ,a,c\mid Y)}{\partial \alpha \,\partial \beta }}=\operatorname {cov} [\ln X,(1-X)]=-\psi _{1}(\alpha +\beta )={\mathcal {I}}_{\alpha ,\beta }=\operatorname {E} \left[-{\frac {1}{N}}{\frac {\partial ^{2}\ln {\mathcal {L}}(\alpha ,\beta ,a,c\mid Y)}{\partial \alpha \,\partial \beta }}\right]=\ln(\operatorname {cov} _{G{X,(1-X)}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02a56af746fb9315340cf382951fe0c3f3640678)
![{\displaystyle {\begin{aligned}\alpha >2:\quad \operatorname {E} \left[-{\frac {1}{N}}{\frac {\partial ^{2}\ln {\mathcal {L}}(\alpha ,\beta ,a,c\mid Y)}{\partial a^{2}}}\right]&={\mathcal {I}}_{a,a}={\frac {\beta (\alpha +\beta -1)}{(\alpha -2)(c-a)^{2}}}\\\beta >2:\quad \operatorname {E} \left[-{\frac {1}{N}}{\frac {\partial ^{2}\ln {\mathcal {L}}(\alpha ,\beta ,a,c\mid Y)}{\partial c^{2}}}\right]&={\mathcal {I}}_{c,c}={\frac {\alpha (\alpha +\beta -1)}{(\beta -2)(c-a)^{2}}}\\\operatorname {E} \left[-{\frac {1}{N}}{\frac {\partial ^{2}\ln {\mathcal {L}}(\alpha ,\beta ,a,c\mid Y)}{\partial a\,\partial c}}\right]&={\mathcal {I}}_{a,c}={\frac {(\alpha +\beta -1)}{(c-a)^{2}}}\\\alpha >1:\quad \operatorname {E} \left[-{\frac {1}{N}}{\frac {\partial ^{2}\ln {\mathcal {L}}(\alpha ,\beta ,a,c\mid Y)}{\partial \alpha \,\partial a}}\right]&={\mathcal {I}}_{\alpha ,a}={\frac {\beta }{(\alpha -1)(c-a)}}\\\operatorname {E} \left[-{\frac {1}{N}}{\frac {\partial ^{2}\ln {\mathcal {L}}(\alpha ,\beta ,a,c\mid Y)}{\partial \alpha \,\partial c}}\right]&={\mathcal {I}}_{\alpha ,c}={\frac {1}{(c-a)}}\\\operatorname {E} \left[-{\frac {1}{N}}{\frac {\partial ^{2}\ln {\mathcal {L}}(\alpha ,\beta ,a,c\mid Y)}{\partial \beta \,\partial a}}\right]&={\mathcal {I}}_{\beta ,a}=-{\frac {1}{(c-a)}}\\\beta >1:\quad \operatorname {E} \left[-{\frac {1}{N}}{\frac {\partial ^{2}\ln {\mathcal {L}}(\alpha ,\beta ,a,c\mid Y)}{\partial \beta \,\partial c}}\right]&={\mathcal {I}}_{\beta ,c}=-{\frac {\alpha }{(\beta -1)(c-a)}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/636646f51bdb1a3193b1721483878e98f4f19c3e)
![{\displaystyle {\mathcal {I}}_{\alpha ,a}={\frac {\operatorname {E} \left[{\frac {1-X}{X}}\right]}{c-a}}={\frac {\beta }{(\alpha -1)(c-a)}}{\text{ if }}\alpha >1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e670565bb8d06bace69cf892864520f5c83b5449)
![{\displaystyle {\mathcal {I}}_{\beta ,c}=-{\frac {\operatorname {E} \left[{\frac {X}{1-X}}\right]}{c-a}}=-{\frac {\alpha }{(\beta -1)(c-a)}}{\text{ if }}\beta >1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94f9b7788a4f19e1cbc765ab8fc85a7ad55dec4f)
![{\displaystyle {\begin{aligned}\alpha >2:\quad {\mathcal {I}}_{a,a}&=\operatorname {var} \left[{\frac {1}{X}}\right]\left({\frac {\alpha -1}{c-a}}\right)^{2}=\operatorname {var} \left[{\frac {1-X}{X}}\right]\left({\frac {\alpha -1}{c-a}}\right)^{2}={\frac {\beta (\alpha +\beta -1)}{(\alpha -2)(c-a)^{2}}}\\\beta >2:\quad {\mathcal {I}}_{c,c}&=\operatorname {var} \left[{\frac {1}{1-X}}\right]\left({\frac {\beta -1}{c-a}}\right)^{2}=\operatorname {var} \left[{\frac {X}{1-X}}\right]\left({\frac {\beta -1}{c-a}}\right)^{2}={\frac {\alpha (\alpha +\beta -1)}{(\beta -2)(c-a)^{2}}}\\{\mathcal {I}}_{a,c}&=\operatorname {cov} \left[{\frac {1}{X}},{\frac {1}{1-X}}\right]{\frac {(\alpha -1)(\beta -1)}{(c-a)^{2}}}=\operatorname {cov} \left[{\frac {1-X}{X}},{\frac {X}{1-X}}\right]{\frac {(\alpha -1)(\beta -1)}{(c-a)^{2}}}={\frac {(\alpha +\beta -1)}{(c-a)^{2}}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1f89730020364bb58791ca0eb47d0de25c896c2)
_Uniform_distribution_-_J._Rodal.png/500px-Beta(1,1)_Uniform_distribution_-_J._Rodal.png)
_-_J._Rodal.png/500px-Beta_distribution_for_3_different_prior_probability_functions,_skewed_case_sample_size_=_(4,12,40)_-_J._Rodal.png)
![{\displaystyle {\begin{aligned}{\sqrt {{\mathcal {I}}(p)}}&={\sqrt {\operatorname {E} \!\left[\left({\frac {d}{dp}}\ln({\mathcal {L}}(p\mid H))\right)^{2}\right]}}\\[6pt]&={\sqrt {\operatorname {E} \!\left[\left({\frac {H}{p}}-{\frac {1-H}{1-p}}\right)^{2}\right]}}\\[6pt]&={\sqrt {p^{1}(1-p)^{0}\left({\frac {1}{p}}-{\frac {0}{1-p}}\right)^{2}+p^{0}(1-p)^{1}\left({\frac {0}{p}}-{\frac {1}{1-p}}\right)^{2}}}\\&={\frac {1}{\sqrt {p(1-p)}}}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/324e2e5a963eed1652cf2d3a9ba95e5491b5df95)
![{\displaystyle {\begin{aligned}&{\text{posterior probability density}}(x=p\mid s,n-s)\\[6pt]={}&{\frac {\operatorname {priorprobabilitydensity} (x=p;\alpha \operatorname {prior} ,\beta \operatorname {prior} ){\mathcal {L}}(s,f\mid x=p)}{\int _{0}^{1}{\text{prior probability density}}(x=p;\alpha \operatorname {prior} ,\beta \operatorname {prior} ){\mathcal {L}}(s,f\mid x=p)\,dx}}\\[6pt]={}&{\frac {{n \choose s}x^{s+\alpha \operatorname {prior} -1}(1-x)^{n-s+\beta \operatorname {prior} -1}/\mathrm {B} (\alpha \operatorname {prior} ,\beta \operatorname {prior} )}{\int _{0}^{1}\left({n \choose s}x^{s+\alpha \operatorname {prior} -1}(1-x)^{n-s+\beta \operatorname {prior} -1}/\mathrm {B} (\alpha \operatorname {prior} ,\beta \operatorname {prior} )\right)\,dx}}\\[6pt]={}&{\frac {x^{s+\alpha \operatorname {prior} -1}(1-x)^{n-s+\beta \operatorname {prior} -1}}{\int _{0}^{1}\left(x^{s+\alpha \operatorname {prior} -1}(1-x)^{n-s+\beta \operatorname {prior} -1}\right)\,dx}}\\[6pt]={}&{\frac {x^{s+\alpha \operatorname {prior} -1}(1-x)^{n-s+\beta \operatorname {prior} -1}}{\mathrm {B} (s+\alpha \operatorname {prior} ,n-s+\beta \operatorname {prior} )}}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/569ad0317cf545ff98538c8a845f216120e87c08)
![{\displaystyle {\begin{aligned}\mu (X)&={\frac {a+4b+c}{6}}\\[8pt]\sigma (X)&={\frac {c-a}{6}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a89a68d1250ebe659be15e88edb5a9eb3e0cf87)
