# Get e-book The Estimation Of Probabilities: An Essay on Modern Bayesian Methods

Curator: David Spiegelhalter. Eugene M. Bayesian statistics is a system for describing epistemological uncertainty using the mathematical language of probability. In the 'Bayesian paradigm,' degrees of belief in states of nature are specified; these are non-negative, and the total belief in all states of nature is fixed to be one. Bayesian statistical methods start with existing 'prior' beliefs, and update these using data to give 'posterior' beliefs, which may be used as the basis for inferential decisions. In , Thomas Bayes published a paper on the problem of induction , that is, arguing from the specific to the general.

Modern 'Bayesian statistics' is still based on formulating probability distributions to express uncertainty about unknown quantities. These can be underlying parameters of a system induction or future observations prediction. While an innocuous theory, practical use of the Bayesian approach requires consideration of complex practical issues, including the source of the prior distribution, the choice of a likelihood function, computation and summary of the posterior distribution in high-dimensional problems, and making a convincing presentation of the analysis.

Bayes theorem can be thought of as way of coherently updating our uncertainty in the light of new evidence.

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The use of a probability distribution as a 'language' to express our uncertainty is not an arbitrary choice: it can in fact be determined from deeper principles of logical reasoning or rational behavior; see Jaynes or Lindley In particular, De Finetti showed that making a qualitative assumptions of exchangeability of binary observations i.

Suppose a hospital has around beds occupied each day, and we want to know the underlying risk that a patient will be infected by MRSA methicillin-resistant Staphylococcus aureus.

### You don't always want what you get.

However, other evidence about the underlying risk may exist, such as the previous year's rates or rates in similar hospitals which may be included as part of a hierarchical model see below. Figure 1 also shows a density proportional to the likelihood function, under an assumed Poisson model. Figure 1 shows that this posterior is primarily influenced by the likelihood function but is 'shrunk' towards the prior distribution to reflect that the expectation based on external evidence was of a higher rate than that actually observed.

This can be thought of as an automatic adjustment for ' Regression to the mean ', in that the prior distribution will tend to counteract chance highs or lows in the data. While progress in Objective Bayes methods has been made for simple situations, a universal theory of priors that represent zero or minimal information has been elusive. A complete alternative is the fully subjectivist position, which compels one to elicit priors on all parameters based on the personal judgement of appropriate individuals.

A pragmatic compromise recognizes that Bayesian statistical analyses must usually be justified to external bodies and therefore the prior distribution should, as far as possible, be based on convincing external evidence or at least be guaranteed to be weakly informative: of course, exactly the same holds for the choice of functional form for the sampling distribution which will also be a subject of judgement and will need to be justified. Bayesian analysis is perhaps best seen as a process for obtaining posterior distributions or predictions based on a range of assumptions about both prior distributions and likelihoods: arguing in this way, sensitivity analysis and reasoned justification for both prior and likelihood become vital.

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Sets of prior distributions can themselves share unknown parameters, forming hierarchical models. These feature strongly within applied Bayesian analysis and provide a powerful basis for pooling evidence from multiple sources in order to reach more precise conclusions. Essentially a compromise is reached between the two extremes of assuming the sources are estimating a precisely the same, or b totally unrelated, parameters. The degree of pooling is itself estimated from the data according to the similarity of the sources, but this does not avoid the need for careful judgement about whether the sources are indeed exchangeable, in the sense that we have no external reasons to believe that certain sources are systematically different from others.

One of the strengths of the Bayesian paradigm is its ease in making predictions.

For inference, a full report of the posterior distribution is the correct and final conclusion of a statistical analysis. However, this may be impractical, particularly when the posterior is high-dimensional. Instead, posterior summaries are commonly reported, for example the posterior mean and variance, or particular tail areas. If the analysis is performed with the goal of making a specific decision, measures of utility , or loss functions can be used to derive the posterior summary that is the 'best' decision, given the data.

In Decision Theory , the loss function describes how bad a particular decision would be, given a true state of nature.

Given a particular posterior, the Bayes rule is the decision which minimizes the expected loss with respect to that posterior. If a rule is admissible meaning that there is no rule with strictly greater utility, for at least some state of nature it can be shown to be a Bayes rule for some proper prior and utility function. Many intuitively-reasonable summaries of posteriors can also be motivated as Bayes rules. As noted, for example, by Schervish , quantile-based credible intervals can be justified as a Bayes rule for a bivariate decision problem, and Highest Posterior Density intervals can be justified as a Bayes rule for a set-valued decision problem.

As a specific example, suppose we had to provide a point prediction for the number of MRSA cases in the next 6 months. For every case that we over-estimate, we will lose 10 units of wasted resources, but for every case that we under-estimate we will lose 50 units through having to make emergency provision. Bayesian analysis requires evaluating expectations of functions of random quantities as a basis for inference, where these quantities may have posterior distributions which are multivariate or of complex form or often both.

This meant that for many years Bayesian statistics was essentially restricted to conjugate analysis, where the mathematical form of the prior and likelihood are jointly chosen to ensure that the posterior may be evaluated with ease. Numerical integration methods based on analytic approximations or quadrature were developed in 70s and 80s with some success, but a revolutionary change occurred in the early s with the adoption of indirect methods, notably Monte Carlo Markov Chain. Realizations from the posterior used in Monte Carlo methods need not be independent, or generated directly.

If the conditional distribution of each parameter is known conditional on all other parameters , one simple way to generate a possibly-dependent sample of data points is via Gibbs Sampling. This algorithm generates one parameter at a time; as it sequentially updates each parameter, the entire parameter space is explored. It is appropriate to start from multiple starting points in order to check convergence, and in the long-run, the 'chains' of realizations produced will reflect the posterior of interest.

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More general versions of the same argument include the Metropolis-Hastings algorithm ; developing practical algorithms to approximate posterior distributions for complex problems remains an active area of research. Explicitly Bayesian statistical methods tend to be used in three main situations.

The first is where one has no alternative but to include quantitative prior judgments, due to lack of data on some aspect of a model, or because the inadequacies of some evidence has to be acknowledged through making assumptions about the biases involved. These situations can occur when a policy decision must be made on the basis of a combination of imperfect evidence from multiple sources, an example being the encouragement of Bayesian methods by the Food and Drug Administration FDA division responsible for medical devices.

The second situation is with moderate-size problems with multiple sources of evidence, where hierarchical models can be constructed on the assumption of shared prior distributions whose parameters can be estimated from the data. Common application areas include meta-analysis, disease mapping, multi-centre studies, and so on.

Srinivasan: State estimation by orthogonal expansion of probability distributions. IEEE Trans. Control AC , Bucy, K. Senne: Digital synthesis of non-linear filters. Automatica 7 , Sorenson, D. Alspach: Recursive Bayesian estimation using Gaussian sums. Alspach: Gaussian sum approximations in nonlinear filtering and control. In: Esti- mation Theory D.

Lainiotis, ed. Center: Practical nonlinear filtering of discrete observations by generalized least- squares approximation of the conditional probability distribution.

In: Proc. Jan: Spline filters. Lainiotis, J. Deshpande: Parameter estimation using splines. In: Estimation Theory D.

## The Estimation Of Probabilities

Sorenson: On the development of practical nonlinear filters. Wang, R. Klein: Implementation of nonlinear estimators using monospline. Lindley: Approximate Bayesian methods. In: Bayesian Statistics J. DeGroot, D. V, Lindley and A. Smith, eds.