The Information function computes the Information function of a random variable or a distribution.  The Information function is defined as the second derivative of the log likelihood function with respect to all parameters or to one particular parameter.

The first parameter R can be a distribution (see Statistics[Distribution]), a random variable, or an algebraic expression involving random variables (see Statistics[RandomVariable]).

The second parameter V can be an Array of data sample or a symbol representing an Array of data samples (in which case the option samplesize must be specified).

The options argument can contain one or more of the options shown below.

samplesize=deduce or posint -- If this option is set to 'deduce' (default) the information function attempts to automatically determine the number of data samples provided in V.  This parameter must be specified if the number of samples in V is not immediately detectable.

param=all or name -- If this option is set to 'all' (default) the information of all parameters for this distribution is calculated and the result is returned as a Vector.  Otherwise, this option specifies the parameter which should be used to calculate the information.

ignore=truefalse -- This option is used to specify how to handle non-numeric data. If ignore is set to true all non-numeric items in V will be ignored.

weights=rtable -- Vector of weights (one-dimensional rtable). If weights are given, the Information function will scale each data point to have given weight. Note that the weights provided must have type,realcons and the results are floating-point, even if the problem is specified with exact values. Both the data array and the weights array must have the same number of elements.

$\mathrm{with}\left(\mathrm{Statistics}\right)\:$

$\mathrm{Information}\left(\mathrm{Normal}\left(\mathrm{\mu }\,\mathrm{\sigma }\right)\,A\,\mathrm{samplesize}=4\,\mathrm{param}=\mathrm{\sigma }\right)$

$\frac{4}{{\mathrm{\sigma }}^2}-\frac{3A_1^2}{{\mathrm{\sigma }}^4}-\frac{12{\mathrm{\mu }}^2}{{\mathrm{\sigma }}^4}-\frac{3A_2^2}{{\mathrm{\sigma }}^4}-\frac{3A_3^2}{{\mathrm{\sigma }}^4}-\frac{3A_4^2}{{\mathrm{\sigma }}^4}+\frac{6A_1\mathrm{\mu }}{{\mathrm{\sigma }}^4}+\frac{6A_2\mathrm{\mu }}{{\mathrm{\sigma }}^4}+\frac{6A_3\mathrm{\mu }}{{\mathrm{\sigma }}^4}+\frac{6A_4\mathrm{\mu }}{{\mathrm{\sigma }}^4}$

$N≔\mathrm{RandomVariable}\left(\mathrm{Normal}\left(5\,1\right)\right)\:$

$S≔\mathrm{Sample}\left(N\,4\right)\:$

$\mathrm{Information}\left(\mathrm{Normal}\left(\mathrm{\mu }\,\mathrm{\sigma }\right)\,S\,\mathrm{param}=\mathrm{\mu }\right)$

$-\frac{4}{{\mathrm{\sigma }}^2}$

$S\left[4\right]≔\mathrm{undefined}\:$

$\mathrm{Information}\left(\mathrm{Normal}\left(\mathrm{\mu }\,2\right)\,S\right)$

$Float\left(\mathrm{undefined}\right)$

$\mathrm{Information}\left(\mathrm{Normal}\left(\mathrm{\mu }\,2\right)\,S\,\mathrm{ignore}=\mathrm{true}\right)$

$-\frac{3}{4}$

$S≔\mathrm{Sample}\left(N\,4\right)\:$

$W≔⟨2\,2\,0\,0⟩\:$

$\mathrm{Information}\left(\mathrm{Normal}\left(\mathrm{\mu }\,1\right)\,S\,\mathrm{weights}=W\,\mathrm{param}=\mathrm{\mu }\right)$

$\mathrm{-4}$