Fisher information

Ronald Fisher

Francis Ysidro Edgeworth

In mathematical statistics, the Fisher information (sometimes simply called information) is a way of measuring the amount of information that an observable random variable X carries about an unknown parameter θ of a distribution that models X. Formally, it is the variance of the score, or the expected value of the observed information. In Bayesian statistics, the asymptotic distribution of the posterior mode depends on the Fisher information and not on the prior (according to the Bernstein–von Mises theorem, which was anticipated by Laplace for exponential families). The role of the Fisher information in the asymptotic theory of maximum-likelihood estimation was emphasized by the statistician Ronald Fisher (following some initial results by Francis Ysidro Edgeworth). The Fisher information is also used in the calculation of the Jeffreys prior, which is used in Bayesian statistics.

The Fisher-information matrix is used to calculate the covariance matrices associated with maximum-likelihood estimates. It can also be used in the formulation of test statistics, such as the Wald test.

Statistical systems of a scientific nature (physical, biological, etc.) whose likelihood functions obey shift invariance have been shown to obey maximum Fisher information. The level of the maximum depends upon the nature of the system constraints.

Definition

The Fisher information is a way of measuring the amount of information that an observable random variable X carries about an unknown parameter θ upon which the probability of X depends. The probability function for X, which is also the likelihood function for θ, is a function f(X; θ); it is the probability mass (or probability density) of the random variable X conditional on the value of θ. The partial derivative with respect to θ of the natural logarithm of the likelihood function is called the score.

Under certain regularity conditions, it can be shown that the first moment of the score (that is, its expected value) is 0:

$\operatorname{E} \left[\left. \frac{\partial}{\partial\theta} \log f(X;\theta)\right|\theta \right] = \operatorname{E} \left[\left. \frac{\frac{\partial}{\partial\theta} f(X;\theta)}{f(X; \theta)}\right|\theta \right] = \int \frac{\frac{\partial}{\partial\theta} f(x;\theta)}{f(x; \theta)} f(x;\theta)\; \mathrm{d}x =$

$= \int \frac{\partial}{\partial\theta} f(x;\theta)\; \mathrm{d}x = \frac{\partial}{\partial\theta} \int f(x; \theta)\; \mathrm{d}x = \frac{\partial}{\partial\theta} \; 1 = 0.$

The second moment is called the Fisher information:

$\mathcal{I}(\theta)=\operatorname{E} \left[\left. \left(\frac{\partial}{\partial\theta} \log f(X;\theta)\right)^2\right|\theta \right] = \int \left(\frac{\partial}{\partial\theta} \log f(x;\theta)\right)^2 f(x; \theta)\; \mathrm{d}x\,,$

where, for any given value of θ, the expression E[...|θ] denotes the conditional expectation over values for X with respect to the probability function f(x; θ) given θ. Note that $0 \leq \mathcal{I}(\theta) < \infty$. A random variable carrying high Fisher information implies that the absolute value of the score is often high. The Fisher information is not a function of a particular observation, as the random variable X has been averaged out.

Since the expectation of the score is zero, the Fisher information is also the variance of the score.

If log f(x; θ) is twice differentiable with respect to θ, and under certain regularity conditions, then the Fisher information may also be written as

$\mathcal{I}(\theta) = - \operatorname{E} \left[\left. \frac{\partial^2}{\partial\theta^2} \log f(X;\theta)\right|\theta \right]\,,$

since

$\frac{\partial^2}{\partial\theta^2} \log f(X;\theta) = \frac{\frac{\partial^2}{\partial\theta^2} f(X;\theta)}{f(X; \theta)} \;-\; \left( \frac{\frac{\partial}{\partial\theta} f(X;\theta)}{f(X; \theta)} \right)^2 = \frac{\frac{\partial^2}{\partial\theta^2} f(X;\theta)}{f(X; \theta)} \;-\; \left( \frac{\partial}{\partial\theta} \log f(X;\theta)\right)^2$

and

$\operatorname{E} \left[\left. \frac{\frac{\partial^2}{\partial\theta^2} f(X;\theta)}{f(X; \theta)}\right|\theta \right] = \cdots = \frac{\partial^2}{\partial\theta^2} \int f(x; \theta)\; \mathrm{d}x = \frac{\partial^2}{\partial\theta^2} \; 1 = 0.$

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

$\mathcal{I}_{X,Y}(\theta) = \mathcal{I}_X(\theta) + \mathcal{I}_Y(\theta).$

This result follows from the elementary fact that if random variables are independent, the variance of their sum is the sum of their variances. In particular, the information in a random sample of size n is n times that in a sample of size 1, when observations are independent and identically distributed.

The information provided by a sufficient statistic is the same as that of the sample X. This may be seen by using Neyman'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:

$\frac{\partial}{\partial\theta} \log \left[f(X ;\theta)\right] = \frac{\partial}{\partial\theta} \log \left[g(T(X);\theta)\right]$

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

$\mathcal{I}_T(\theta) \leq \mathcal{I}_X(\theta)$

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

Informal derivation of the Cramér–Rao bound

The Cramér–Rao bound states that the inverse of the Fisher information is a lower bound on the variance of any unbiased estimator of θ. H.L. Van Trees (1968) and B. Roy Frieden (2004) provide the following method of deriving the Cramér–Rao bound, a result which describes use of the Fisher information, informally:

Consider an unbiased estimator $\hat\theta(X)$. Mathematically, we write

$\operatorname{E}\left[ \left. \hat\theta(X) - \theta \right| \theta \right] = \int \left[ \hat\theta(x) - \theta \right] \cdot f(x ;\theta) \, \mathrm{d}x = 0.$

The likelihood function f(X; θ) describes the probability that we observe a given sample x given a known value of θ. If f is sharply peaked with respect to changes in θ, it is easy to intuit the "correct" value of θ given the data, and hence the data contains a lot of information about the parameter. If the likelihood f is flat and spread-out, then it would take many, many samples of X to estimate the actual "true" value of θ. Therefore, we would intuit that the data contain much less information about the parameter.

Now, we use the product rule to differentiate the unbiased-ness condition above to get

$\frac{\partial}{\partial\theta} \int \left[ \hat\theta(x) - \theta \right] \cdot f(x ;\theta) \, \mathrm{d}x = \int \left(\hat\theta(x)-\theta\right) \frac{\partial f}{\partial\theta} \, \mathrm{d}x - \int f \, \mathrm{d}x = 0.$

We now make use of two facts. The first is that the likelihood f is just the probability of the data given the parameter. Since it is a probability, it must be normalized, implying that

$\int f \, \mathrm{d}x = 1.$

Second, we know from basic calculus that

$\frac{\partial f}{\partial\theta} = f \, \frac{\partial \log f}{\partial\theta}.$

Using these two facts in the above let us write

$\int \left(\hat\theta-\theta\right) f \, \frac{\partial \log f}{\partial\theta} \, \mathrm{d}x = 1.$

Factoring the integrand gives

$\int \left(\left(\hat\theta-\theta\right) \sqrt{f} \right) \left( \sqrt{f} \, \frac{\partial \log f}{\partial\theta} \right) \, \mathrm{d}x = 1.$

If we square the equation, the Cauchy–Schwarz inequality lets us write

$\left[ \int \left(\hat\theta - \theta\right)^2 f \, \mathrm{d}x \right] \cdot \left[ \int \left( \frac{\partial \log f}{\partial\theta} \right)^2 f \, \mathrm{d}x \right] \geq 1.$

The right-most factor is defined to be the Fisher Information

$\mathcal{I}\left(\theta\right) = \int \left( \frac{\partial \log f}{\partial\theta} \right)^2 f \, \mathrm{d}x.$

The left-most factor is the expected mean-squared error of the estimator θ^{^}, since

$\operatorname{E}\left[ \left. \left( \hat\theta\left(X\right) - \theta \right)^2 \right| \theta \right] = \int \left(\hat\theta - \theta\right)^2 f \, \mathrm{d}x.$

Notice that the inequality tells us that, fundamentally,

$\operatorname{Var}\left(\hat\theta\right) \, \geq \, \frac{1}{\mathcal{I}\left(\theta\right)}.$

In other words, the precision to which we can estimate θ is fundamentally limited by the Fisher Information of the likelihood function.

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 heads being θ and the probability of tails 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.

$\begin{align} \mathcal{I}(\theta) & = -\operatorname{E} \left[ \left. \frac{\partial^2}{\partial\theta^2} \log(f(A;\theta)) \right| \theta \right] \qquad (1) \\ & = -\operatorname{E} \left[ \left. \frac{\partial^2}{\partial\theta^2} \log \left( \theta^A(1-\theta)^B\frac{(A+B)!}{A!B!} \right) \right| \theta \right] \qquad (2) \\ & = -\operatorname{E} \left[ \left. \frac{\partial^2}{\partial\theta^2} \left( A \log (\theta) + B \log(1-\theta) \right) \right| \theta \right] \qquad (3) \\ & = -\operatorname{E} \left[ \left. \frac{\partial}{\partial\theta} \left( \frac{A}{\theta} - \frac{B}{1-\theta} \right) \right| \theta \right] \qquad (4) \\ & = +\operatorname{E} \left[ \left. \frac{A}{\theta^2} + \frac{B}{(1-\theta)^2} \right| \theta \right] \qquad (5) \\ & = \frac{n\theta}{\theta^2} + \frac{n(1-\theta)}{(1-\theta)^2} \qquad (6) \\ & \text{since the expected value of }A\text{ given }\theta\text{ is }n\theta,\text{ etc.} \\ & = \frac{n}{\theta(1-\theta)} \qquad (7) \end{align}$

(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 natural logarithm 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,

$\mathcal{I}(\theta) = \frac{n}{\theta(1-\theta)},$

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 N × 1 vector $\theta = \begin{bmatrix} \theta_{1}, \theta_{2}, \dots , \theta_{N} \end{bmatrix}^{\mathrm T},$ then the Fisher information takes the form of an N × N matrix, the Fisher Information Matrix (FIM), with typical element

${\left(\mathcal{I} \left(\theta \right) \right)}_{i, j} = \operatorname{E} \left[\left. \left(\frac{\partial}{\partial\theta_i} \log f(X;\theta)\right) \left(\frac{\partial}{\partial\theta_j} \log f(X;\theta)\right) \right|\theta\right].$

The FIM is a N × N positive semidefinite symmetric matrix, defining a Riemannian metric on the N-dimensional parameter space, thus connecting Fisher information to differential geometry. In that context, this metric is known as the Fisher information metric, and the topic is called information geometry.

Under certain regularity conditions, the Fisher Information Matrix may also be written as

${\left(\mathcal{I} \left(\theta \right) \right)}_{i, j} = - \operatorname{E} \left[\left. \frac{\partial^2}{\partial\theta_i \, \partial\theta_j} \log f(X;\theta) \right|\theta\right]\,.$

The metric is interesting in several ways; it can be derived as the Hessian of the relative entropy; it can be understood as a metric induced from the Euclidean metric, after appropriate change of variable; in its complex-valued form, it is the Fubini–Study metric.

Orthogonal parameters

We say that two parameters θ_i and θ_j are orthogonal if the element of the ith row and jth column of the Fisher information matrix is zero. Orthogonal parameters are easy to deal with in the sense that their maximum likelihood estimates are independent and can be calculated separately. When dealing with research problems, it is very common for the researcher to invest some time searching for an orthogonal parametrization of the densities involved in the problem.

Multivariate normal distribution

The FIM for a N-variate multivariate normal distribution has a special form. Let $\mu(\theta) = \begin{bmatrix} \mu_{1}(\theta), \mu_{2}(\theta), \dots , \mu_{N}(\theta) \end{bmatrix}^\mathrm{T},$ and let Σ(θ) be the covariance matrix. Then the typical element $\mathcal{I}_{m,n}$, 0 ≤ m, n < K, of the FIM for X ∼ N(μ(θ), Σ(θ)) is:

$\mathcal{I}_{m,n} = \frac{\partial \mu^\mathrm{T}}{\partial \theta_m} \Sigma^{-1} \frac{\partial \mu}{\partial \theta_n} + \frac{1}{2} \operatorname{tr} \left( \Sigma^{-1} \frac{\partial \Sigma}{\partial \theta_m} \Sigma^{-1} \frac{\partial \Sigma}{\partial \theta_n} \right),$

where $(..)^\mathrm{T}$ denotes the transpose of a vector, tr(..) denotes the trace of a square matrix, and:

• $\frac{\partial \mu}{\partial \theta_m} = \begin{bmatrix} \frac{\partial \mu_1}{\partial \theta_m} & \frac{\partial \mu_2}{\partial \theta_m} & \cdots & \frac{\partial \mu_N}{\partial \theta_m} \end{bmatrix}^\mathrm{T};$

• $\frac{\partial \Sigma}{\partial \theta_m} = \begin{bmatrix} \frac{\partial \Sigma_{1,1}}{\partial \theta_m} & \frac{\partial \Sigma_{1,2}}{\partial \theta_m} & \cdots & \frac{\partial \Sigma_{1,N}}{\partial \theta_m} \\ \\ \frac{\partial \Sigma_{2,1}}{\partial \theta_m} & \frac{\partial \Sigma_{2,2}}{\partial \theta_m} & \cdots & \frac{\partial \Sigma_{2,N}}{\partial \theta_m} \\ \\ \vdots & \vdots & \ddots & \vdots \\ \\ \frac{\partial \Sigma_{N,1}}{\partial \theta_m} & \frac{\partial \Sigma_{N,2}}{\partial \theta_m} & \cdots & \frac{\partial \Sigma_{N,N}}{\partial \theta_m} \end{bmatrix}.$

Note that a special, but very common, case is the one where Σ(θ) = Σ, a constant. Then

$\mathcal{I}_{m,n} = \frac{\partial \mu^\mathrm{T}}{\partial \theta_m} \Sigma^{-1} \frac{\partial \mu}{\partial \theta_n}.\$

In this case the Fisher information matrix may be identified with the coefficient matrix of the normal equations of least squares estimation theory.

Another special case is that the mean and covariance depends on two different vector parameters, say, β and θ. This is especially popular in the analysis of spatial data, which uses a linear model with correlated residuals. We have

$\mathcal{I}\left( \beta ,\theta \right)=\text{diag}\left( \mathcal{I}\left( \beta \right),\mathcal{I}\left( \theta \right) \right)$

where

$\mathcal{I}{{\left( \beta \right)}_{m,n}}=\frac{\partial {{\mu }^{\text{T}}}}{\partial {{\beta }_{m}}}{{\Sigma }^{-1}}\frac{\partial \mu }{\partial {{\beta }_{n}}}$,

$\mathcal{I}{{\left( \theta \right)}_{m,n}}=\frac{1}{2}\operatorname{tr}\left( {{\Sigma }^{-1}}\frac{\partial \Sigma }{\partial {{\theta }_{m}}}{{\Sigma }^{-1}}\frac{\partial \Sigma }{\partial {{\theta }_{n}}} \right)$

The proof of this special case is given in literature. Using the same technique in this paper, it's not difficult to prove the original result.

Properties

Reparametrization

The Fisher information depends on the parametrization of the problem. If θ and η are two scalar parametrizations of an estimation problem, and θ is a continuously differentiable function of η, then

${\mathcal I}_\eta(\eta) = {\mathcal I}_\theta(\theta(\eta)) \left( \frac{{\mathrm d} \theta}{{\mathrm d} \eta} \right)^2$

where ${\mathcal I}_\eta$ and ${\mathcal I}_\theta$ are the Fisher information measures of η and θ, respectively.

In the vector case, suppose ${\boldsymbol \theta}$ and ${\boldsymbol \eta}$ are k-vectors which parametrize an estimation problem, and suppose that ${\boldsymbol \theta}$ is a continuously differentiable function of ${\boldsymbol \eta}$, then,

${\mathcal I}_{\boldsymbol \eta}({\boldsymbol \eta}) = {\boldsymbol J}^{\mathrm T} {\mathcal I}_{\boldsymbol \theta} ({\boldsymbol \theta}({\boldsymbol \eta})) {\boldsymbol J}$

where the (i, j)th element of the k × k Jacobian matrix $\boldsymbol J$ is defined by

$J_{ij} = \frac{\partial \theta_i}{\partial \eta_j}\,,$

and where ${\boldsymbol J}^{\mathrm T}$ is the matrix transpose of ${\boldsymbol J}$.

In information geometry, this is seen as a change of coordinates on a Riemannian manifold, and the intrinsic properties of curvature are unchanged under different parametrization. In general, the Fisher information matrix provides a Riemannian metric (more precisely, the Fisher-Rao metric) for the manifold of thermodynamic states, and can be used as an information-geometric complexity measure for a classification of phase transitions, e.g., the scalar curvature of the thermodynamic metric tensor diverges at (and only at) a phase transition point.

In the thermodynamic context, the Fisher information matrix is directly related to the rate of change in the corresponding order parameters. In particular, such relations identify second-order phase transitions via divergences of individual elements of the Fisher information matrix.

Applications

Optimal design of experiments

Fisher information is widely used in optimal experimental design. Because of the reciprocity of estimator-variance and Fisher information, minimizing the variance corresponds to maximizing the information.

When the linear (or linearized) statistical model has several parameters, the mean of the parameter-estimator is a vector and its variance is a matrix. The inverse matrix of the variance-matrix is called the "information matrix". Because the variance of the estimator of a parameter vector is a matrix, the problem of "minimizing the variance" is complicated. Using statistical theory, statisticians compress the information-matrix using real-valued summary statistics; being real-valued functions, these "information criteria" can be maximized.

Traditionally, statisticians have evaluated estimators and designs by considering some summary statistic of the covariance matrix (of a mean–unbiased estimator), usually with positive real values (like the determinant or matrix trace). Working with positive real-numbers brings several advantages: If the estimator of a single parameter has a positive variance, then the variance and the Fisher information are both positive real numbers; hence they are members of the convex cone of nonnegative real numbers (whose nonzero members have reciprocals in this same cone). For several parameters, the covariance-matrices and information-matrices are elements of the convex cone of nonnegative-definite symmetric matrices in a partially ordered vector space, under the Loewner (Löwner) order. This cone is closed under matrix-matrix addition, under matrix-inversion, and under the multiplication of positive real-numbers and matrices. An exposition of matrix theory and the Loewner-order appears in Pukelsheim.

The traditional optimality-criteria are the information-matrix's invariants; algebraically, the traditional optimality-criteria are functionals of the eigenvalues of the (Fisher) information matrix: see optimal design.

Jeffreys prior in Bayesian statistics

In Bayesian statistics, the Fisher information is used to calculate the Jeffreys prior, which is a standard, non-informative prior for continuous distribution parameters.

Computational neuroscience

The Fisher information has been used to find bounds on the accuracy of neural codes. In that case X are typically the joint responses of many neurons representing a low dimensional variable θ (such as a stimulus parameter). In particular the role of correlations in the noise of the neural responses has been studied.

Relation to relative entropy

See also: Fisher information metric

Fisher information is related to relative entropy. Consider a family of probability distributions $f(x; \theta)$ where $\theta$ is a parameter which lies in a range of values. Then the relative entropy, or Kullback–Leibler divergence, between two distributions in the family can be written as

$D(\theta'||\theta) = \int f(x; \theta')\log\frac{f(x;\theta')}{f(x; \theta)} \mathrm{d}x$

And the Fisher information is:

$\mathcal{I}(\theta) = \left(\frac{\mathrm{d}^2}{\mathrm{d}\theta'_i \mathrm{d}\theta'_j}D(\theta'||\theta)\right)_{\theta=\theta'}$

If we consider $\theta$ fixed, the relative entropy between two distributions of the same family is minimized at $\theta'=\theta$. For $\theta'$ close to $\theta$ one may expand the previous expression in a series up to second order:

$D(\theta'||\theta) = \frac{1}{2}(\theta'-\theta)^\top\underbrace{\left(\frac{\mathrm{d}^2}{\mathrm{d}\theta'_i \mathrm{d}\theta'_j}D(\theta'||\theta)\right)_{\theta=\theta'}}_{\text{Fisher info.}}(\theta'-\theta)+\cdots$

Thus the Fisher information represents the curvature of the relative entropy.

Schervish (1995: §2.3) says the following.

One advantage Kullback-Leibler information has over Fisher information is that it is not affected by changes in parameterization. Another advantage is that Kullback-Leibler information can be used even if the distributions under consideration are not all members of a parametric family.

...
Another advantage to Kullback-Leibler information is that no smoothness conditions on the densities … are needed.

History

The Fisher information was discussed by several early statisticians, notably F. Y. Edgeworth. For example, Savage says: "In it [Fisher information], he [Fisher] was to some extent anticipated (Edgeworth 1908–9 esp. 502, 507–8, 662, 677–8, 82–5 and references he [Edgeworth] cites including Pearson and Filon 1898 [. . .])." There are a number of early historical sources and a number of reviews of this early work.

See also

• Observed information
• Fisher information metric
• Formation matrix
• Information geometry
• Jeffreys prior
• Cramér–Rao bound

Other measures employed in information theory:

• Entropy (information theory)
• Kullback–Leibler divergence
• Self-information

References

• Frieden, B. R. (2004) Science from Fisher Information: A Unification. Cambridge Univ. Press. ISBN 0-521-00911-1.