Jeffreys prior explained

In Bayesian statistics, the Jeffreys prior is a non-informative prior distribution for a parameter space. Named after Sir Harold Jeffreys,^[1] its density function is proportional to the square root of the determinant of the Fisher information matrix:

$$p\backslash left(\backslash theta\; \backslash right)\; \backslash propto\; \backslash left|\; I\; (\backslash theta)\; \backslash right|^\; .\backslash ,$$

It has the key feature that it is invariant under a change of coordinates for the parameter vector $\backslash theta$. That is, the relative probability assigned to a volume of a probability space using a Jeffreys prior will be the same regardless of the parameterization used to define the Jeffreys prior. This makes it of special interest for use with scale parameters.^[2] As a concrete example, a Bernoulli distribution can be parametrized by the probability of occurrence, or by the odds ratio. A naive uniform prior in this case is not invariant to this reparametrization, but the Jeffreys prior is.

In maximum likelihood estimation of exponential family models, penalty terms based on the Jeffreys prior were shown to reduce asymptotic bias in point estimates.^{[3] [4]}

Reparameterization

One-parameter case

If $\backslash theta$ and $\backslash varphi$ are two possible parametrizations of a statistical model, and $\backslash theta$ is a continuously differentiable function of $\backslash varphi$, we say that the prior $p\_\backslash theta(\backslash theta)$ is "invariant" under a reparametrization if$$p\_\backslash varphi(\backslash varphi)\; =\; p\_\backslash theta(\backslash theta)\; \backslash left|\backslash frac\backslash right|,$$that is, if the priors $p\_\backslash theta(\backslash theta)$ and $p\_\backslash varphi(\backslash varphi)$ are related by the usual change of variables theorem.

Since the Fisher information transforms under reparametrization as $$I\_\backslash varphi(\backslash varphi)\; =\; I\_\backslash theta(\backslash theta)\; \backslash left(\backslash frac\; \backslash right)^2,$$defining the priors as $p\_\backslash varphi(\backslash varphi)\; \backslash propto\; \backslash sqrt$ and $p\_\backslash theta(\backslash theta)\; \backslash propto\; \backslash sqrt$ gives us the desired "invariance".^[5]

Multiple-parameter case

Analogous to the one-parameter case, let $\backslash vec\backslash theta$ and $\backslash vec\backslash varphi$ be two possible parametrizations of a statistical model, with $\backslash vec\backslash theta$ a continuously differentiable function of $\backslash vec\backslash varphi$. We call the prior $p\_\backslash theta(\backslash vec\backslash theta)$ "invariant" under reparametrization if$$p\_\backslash varphi(\backslash vec\backslash varphi)\; =\; p\_\backslash theta(\backslash vec\backslash theta)\; ~|\backslash det\; J|\backslash ,,$$where $J$ is the Jacobian matrix with entries$$J\_\; =\; \backslash frac\; .$$Since the Fisher information matrix transforms under reparametrization as $$I\_\backslash varphi(\backslash vec\backslash varphi)\; =\; J^T\; I\_\backslash theta(\backslash vec\backslash theta)\; J,$$we have that$$\backslash det\; I\_\backslash varphi(\backslash varphi)\; =\; \backslash det\; I\_\backslash theta(\backslash theta)\; (\backslash det\; J)^2$$and thus defining the priors as $p\_\backslash varphi(\backslash vec\backslash varphi)\; \backslash propto\; \backslash sqrt$ and $p\_\backslash theta(\backslash vec\backslash theta)\; \backslash propto\; \backslash sqrt$ gives us the desired "invariance".

Attributes

From a practical and mathematical standpoint, a valid reason to use this non-informative prior instead of others, like the ones obtained through a limit in conjugate families of distributions, is that the relative probability of a volume of the probability space is not dependent upon the set of parameter variables that is chosen to describe parameter space.

Sometimes the Jeffreys prior cannot be normalized, and is thus an improper prior. For example, the Jeffreys prior for the distribution mean is uniform over the entire real line in the case of a Gaussian distribution of known variance.

Use of the Jeffreys prior violates the strong version of the likelihood principle, which is accepted by many, but by no means all, statisticians. When using the Jeffreys prior, inferences about $\backslash vec\backslash theta$ depend not just on the probability of the observed data as a function of $\backslash vec\backslash theta$, but also on the universe of all possible experimental outcomes, as determined by the experimental design, because the Fisher information is computed from an expectation over the chosen universe. Accordingly, the Jeffreys prior, and hence the inferences made using it, may be different for two experiments involving the same $\backslash vec\backslash theta$ parameter even when the likelihood functions for the two experiments are the same—a violation of the strong likelihood principle.

Minimum description length

In the minimum description length approach to statistics the goal is to describe data as compactly as possible where the length of a description is measured in bits of the code used. For a parametric family of distributions one compares a code with the best code based on one of the distributions in the parameterized family. The main result is that in exponential families, asymptotically for large sample size, the code based on the distribution that is a mixture of the elements in the exponential family with the Jeffreys prior is optimal. This result holds if one restricts the parameter set to a compact subset in the interior of the full parameter space. If the full parameter is used a modified version of the result should be used.

Examples

The Jeffreys prior for a parameter (or a set of parameters) depends upon the statistical model.

Gaussian distribution with mean parameter

For the Gaussian distribution of the real value $x$$$f(x\backslash mid\backslash mu)\; =\; \backslash frac$$with $\backslash sigma$ fixed, the Jeffreys prior for the mean $\backslash mu$ isThat is, the Jeffreys prior for $\backslash mu$ does not depend upon $\backslash mu$; it is the unnormalized uniform distribution on the real line — the distribution that is 1 (or some other fixed constant) for all points. This is an improper prior, and is, up to the choice of constant, the unique translation-invariant distribution on the reals (the Haar measure with respect to addition of reals), corresponding to the mean being a measure of location and translation-invariance corresponding to no information about location.

Gaussian distribution with standard deviation parameter

For the Gaussian distribution of the real value $x$$$f(x\backslash mid\backslash sigma)\; =\; \backslash frac,$$with $\backslash mu$ fixed, the Jeffreys prior for the standard deviation $\backslash sigma\; >\; 0$ isEquivalently, the Jeffreys prior for $\backslash log\; \backslash sigma\; =\; \backslash int\; d\backslash sigma/\backslash sigma$ is the unnormalized uniform distribution on the real line, and thus this distribution is also known as the . Similarly, the Jeffreys prior for $\backslash log\; \backslash sigma^2\; =\; 2\; \backslash log\; \backslash sigma$ is also uniform. It is the unique (up to a multiple) prior (on the positive reals) that is scale-invariant (the Haar measure with respect to multiplication of positive reals), corresponding to the standard deviation being a measure of scale and scale-invariance corresponding to no information about scale. As with the uniform distribution on the reals, it is an improper prior.

Poisson distribution with rate parameter

For the Poisson distribution of the non-negative integer $n$,$$f(n\; \backslash mid\; \backslash lambda)\; =\; e^\backslash frac,$$the Jeffreys prior for the rate parameter $\backslash lambda\; \backslash ge\; 0$ isEquivalently, the Jeffreys prior for $\backslash sqrt\backslash lambda\; =\; \backslash int\; d\backslash lambda/\backslash sqrt\backslash lambda$ is the unnormalized uniform distribution on the non-negative real line.

Bernoulli trial

For a coin that is "heads" with probability $\backslash gamma\; \backslash in\; [0,1]$ and is "tails" with probability $1\; -\; \backslash gamma$, for a given $(H,T)\; \backslash in\; \backslash$ the probability is $\backslash gamma^H\; (1-\backslash gamma)^T$. The Jeffreys prior for the parameter $\backslash gamma$ is

This is the arcsine distribution and is a beta distribution with $\backslash alpha\; =\; \backslash beta\; =\; 1/2$. Furthermore, if $\backslash gamma\; =\; \backslash sin^2(\backslash theta)$ then$$\backslash Pr[\backslash theta]\; =\; \backslash Pr[\backslash gamma]\; \backslash frac\; \backslash propto\; \backslash frac\; ~2\; \backslash sin\; \backslash theta\; \backslash cos\; \backslash theta\; =2\backslash ,.$$That is, the Jeffreys prior for $\backslash theta$ is uniform in the interval $[0,\; \backslash pi\; /\; 2]$. Equivalently, $\backslash theta$ is uniform on the whole circle $[0,\; 2\; \backslash pi]$.

N-sided die with biased probabilities

Similarly, for a throw of an $N$-sided die with outcome probabilities $\backslash vec\; =\; (\backslash gamma\_1,\; \backslash ldots,\; \backslash gamma\_N)$, each non-negative and satisfying $\backslash sum\_^N\; \backslash gamma\_i\; =\; 1$, the Jeffreys prior for $\backslash vec$ is the Dirichlet distribution with all (alpha) parameters set to one half. This amounts to using a pseudocount of one half for each possible outcome.

Equivalently, if we write $\backslash gamma\_i\; =\; \backslash varphi\_i^2$ for each $i$, then the Jeffreys prior for $\backslash vec$ is uniform on the $(N\; -\; 1)$-dimensional unit sphere (i.e., it is uniform on the surface of an $N$-dimensional unit ball).

Generalizations

Probability-matching prior

In 1963, Welch and Peers showed that for a scalar parameter θ the Jeffreys prior is "probability-matching" in the sense that posterior predictive probabilities agree with frequentist probabilities and credible intervals of a chosen width coincide with frequentist confidence intervals.^[6] In a follow-up, Peers showed that this was not true for the multi-parameter case,^[7] instead leading to the notion of probability-matching priors with are only implicitly defined as the probability distribution solving a certain partial differential equation involving the Fisher information.^[8]

α-parallel prior

Using tools from information geometry, the Jeffreys prior can be generalized in pursuit of obtaining priors that encode geometric information of the statistical model, so as to be invariant under a change of the coordinate of parameters.^[9] A special case, the so-called Weyl prior, is defined as a volume form on a Weyl manifold.^[10]  

Further reading

• Kass RE, Wasserman L . The Selection of Prior Distributions by Formal Rules . Journal of the American Statistical Association . 91 . 1996 . 435 . 1343–1370 . 10.1080/01621459.1996.10477003 .
• Book: Lee, Peter M. . Bayesian Statistics: An Introduction . Wiley . 4th . 2012 . 978-1-118-33257-3 . Jeffreys’ rule . 96–102 .

Notes and References

1. Jeffreys H . An invariant form for the prior probability in estimation problems . Proceedings of the Royal Society of London . Series A, Mathematical and Physical Sciences . 186 . 1007 . 453–461 . 1946 . 20998741 . 10.1098/rspa.1946.0056 . Harold Jeffreys . 97883 . 1946RSPSA.186..453J .
2. Jaynes ET . Prior probabilities. . IEEE Transactions on Systems Science and Cybernetics . September 1968 . 4 . 3 . 227–241 . 10.1109/TSSC.1968.300117 .
3. Book: Firth, David . Bias reduction, the Jeffreys prior and GLIM . Advances in GLIM and Statistical Modelling . Ludwig . Fahrmeir . Brian . Francis . Robert . Gilchrist . Gerhard . Tutz . New York . Springer . 1992 . 91–100 . 0-387-97873-9 . 10.1007/978-1-4612-2952-0_15 .
4. David . Magis . A Note on Weighted Likelihood and Jeffreys Modal Estimation of Proficiency Levels in Polytomous Item Response Models . . 80 . 200–204 . 2015 . 10.1007/s11336-013-9378-5 .
5. Robert CP, Chopin N, Rousseau J . Harold Jeffreys's Theory of Probability Revisited . Statistical Science . 2009 . 24 . 2 . 10.1214/09-STS284. free . 0804.3173 .
6. On Formulae for Confidence Points Based on Integrals of Weighted Likelihoods . B. L. . Welch . H. W. . Peers . Journal of the Royal Statistical Society . Series B (Methodological) . 25 . 2 . 1963 . 318–329 . 10.1111/j.2517-6161.1963.tb00512.x .
7. On Confidence Points and Bayesian Probability Points in the Case of Several Parameters . H. W. . Peers . Journal of the Royal Statistical Society . Series B (Methodological) . 27 . 1 . 1965 . 9–16 . 10.1111/j.2517-6161.1965.tb00581.x .
8. Probability matching priors: a review . Catia . Scricciolo . Journal of the Italian Statistical Society . 8 . 83 . 1999 . 10.1007/BF03178943 .
9. J. . Takeuchi . S. . Amari . α-parallel prior and its properties . IEEE Transactions on Information Theory . 51 . 3 . 2005 . 1011–1023 . 10.1109/TIT.2004.842703 .
10. Ruichao . Jiang . Javad . Tavakoli . Yiqiang . Zhao . Weyl Prior and Bayesian Statistics . 2020 . Entropy . 22 . 4 . 467 . 10.3390/e22040467 . free . 7516948 .