page_title: Beta distribution
section_title: Four parameters
hierarchical_section_title: Beta distribution / Statistical Inference / Parameter estimation / Fisher information matrix / Four parameters
caption_attribution_description: English: Fisher Information I(a,a) for alpha=beta vs range (c-a) and exponent alpha=beta - J. Rodal
context_page_description: In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] parameterized by two positive shape parameters, denoted by α and β, that appear as exponents of the random variable and control the shape of the distribution. The generalization to multiple variables is called a Dirichlet distribution.
  The beta distribution has been applied to model the behavior of random variables limited to intervals of finite length in a wide variety of disciplines.
  In Bayesian inference, the beta distribution is the conjugate prior probability distribution for the Bernoulli, binomial, negative binomial and geometric distributions. The beta distribution is a suitable model for the random behavior of percentages and proportions.
  The formulation of the beta distribution discussed here is also known as the beta distribution of the first kind, whereas beta distribution of the second kind is an alternative name for the beta prime distribution.
context_section_description: 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 N iid 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 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 N i.i.d. samples is N times the individual Fisher information (eq. 11.279, page 394 of Cover and Thomas).  (Aryal and Nadarajah 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 the section titled "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)  and its mirror image, scaled by the range (c − a).
  Als
caption: Fisher Information I(a,a) for α = β vs range (c − a) and exponent α = β
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