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Bayesian Statistics

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Encyclopedia of Complexity and Systems Science

Definition of the Subject and Introduction

Statistics may be defined as the study of uncertainty: how to measure it, and how to make choices in the face ofit. Uncertainty is quantified via probability, of which there are two leading paradigms, frequentist (discussed in Sect. “ Comparison with the Frequentist StatisticalParadigm”) and Bayesian. In the Bayesian approach to probability the primitive constructs aretrue-false propositions A whose truth status is uncertain, and the probabilityof A is the numerical weight of evidence in favor of A, constrained toobey a set of axioms to ensure that Bayesian probabilities are coherent (internally logically consistent).

The discipline of statistics may be divided broadly into four activities: description (graphical and numericalsummaries of a data set y, without attempting to reason outward from it; this activity is almost entirelynon‐probabilistic and will not be discussed further here), inference(drawing probabilistic conclusions about...

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Abbreviations

Bayes' theorem ; prior, likelihood and posterior distributions:

Given (a) θ, something of interest which is unknown to the person making an uncertainty assessment, conveniently referred to as You, (b) y, an information source which is relevant to decreasing Your uncertainty about θ, (c) a desire to learn about θ from y in a way that is both internally and externally logically consistent, and (d) \( { \mathcal{B} } \), Your background assumptions and judgments about how the world works, as these assumptions and judgments relate to learning about θ from y, it can be shown that You are compelled in this situation to reason within the standard rules of probability as the basis of Your inferences about θ, predictions of future data \( { y^{*} } \), and decisions in the face of uncertainty (see below for contrasts between inference, prediction and decision‐making), and to quantify Your uncertainty about any unknown quantities through conditional probability distributions. When inferences about θ are the goal, Bayes' Theorem provides a means of combining all relevant information internal and external to y:

$$ p(\theta | y,\mathcal{B})=c\,p(\theta | \mathcal{B})\,l (\theta | y, \mathcal{B})\:. $$
(1)

Here, for example in the case in which θ is a real‐valued vector of length k, (a) \( { p(\theta | \mathcal{B}) } \) is Your prior distribution about θ given \( { \mathcal{B} } \) (in the form of a probability density function), which quantifies all relevant information available to You about θ external to y, (b) c is a positive normalizing constant, chosen to make the density on the left side of the equation integrate to 1, (c) \( { l(\theta | y, \mathcal{B}) } \) is Your likelihood distribution for θ given y and \( { \mathcal{B} } \), which is defined to be a density‐normalized multiple of Your sampling distribution \( { p (\cdot | \theta, \mathcal{B}) } \) for future data values \( { y^{*} } \) given θ and \( { \mathcal{B} } \), but re‐interpreted as a function of θ for fixed y, and (d) \( { p(\theta | y, \mathcal{B}) } \) is Your posterior distribution about θ given y and \( { \mathcal{B} } \), which summarizes Your current total information about θ and solves the basic inference problem.

Bayesian parametric and non‐parametric modeling:

(1) Following de Finetti [23], a Bayesian statistical model is a joint predictive distribution \( { p(y_1,\dots ,\, y_n) } \) for observable quantities y i that have not yet been observed, and about which You are therefore uncertain. When the y i are real‐valued, often You will not regard them as probabilistically independent (informally, the y i are independent if information about any of them does not help You to predict the others); but it may be possible to identify a parameter vector \( { \theta=(\theta_1,\dots ,\theta_k) } \) such that You would judge the y i conditionally independent given θ, and would therefore be willing to model them via the relation

$$ p(y_1,\dots ,y_n | \theta)=\prod_{i=1}^n p(y_i | \theta)\:. $$
(2)

When combined with a prior distribution \( { p(\theta) } \) on θ that is appropriate to the context, this is Bayesian parametric modeling, in which \( { p(y_i | \theta) } \) will often have a standard distributional form (such as binomial, Poisson or Gaussian). (2) When a (finite) parameter vector that induces conditional independence cannot be found, if You judge your uncertainty about the real‐valued y i exchangeable (see below), then a representation theorem of de Finetti [21] states informally that all internally logically consistent predictive distributions \( { p(y_1,\dots,\,y_n) } \) can be expressed in a way that is equivalent to the hierarchical model (see below)

$$ \begin{aligned} (F | \mathcal{B}) \,\,\,\,\,\, \sim & \,\,\,p(F | \mathcal{B}) \\ (y_i | F, \mathcal{B}) \,\, \stackrel{\textrm{\tiny IID}}{\sim} & \,\,\,\,\,\,\,\,F\:, \end{aligned} $$
(3)

where (a) F is the cumulative distribution function (CDF) of the underlying process \( { (y_1 , y_2 ,\,\dots) } \) from which You are willing to regard \( { p (y_1 , \dots , y_n) } \) as (in effect) like a random sample and (b) \( { p (F | \mathcal{B}) } \) is Your prior distribution on the space \( { \mathcal{F} } \) of all CDFs on the real line. This (placing probability distributions on infinite‐dimensional spaces such as \( { \mathcal{F} } \)) is Bayesian non‐parametric modeling, in which priors involving Dirichlet processes and/or Pólya trees (see Sect. “Inference: Parametric and Non‐Parametric Modeling of Count Data”) are often used.

Exchangeability :

A sequence \( { y = (y_1, \ldots,\, y_n) } \) of random variables (for \( { n \ge 1 } \)) is (finitely) exchangeable if the joint probability distribution \( { p (y_1, \dots,\, y_n) } \) of the elements of y is invariant under permutation of the indices \( { (1, \dots,\, n) } \), and a countably infinite sequence \( { (y_1, y_2, \dots) } \) is (infinitely) exchangeable if every finite subsequence is finitely exchangeable.

Hierarchical modeling :

Often Your uncertainty about something unknown to You can be seen to have a nested or hierarchical character. One class of examples arises in cluster sampling in fields such as education and medicine, in which students (level 1) are nested within classrooms (level 2) and patients (level 1) within hospitals (level 2); cluster sampling involves random samples (and therefore uncertainty) at two or more levels in such a data hierarchy (examples of this type of hierarchical modeling are given in Sect. “Strengths and Weaknesses of the Two Approaches”). Another, quite different, class of examples of Bayesian hierarchical modeling is exemplified by equation (3) above, in which is was helpful to decompose Your overall predictive uncertainty about \( { (y_1 , \dots ,\, y_n) } \) into (a) uncertainty about F and then (b) uncertainty about the y i given F (examples of this type of hierarchical modeling appear in Sect. “Inference and Prediction: Binary Outcomes with No Covariates” and “Inference: Parametric and Non‐Parametric Modeling of Count Data”).

Inference, prediction and decision‐making; samplesand populations:

Given a data source y, inference involves drawing probabilistic conclusions about the underlying process that gave rise to y, prediction involves summarizing uncertainty about future observable data values \( { y^{*} } \), and decision‐making involves looking for optimal behavioral choices in the face of uncertainty (about either the underlying process, or the future, or both). In some cases inference takes the form of reasoning backwards from a sample of data values to a population: a (larger) universe of possible data values from which You judge that the sample has been drawn in a manner that is representative (i. e., so that the sampled and unsampled values in the population are (likely to be) similar in relevant ways).

Mixture modeling:

Given y, unknown to You, and \( { \mathcal{B} } \), Your background assumptions and judgments relevant to y, You have a choice: You can either model (Your uncertainty about) y directly, through the probability distribution \( { p(y|\mathcal{B}) } \), or (if that is not feasible) You can identify a quantity x upon which You judge y to depend and model y hierarchically, in two stages: first by modeling x, through the probability distribution \( { p (x | \mathcal{B}) } \), and then by modeling y given x, through the probability distribution \( { p(y|x,\mathcal{B}) } \):

$$ p (y | \mathcal{B}) = \int_\mathcal{X} p (y | x , \mathcal{B}) \, p (x | \mathcal{B}) \, \text{d} x\:, $$
(4)

where \( { \mathcal{X} } \) is the space of possible values of x over which Your uncertainty is expressed. This is mixture modeling , a special case of hierarchical modeling (see above). In hierarchical notation (4) can be re‐expressed as

$$ y = \left\{\begin{array}{c} x \\ (y | x) \end{array}\right\}\:. $$
(5)

Examples of mixture modeling in this article include (a) equation (3) above, with F playing the role of x; (b) the basic equation governing Bayesian prediction, discussed in Sect. “The Bayesian Statistical Paradigm”; (c) Bayesian model averaging (Sect. “The Bayesian Statistical Paradigm”); (d) de Finetti's representation theorem for binary outcomes (Sect. “Inference and Prediction: Binary Outcomes with No Covariates”); (e) random‐effects parametric and non‐parametric modeling of count data (Sect. “Inference: Parametric and Non‐Parametric Modeling of Count Data”); and (f) integrated likelihoods in Bayes factors (Sect. “Decision‐Making: Variable Selection in Generalized Linear Models; Bayesian Model Selection”).

Probability – frequentist and Bayesian:

In the frequentist probability paradigm, attention is restricted to phenomena that are inherently repeatable under (essentially) identical conditions; then, for an event A of interest, \( { P_f (A) } \) is the limiting relative frequency with which A occurs in the (hypothetical) repetitions, as the number of repetitions \( { n \rightarrow \infty }\). By contrast, YourBayesian probability \( { P_B (A|\mathcal{B}) } \) is the numerical weightof evidence, given Your background information \( { \mathcal{B} } \) relevantto A, in favor of a true-false proposition A whose truth status is uncertain to You, obeying a series of reasonable axioms to ensure that Your Bayesian probabilities are internally logically consistent.

Utility:

To ensure internal logical consistency, optimal decision‐making proceeds by (a) specifying a utility function \( { U (a , \theta_0) } \) quantifying the numerical value associated with taking action a if the unknown is really θ0 and (b) maximizing expected utility, where the expectation is taken over uncertainty in θ as quantified by the posterior distribution \( { p (\theta | y , \mathcal{B}) } \).

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Authors and Affiliations

  1. Department of Applied Mathematics and Statistics, Baskin School of Engineering, University of California, Santa Cruz, USA

    David Draper

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  1. David Draper
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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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© 2009 Springer-Verlag

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Draper, D. (2009). Bayesian Statistics. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_31

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