Fisher information

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Ronald Fisher

In mathematical statistics, the Fisher information (sometimes simply called information[1]) 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).[2] 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.[3] The level of the maximum depends upon the nature of the system constraints.

Definition[edit]

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. Let f(X; θ) be the probability density function (or probability mass function) for X conditional on the value of θ. This is also the likelihood function for θ. It 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 indicate the “correct” value of θ from the data, or equivalently, that the data X provides a lot of information about the parameter θ. If the likelihood f is flat and spread-out, then it would take many, many samples like X to estimate the actual “true” value of θ that would be obtained using the entire population being sampled. This suggests studying some kind of variance with respect to θ.

Formally, 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 expected value (the first moment) of the score is 0:[4]

The variance (which equals the second raw moment) is defined to be the Fisher information:

Note that . 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.

If log f(x; θ) is twice differentiable with respect to θ, and under certain regularity conditions[4], then the Fisher information may also be written as[5]

since

and

Thus, the Fisher information may be seen as the curvature of the support curve (the graph of the log-likelihood). Near the maximum likelihood estimate, low Fisher information therefore indicates that the maximum appears "blunt", that is, the maximum is shallow and there are many nearby values with a similar log-likelihood. Conversely, high Fisher information indicates that the maximum is sharp.

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

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

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

for some functions g and h. The independence of h(X) from θ implies

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

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

Informal derivation of the Cramér–Rao bound[edit]

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, we begin by considering an unbiased estimator . Mathematically, "unbiased" means that

This expression is zero independent of θ, so its partial derivative with respect to θ must also be zero. By the product rule, this partial derivative is also equal to

For each θ, the likelihood function is a probability density function, and therefore . A basic computation implies that

Using these two facts in the above lets us write

Factoring the integrand gives

If we square the expression in the integral, the Cauchy–Schwarz inequality lets us write

The second bracketed factor is defined to be the Fisher Information, while the first bracketed factor is the expected mean-squared error of the estimator . By rearranging, the inequality tells us that

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[edit]

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.

Let X be a Bernoulli trial. The Fisher information contained in X may be calculated to be

Because Fisher information is additive, the Fisher information contained in n independent Bernoulli trials is therefore

This is the reciprocal of the variance of the mean number of successes in n Bernoulli trials, so in this case, the Cramér–Rao bound is an equality.

Matrix form[edit]

When there are N parameters, so that θ is a N × 1 vector then the Fisher information takes the form of an N × N matrix. This matrix is called the Fisher information matrix (FIM) and has typical element

The FIM is a N × N positive semidefinite symmetric matrix. If it is positive definite, then it defines a Riemannian metric on the N-dimensional parameter space. The topic information geometry uses this to connects Fisher information to differential geometry, and in that context, this metric is known as the Fisher information metric.

Under certain regularity conditions, the Fisher information matrix may also be written as

The result is interesting in several ways:

Orthogonal parameters[edit]

We say that two parameters θi and θj 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.[citation needed]

Multivariate normal distribution[edit]

The FIM for a N-variate multivariate normal distribution, has a special form. Let the K-dimensional vector of parameters be and the vector of random normal variables be . Assume that the mean values of these random variables are , and let be the covariance matrix. Then, for , the (m, n) entry of the FIM is:

where denotes the transpose of a vector, tr(·) denotes the trace of a square matrix, and:

Note that a special, but very common, case is the one where , a constant. Then

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 occurs when the mean and covariance depend on two different vector parameters, say, β and θ. This is especially popular in the analysis of spatial data, which often uses a linear model with correlated residuals. In this case,[6]

where

Properties[edit]

Reparametrization[edit]

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

where and are the Fisher information measures of η and θ, respectively.[7]

In the vector case, suppose and are k-vectors which parametrize an estimation problem, and suppose that is a continuously differentiable function of , then,[8]

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

and where is the matrix transpose of

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.[9]

In the thermodynamic context, the Fisher information matrix is directly related to the rate of change in the corresponding order parameters.[10] In particular, such relations identify second-order phase transitions via divergences of individual elements of the Fisher information matrix.

Applications[edit]

Optimal design of experiments[edit]

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 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 an 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 addition and inversion, as well as under the multiplication of positive real numbers and matrices. An exposition of matrix theory and Loewner order appears in Pukelsheim.[11]

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

Jeffreys prior in Bayesian statistics[edit]

In Bayesian statistics, the Fisher information is used to calculate the Jeffreys prior, which is a standard, non-informative prior for continuous distribution parameters.[12]

Computational neuroscience[edit]

The Fisher information has been used to find bounds on the accuracy of neural codes. In that case, X is 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.

Derivation of physical laws[edit]

Fisher information plays a central role in a controversial principle put forward by Frieden as the basis of physical laws, a claim that has been disputed.[13]

Machine learning[edit]

The Fisher information is used in machine learning techniques such as elastic weight consolidation, which reduces catastrophic forgetting in artificial neural networks.

Relation to relative entropy[edit]

Fisher information is related to relative entropy.[14] Consider a family of probability distributions where 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

while the Fisher information matrix is:

If is fixed, then the relative entropy between two distributions of the same family is minimized at . For close to , one may expand the previous expression in a series up to second order:

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[edit]

The Fisher information was discussed by several early statisticians, notably F. Y. Edgeworth.[15] For example, Savage[16] 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[17] and a number of reviews of this early work.[18][19][20]

See also[edit]

Other measures employed in information theory:

Notes[edit]

  1. ^ Lehmann & Casella, p. 115
  2. ^ Lucien Le Cam (1986) Asymptotic Methods in Statistical Decision Theory: Pages 336 and 618–621 (von Mises and Bernstein).
  3. ^ Frieden & Gatenby (2013)
  4. ^ a b Suba Rao. "Lectures on statistical inference" (PDF). 
  5. ^ Lehmann & Casella, eq. (2.5.16), Lemma 5.3, p.116.
  6. ^ Mardia, K. V.; Marshall, R. J. (1984). "Maximum likelihood estimation of models for residual covariance in spatial regression". Biometrika. 71 (1): 135–46. doi:10.1093/biomet/71.1.135. 
  7. ^ Lehmann & Casella, eq. (2.5.11).
  8. ^ Lehmann & Casella, eq. (2.6.16)
  9. ^ Janke, W.; Johnston, D. A.; Kenna, R. (2004). "Information Geometry and Phase Transitions". Physica A. 336 (1–2): 181. doi:10.1016/j.physa.2004.01.023. 
  10. ^ Prokopenko, M.; Lizier, Joseph T.; Lizier, J. T.; Obst, O.; Wang, X. R. (2011). "Relating Fisher information to order parameters". Physical Review E. 84 (4): 041116. doi:10.1103/PhysRevE.84.041116. 
  11. ^ Pukelsheim, Friedrick (1993). Optimal Design of Experiments. New York: Wiley. ISBN 0-471-61971-X. 
  12. ^ Bernardo, Jose M.; Smith, Adrian F. M. (1994). Bayesian Theory. New York: John Wiley & Sons. ISBN 0-471-92416-4. 
  13. ^ Streater, R. F. (2007). Lost Causes in and beyond Physics. Springer. p. 69. ISBN 3-540-36581-8. 
  14. ^ Gourieroux & Montfort (1995), page 87
  15. ^ Savage (1976)
  16. ^ Savage(1976), page 156
  17. ^ Edgeworth (September 1908, December 1908)
  18. ^ Pratt (1976)
  19. ^ Stigler (1978, 1986, 1999)
  20. ^ Hald (1998, 1999)

References[edit]