Fisher information

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In statistics and information theory, the Fisher information (denoted [\mathcal(\theta)]) is the variance of the score. It was first proposed by the statistician R.A. Fisher, and thus is named in his honor.

Definition

Fisher information is the amount of information that an observable random variable X carries about an unobservable parameter θ upon which the probability distribution of X depends. Since the expectation of the score is zero, the variance is also the second moment of the score, and the Fisher information can be written

[\mathcal(\theta)=\mathrm\left[ left[ frac ln f(X;theta) right]^2right],]

where f is the probability density function of random variable X and, consequently, [0 \leq \mathcal(\theta) < \infty]. The Fisher information is thus the expectation of the square of the score. A random variable carrying high Fisher information implies that the absolute value of the score is frequently high. (Remember that the expectation of the score is zero.)

Note that the information as defined above is not a function of a particular observation, as the random variable X has been averaged out. The concept of information is useful when comparing two methods of observing some random process.

If the following regularity condition is met:

[\int \fracf(X ; \theta ) \, d\theta = 0,]

then the Fisher information may also be written as:

[\mathcal(\theta) = - \mathrm \left[ frac ln f(X;theta) right].]

Thus Fisher information is the the negative of expectation of the second derivative of the log of f with respect to θ. Information may thus be seen to be a measure of the "sharpness" of the support curve near the maximum likelihood estimate of θ. A "blunt" support curve (one with a shallow maximum) would have low expected second derivative, and thus low information; while a sharp one would have a high expected second derivative and thus high information.

Information is additive, in that the information yielded by two independent experiments is the sum of the information from each experiment separately:

[ \mathcal_(\theta) = \mathcal_X(\theta) + \mathcal_Y(\theta). ]

This result follows from the elementary fact that if random variables are independent, the variance of their sum is the sum of their variances. Hence the information in a random sample of size n is n times that in a sample of size 1 (if observations are independent).

The information provided by a sufficient statistic is same as that of the sample X. This may be seen by using Fisher's factorization criterion for a sufficient statistic. If T(X) is sufficient for θ, then

[ f(X;\theta) = g(T(X), \theta) h(X) ]

for some functions g and h. See sufficient statistic for a more detailed explanation. The equality of information then follows from the following fact:

[ \frac \ln \left[f(X ;theta)right]= \frac \ln \left[g(T(X);theta)right] ]

which follows from the definition of Fisher information, and the independence of h(X) from θ. More generally, if T = t(X) is a statistic, then

[\mathcal_T(\theta)\leq\mathcal_X(\theta)]

with equality if and only if T is a sufficient statistic.

The Cramér-Rao inequality states that the reciprocal of the Fisher information is an asymptotic lower bound on the variance of any unbiased estimator of θ.

Single parameter Bernoulli experiment

A Bernoulli trial is a random variable with two possible outcomes, "success" and "failure", with "success" having a probability of θ. The outcome can be thought of as determined by a coin toss, with the probability of obtaining a "head" being θ and the probability of obtaining a "tail" being 1 - θ.

The Fisher information contained in n independent Bernoulli trials may be calculated as follows. In the following, A represents the number of successes, B the number of failures, and n = A + B is the total number of trials.

[\mathcal(\theta)=-\mathrm\left[ frac ln(f(A;theta))right] (1)]

:[=-\mathrm\left[ frac ln left[ theta^A(1-theta)^Bfrac right]right] (2)]

:[=-\mathrm\left[ frac left[ A ln (theta) + B ln(1-theta) right]right] (3)]

:[=-\mathrm\left[ frac left[ frac - frac right]right]] (on differentiating ln x, see logarithm) (4)

:[=+\mathrm\left[ frac + fracright] (5)]

:[=\frac + \frac] (as the expected value of A = nθ, etc.) (6)

:[= \frac (7)]

(1) defines Fisher information. (2) invokes the fact that the information in a sufficient statistic is the same as that of the sample itself. (3) expands the log term and drops a constant. (4) and (5) differentiate with respect to θ. (6) replaces A and B with their expectations. (7) is algebra.

The end result, namely,

[\mathcal(\theta) = \frac,]

is the reciprocal of the variance of the mean number of successes in n Bernoulli trials, as expected (see last sentence of the preceding section).

Matrix form

When there are N parameters, so that θ is a Nx1 vector [\theta = \begin \theta_, \theta_, \cdots , \theta_ \end,], then the Fisher information takes the form of an NxN matrix, the Fisher information matrix (FIM), with typical element:

[ \left(\theta \right) \right)}_=\mathrm\left[ frac ln f(X;theta) frac ln f(X;theta)right].]

The FIM is a NxN positive definite symmetric matrix, defining a metric on the N-dimensional parameter space. Exploring this topic requires differential geometry.

Multivariate normal distribution

The FIM for a N-variate multivariate normal distribution has a special form. Let [\mu(\theta) = \begin \mu_(\theta), \mu_(\theta), \cdots , \mu_(\theta) \end,] and let [\Sigma(\theta)] be the covariance matrix of [\mu(\theta)]. Then the typical element [\mathcal_], 0≤m,n<N, of the FIM for [X \sim N(\mu(\theta), \Sigma(\theta))] is:

[\mathcal_=\frac\Sigma^\frac+\frac\mathrm\left( \Sigma^ \frac \Sigma^ \frac\right),]

where [(..)^\top] denotes the transpose of a vector, [\mathrm(..)] denotes the trace of a square matrix, and:

• [\frac=\begin \frac & \frac & \cdots & \frac &\end;]

• [\frac=\begin \frac} & \frac} & \cdots & \frac} \\ \\ \frac} & \frac} & \cdots & \frac} \\ \\ \vdots & \vdots & \ddots & \vdots \\ \\ \frac} & \frac} & \cdots & \frac}\end.]

See also

• Cramér-Rao inequality

Other measures employed in information theory:

• Self-information
• Kullback-Leibler divergence
• Shannon entropy

 

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