Bayesian inference

In statistics, Bayesian inference is a method of inference in which the rule of Bayes updates the probability of an event as additional evidence is learned. Bayesian updating is an important technique throughout statistics, and especially in mathematical statistics: Exhibiting a Bayesian derivation for a statistical method automatically ensures that the method is not always worse than a competing method. Bayesian updating is especially important in the dynamic analysis of a sequence of data, sequential analysis. Bayesian inference has found application in a range of fields including science, engineering, medicine and law.

In the philosophy of decision theory, Bayesian inference is closely related to discussions of subjective probability, often called "Bayesian probability". Bayesian probability provides a rational method for updating beliefs; however, non-Bayesian updating rules are compatible with rationality, as demonstrated by Richard C. Jeffrey's rule of probability kinematics, according to Ian Hacking and Bas van Fraassen.

Conditional probability and the rule of Bayes

Main article: Bayes' rule

See also: Bayesian probability

Bayesian inference derives a "posterior" probability distribution as a consequence of two antecedents, a "prior" probability and a "likelihood", probability model for the data to be observed. Bayesian inference computes the posterior probability by conditioning, according to the the rule of Bayes:

$P(H|E)={\frac {P(E|H)}{P(E)}}\cdot P(H)$

• $P(H|E)$, the posterior, is the probability in $H$ after $E$ is observed.
• $P(H)$, the prior, is the probability in $H$ before $E$ is observed.
• $\textstyle {\frac {P(E|H)}{P(E)}}$ is a factor representing the impact of $E$ on the probability in $H$. The numerator is called the likelihood.

Bayesian updating: One rational method among many

Bayesian updating is conventional and computationally convenient. However, it is not the only updating rule that can be called "rational" in any serious sense.

Ian Hacking noted that traditional Dutch book arguments did not specify Bayesian updating: they left open the possibility that non-Bayesian updating rules could avoid Dutch books. For example, Hacking writes "And neither the Dutch book argument, nor any other in the personalist arsenal of proofs of the probability axioms, entails the dynamic assumption. Not one entails Bayesianism. So the personalist requires the dynamic assumption to be Bayesian. It is true that in consistency a personalist could abandon the Bayesian model of learning from experience. Salt could lose its savour."

In fact, there are non-Bayesian updating rules that also avoid Dutch books (as discussed in the literature on "probability kinematics" following the publication of Richard C. Jeffrey's rule). The additional hypotheses sufficient to (uniquely) specify Bayesian updating are substantial, complicated, and unsatisfactory.

Inference over exclusive and exhaustive possibilities

If evidence is simultaneously used to update belief over a set of exclusive and exhaustive propositions, Bayesian inference may be thought of as acting on this belief distribution as a whole.

General formulation

Diagram illustrating event space $\Omega$ in general formulation of Bayesian inference. Although this diagram shows discrete models and events, the continuous case may be visualized similarly using probability densities.

Suppose a process is generating independent and identically distributed events $E_{n}$, but the probability distribution is unknown. Let the event space $\Omega$ represent the current state of belief for this process. Each model is represented by event $M_{m}$. The conditional probabilities $P(E_{n}|M_{m})$ are specified to define the models. $P(M_{m})$ is the degree of belief in $M_{m}$. Before the first inference step, $\{P(M_{m})\}$ is a set of initial prior probabilities. These must sum to 1, but are otherwise arbitrary.

Suppose that the process is observed to generate $\textstyle E\in \{E_{n}\}$. For each $M\in \{M_{m}\}$, the prior $P(M)$ is updated to the posterior $P(M|E)$. From Bayes' theorem:

$P(M|E)={\frac {P(E|M)}{\sum _{m}{P(E|M_{m})P(M_{m})}}}\cdot P(M)$

Upon observation of further evidence, this procedure may be repeated.

Multiple observations

For a set of independent and identically distributed observations $\mathbf {E} =\{e_{1},\dots ,e_{n}\}$, it may be shown that repeated application of the above is equivalent to

$P(M|\mathbf {E} )={\frac {P(\mathbf {E} |M)}{\sum _{m}{P(\mathbf {E} |M_{m})P(M_{m})}}}\cdot P(M)$

Where

$P(\mathbf {E} |M)=\prod _{k}{P(e_{k}|M)}.$

This may be used to optimse practical calculations.

Parametric formulation

By parametrizing the space of models, the belief in all models may be updated in a single step. The distribution of belief over the model space may then be thought of as a distribution of belief over the parameter space. The distributions in this section are expressed as continuous, represented by probability densities, as this is the usual situation. The technique is however equally applicable to discrete distributions.

Let the vector $\mathbf {\theta }$ span the parameter space. Let the initial prior distribution over $\mathbf {\theta }$ be $p(\mathbf {\theta } |\mathbf {\alpha } )$, where $\mathbf {\alpha }$ is a set of parameters to the prior itself, or hyperparameters. Let $\mathbf {E} =\{e_{1},\dots ,e_{n}\}$ be a set of independent and identically distributed event observations, where all $e_{i}$ are distributed as $p(e|\mathbf {\theta } )$ for some $\mathbf {\theta }$. Bayes' theorem is applied to find the posterior distribution over $\mathbf {\theta }$:

${\begin{aligned}p(\mathbf {\theta } |\mathbf {E} ,\mathbf {\alpha } )&={\frac {p(\mathbf {E} |\mathbf {\theta } ,\mathbf {\alpha } )}{p(\mathbf {E} |\mathbf {\alpha } )}}\cdot p(\mathbf {\theta } |\mathbf {\alpha } )\\&={\frac {p(\mathbf {E} |\mathbf {\theta } ,\mathbf {\alpha } )}{\int _{\mathbf {\theta } }p(\mathbf {E} |\mathbf {\theta } ,\mathbf {\alpha } )p(\mathbf {\theta } |\mathbf {\alpha } )\,d\mathbf {\theta } }}\cdot p(\mathbf {\theta } |\mathbf {\alpha } )\end{aligned}}$

Where

$p(\mathbf {E} |\mathbf {\theta } ,\mathbf {\alpha } )=\prod _{k}p(e_{k}|\mathbf {\theta } )$

Mathematical properties

Interpretation of factor

$\textstyle {\frac {P(E|M)}{P(E)}}>1\Rightarrow \textstyle P(E|M)>P(E)$. That is, if the model were true, the evidence would be more likely than is predicted by the current state of belief. The reverse applies for a decrease in belief. If the belief does not change, $\textstyle {\frac {P(E|M)}{P(E)}}=1\Rightarrow \textstyle P(E|M)=P(E)$. That is, the evidence is independent of the model. If the model were true, the evidence would be exactly as likely as predicted by the current state of belief.

Cromwell's rule

Main article: Cromwell's rule

If $P(M)=0$ then $P(M|E)=0$. If $P(M)=1$, then $P(M|E)=1$. This can be interpreted to mean that hard convictions are insensitive to counter-evidence.

The former can be proved by inspection of Bayes' theorem. The latter requires consideration that if for some $m$, $P(M_{m})=1$, then for all $k\neq m$, $P(M_{k})=0$. Suppose that $M\equiv M_{m}$ Then,

$P(M|E)={\frac {P(E|M)}{P(E|M)P(M)}}\cdot P(M)=1$.

Asymptotic behaviour of posterior

Consider the behaviour of a belief distribution as it is updated a large number of times with independent and identically distributed trials. For sufficiently nice prior probabilities, the Bernstein-von Mises theorem gives that in the limit of infinite trials and the posterior converges to a Gaussian distribution independent of the initial prior under some conditions firstly outlined and rigorously proven by Joseph Leo Doob in 1948, namely if the random variable in consideration has a finite probability space. The more general results were obtained later by the statistician David A. Freedman who published in two seminal research papers in 1963 and 1965 when and under what circumstances the asymptotic behaviour of posterior is guaranteed. His 1963 paper treats, like Doob (1949), the finite case and comes to a satisfactory conclusion. However, if the random variable has an infinite but countable probability space (i.e. corresponding to a die with infinite many faces) the 1965 paper demonstrates that for a dense subset of priors the Bernstein-von Mises theorem is not applicable. In this case there is almost surely no asymptotic convergence. Later in the eighties and nineties Freedman and Persi Diaconis continued to work on the case of infinite countable probability spaces. To summarise, there may be insufficient trials to suppress the effects of the initial choice, and especially for large (but finite) systems the convergence might be very slow.

Conjugate priors

Main article: Conjugate prior

In paramaterized form, the prior distribution is often assumed to come from a family of distributions called conjugate priors. The usefulness of a conjugate prior is that the corresponding posterior distribution will be in the same family, and the calculation may be expressed in closed form.

Estimates of parameters and predictions

It is often desired to make a point estimate of a parameter or variable from a posterior distribution. One method of achieving this is to take the most likely value, a maximum a posteriori estimate:

$\theta _{\text{MAP}}=\arg \max _{\theta }p(\theta |\mathbf {E} ,\alpha )$

Alternatively, the expectation over the whole distribution may be used. This may be viewed as more fully respecting the Bayesian philosophy, as any value with non-zero posterior probability is still believed to be potentially correct.

${\tilde {\theta }}=\mathbb {E} [\theta ]=\int _{\theta }\theta \,p(\theta |\mathbf {E} ,\alpha )\,d\theta$

The predictive density of a new observation $e$ is determined by

$p(e|\mathbf {E} ,\alpha )=\int _{\theta }p(e,\theta |\mathbf {E} ,\alpha )\,d\theta =\int _{\theta }p(e|\theta )p(\theta |\mathbf {E} ,\alpha )\,d\theta$

Examples

Probability of a hypothesis

Suppose there are two full bowls of cookies. Bowl #1 has 10 chocolate chip and 30 plain cookies, while bowl #2 has 20 of each. Our friend Fred picks a bowl at random, and then picks a cookie at random. We may assume there is no reason to believe Fred treats one bowl differently from another, likewise for the cookies. The cookie turns out to be a plain one. How probable is it that Fred picked it out of bowl #1?

Intuitively, it seems clear that the answer should be more than a half, since there are more plain cookies in bowl #1. The precise answer is given by Bayes' theorem. Let $H_{1}$ correspond to bowl #1, and $H_{2}$ to bowl #2. It is given that the bowls are identical from Fred's point of view, thus $P(H_{1})=P(H_{2})$, and the two must add up to 1, so both are equal to 0.5. The event $E$ is the observation of a plain cookie. From the contents of the bowls, we know that $P(E|H_{1})=30/40=0.75$ and $P(E|H_{2})=20/40=0.5$. Bayes' formula then yields

${\begin{aligned}P(H_{1}|E)&={\frac {P(E|H_{1})\,P(H_{1})}{P(E|H_{1})\,P(H_{1})\;+\;P(E|H_{2})\,P(H_{2})}}\\\\\ &={\frac {0.75\times 0.5}{0.75\times 0.5+0.5\times 0.5}}\\\\\ &=0.6\end{aligned}}$

Before we observed the cookie, the probability we assigned for Fred having chosen bowl #1 was the prior probability, $P(H_{1})$, which was 0.5. After observing the cookie, we must revise the probability to $P(H_{1}|E)$, which is 0.6.

Making a prediction

Example results for archaeology example. This simulation was generated using c=15.2.

An archaeologist is working at a site thought to be from the medieval period, between the 11th century to the 16th century. However, it is uncertain exactly when in this period the site was inhabited. Fragments of pottery are found, some of which are glazed and some of which are decorated. It is expected that if the site were inhabited during the early medieval period, then 1% of the pottery would be glazed and 50% of its area decorated, whereas if it had been inhabited in the late medieval period then 81% would be glazed and 5% of its area decorated. How confident can the archaeologist be in the date of inhabitation as fragments are unearthed?

The degree of belief in the continuous variable $C$ (century) is to be calculated, with the discrete set of events $\{GD,G{\bar {D}},{\bar {G}}D,{\bar {G}}{\bar {D}}\}$ as evidence. Assuming linear variation of glaze and decoration with time, and that these variables are independent,

$P(E=GD|C=c)=(0.01+0.16(c-11))(0.5-0.09(c-11))$

$P(E=G{\bar {D}}|C=c)=(0.01+0.16(c-11))(0.5+0.09(c-11))$

$P(E={\bar {G}}D|C=c)=(0.99-0.16(c-11))(0.5-0.09(c-11))$

$P(E={\bar {G}}{\bar {D}}|C=c)=(0.99-0.16(c-11))(0.5+0.09(c-11))$

Assume a uniform prior of $\textstyle f_{C}(c)=0.2$, and that trials are independent and identically distributed. When a new fragment of type $e$ is discovered, Bayes' theorem is applied to update the degree of belief for each $c$:

$f_{C}(c|E=e)={\frac {P(E=e|C=c)}{P(E=e)}}f_{C}(c)={\frac {P(E=e|C=c)}{\int _{11}^{16}{P(E=e|C=c)f_{C}(c)dc}}}f_{C}(c)$

A computer simulation of the changing belief as 50 fragments are unearthed is shown on the graph. In the simulation, the site was inhabited around 1520, or $c=15.2$. By calculating the area under the relevant portion of the graph for 50 trials, the archaeologist can say that there is practically no chance the site was inhabited in the 11th and 12th centuries, about 1% chance that it was inhabited during the 13th century, 63% chance during the 14th century and 36% during the 15th century. Note that the Bernstein-von Mises theorem asserts here the asymptotic convergence to the "true" distribution because the probability space corresponding to the discrete set of events $\{GD,G{\bar {D}},{\bar {G}}D,{\bar {G}}{\bar {D}}\}$ is finite (see above section on asymptotic behaviour of the posterior).

Relation to decision theory

A decision-theoretic justification of the use of Bayesian inference was given by Abraham Wald, who proved that every Bayesian procedure is admissible. Conversely, every admissible statistical procedure is either a Bayesian procedure or a limit of Bayesian procedures.

Wald's result also established the Bayesian approach as a fundamental technique in such areas of frequentist inference as point estimation, hypothesis testing, and confidence intervals. Wald characterized admissible procedures as Bayesian procedures (and limits of Bayesian procedures), making the Bayesian formalism a central technique in such areas of frequentist statistics as parameter estimation, hypothesis testing, and computing confidence intervals. For example:

• "Under some conditions, all admissible procedures are either Bayes procedures or limits of Bayes procedures (in various senses). These remarkable results, at least in their original form, are due essentially to Wald. They are useful because the property of being Bayes is easier to analyze than admissibility."

• "In decision theory, a quite general method for proving admissibility consists in exhibiting a procedure as a unique Bayes solution."

• "In the first chapters of this work, prior distributions with finite support and the corresponding Bayes procedures were used to establish some of the main theorems relating to the comparison of experiments. Bayes procedures with respect to more general prior distributions have played a very important in the development of statistics, including its asymptotic theory." "There are many problems where a glance at posterior distributions, for suitable priors, yields immediately interesting information. Also, this technique can hardly be avoided in sequential analysis."

• "A useful fact is that any Bayes decision rule obtained by taking a proper prior over the whole parameter space must be admissible"
• "An important area of investigation in the development of admissibility ideas has been that of conventional sampling-theory procedures, and many interesting results have been obtained."

Model selection

See Bayesian model selection

Applications

Computer applications

Bayesian inference has applications in artificial intelligence and expert systems. Bayesian inference techniques have been a fundamental part of computerized pattern recognition techniques since the late 1950s. There is also an ever growing connection between Bayesian methods and simulation-based Monte Carlo techniques since complex models cannot be processed in closed form by a Bayesian analysis, while a graphical model structure may allow for efficient simulation algorithms like the Gibbs sampling and other Metropolis–Hastings algorithm schemes. Recently Bayesian inference has gained popularity amongst the phylogenetics community for these reasons; a number of applications allow many demographic and evolutionary parameters to be estimated simultaneously. In the areas of population genetics and dynamical systems theory, approximate Bayesian computation (ABC) is also becoming increasingly popular.

As applied to statistical classification, Bayesian inference has been used in recent years to develop algorithms for identifying e-mail spam. Applications which make use of Bayesian inference for spam filtering include DSPAM, Bogofilter, SpamAssassin, SpamBayes, and Mozilla. Spam classification is treated in more detail in the article on the naive Bayes classifier.

Solomonoff's Inductive inference is the theory of prediction based on observations; for example, predicting the next symbol based upon a given series of symbols. The only assumption is that the environment follows some unknown but computable probability distribution. It combines two well-studied principles of inductive inference: Bayesian statistics and Occam’s Razor.

In the courtroom

Bayesian inference can be used by jurors to coherently accumulate the evidence for and against a defendant, and to see whether, in totality, it meets their personal threshold for 'beyond a reasonable doubt'. The benefit of a Bayesian approach is that it gives the juror an unbiased, rational mechanism for combining evidence. Bayes' theorem is applied successively to all evidence presented, with the posterior from one stage becoming the prior for the next. A prior probability of guilt is still required. It has been suggested that this could reasonably be the probability that a random person taken from the qualifying population is guilty. Thus, for a crime known to have been committed by an adult male living in a town containing 50,000 adult males, the appropriate initial prior might be 1/50,000.

Adding up evidence.

It may be appropriate to explain Bayes' theorem to jurors in odds form, as betting odds are more widely understood than probabilities. Alternatively, a logarithmic approach, replacing multiplication with addition, might be easier for a jury to handle.

The use of Bayes' theorem by jurors is controversial. In the United Kingdom, a defence expert witness explained Bayes' theorem to the jury in R v Adams. The jury convicted, but the case went to appeal on the basis that no means of accumulating evidence had been provided for jurors who did not wish to use Bayes' theorem. The Court of Appeal upheld the conviction, but it also gave the opinion that "To introduce Bayes' Theorem, or any similar method, into a criminal trial plunges the jury into inappropriate and unnecessary realms of theory and complexity, deflecting them from their proper task."

Gardner-Medwin argues that the criterion on which a verdict in a criminal trial should be based is not the probability of guilt, but rather the probability of the evidence, given that the defendant is innocent (akin to a frequentist p-value). He argues that if the posterior probability of guilt is to be computed by Bayes' theorem, the prior probability of guilt must be known. This will depend on the incidence of the crime, which is an unusual piece of evidence to consider in a criminal trial. Consider the following three propositions:

A The known facts and testimony could have arisen if the defendant is guilty

B The known facts and testimony could have arisen if the defendant is innocent

C The defendant is guilty.

Gardner-Medwin argues that the jury should believe both A and not-B in order to convict. A and not-B implies the truth of C, but the reverse is not true. It is possible that B and C are both true, but in this case he argues that a jury should acquit, even though they know that they will be letting some guilty people go free. See also Lindley's paradox.

Other

• The scientific method is sometimes interpreted as an application of Bayesian inference. In this view, Bayes' rule guides (or should guide) the updating of probabilities about hypotheses conditional on new observations or experiments.
• Bayesian search theory is used to search for lost objects.
• Bayesian inference in phylogeny
• Bayesian tool for methylation analysis

Bayes and Bayesian inference

The problem considered by Bayes in Proposition 9 of his essay, "An Essay towards solving a Problem in the Doctrine of Chances", is the posterior distribution for the parameter a (the success rate) of the binomial distribution.

What is "Bayesian" about Proposition 9 is that Bayes presented it as a probability for the parameter $a$. That is, not only can one compute probabilities for experimental outcomes, but also for the parameter which governs them, and the same algebra is used to make inferences of either kind. Interestingly, Bayes actually states his question in a way that might make the idea of assigning a probability distribution to a parameter palatable to a frequentist. He supposes that a billiard ball is thrown at random onto a billiard table, and that the probabilities p and q are the probabilities that subsequent billiard balls will fall above or below the first ball. By making the binomial parameter $a\,$ depend on a random event, he cleverly escapes a philosophical quagmire that was an issue he most likely was not even aware of.

History

Main article: History of statistics § Bayesian statistics

The term Bayesian refers to Thomas Bayes (1702–1761), who proved a special case of what is now called Bayes' theorem. However, it was Pierre-Simon Laplace (1749–1827) who introduced a general version of the theorem and used it to approach problems in celestial mechanics, medical statistics, reliability, and jurisprudence. Early Bayesian inference, which used uniform priors following Laplace's principle of insufficient reason, was called "inverse probability" (because it infers backwards from observations to parameters, or from effects to causes). After the 1920s, "inverse probability" was largely supplanted by a collection of methods that came to be called frequentist statistics.

In the 20th century, the ideas of Laplace were further developed in two different directions, giving rise to objective and subjective currents in Bayesian practice. In the objective or "non-informative" current, the statistical analysis depends on only the model assumed, the data analysed. and the method assigning the prior, which differs from one objective Bayesian to another objective Bayesian. In the subjective or "informative" current, the specification of the prior depends on the belief (that is, propositions on which the analysis is prepared to act), which can summarize information from experts, previous studies, etc.

In the 1980s, there was a dramatic growth in research and applications of Bayesian methods, mostly attributed to the discovery of Markov chain Monte Carlo methods, which removed many of the computational problems, and an increasing interest in nonstandard, complex applications. Despite growth of Bayesian research, most undergraduate teaching is still based on frequentist statistics. Nonetheless, Bayesian methods are widely accepted and used, such as for example in the field of machine learning.

References

• Bickel, Peter J. and Doksum, Kjell A. (2001). Mathematical Statistics, Volume 1: Basic and Selected Topics (Second (updated printing 2007) ed.). Pearson Prentice–Hall. ISBN 013850363X.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Box, G.E.P. and Tiao, G.C. (1973) Bayesian Inference in Statistical Analysis, Wiley, ISBN 0-471-57428-7
• Edwards, Ward (1968). "Conservatism in Human Information Processing". In Kleinmuntz, B (ed.). Formal Representation of Human Judgment. Wiley.
• Edwards, Ward (1982). "Conservatism in Human Information Processing (excerpted)". In Daniel Kahneman, Paul Slovic and Amos Tversky (ed.). Judgment under uncertainty: Heuristics and biases. Cambridge University Press.
• Jaynes E.T. (2003) Probability Theory: The Logic of Science, CUP. ISBN 978-0-521-59271-0 (Link to Fragmentary Edition of March 1996).
• Howson, C. and Urbach, P. (2005). Scientific Reasoning: the Bayesian Approach (3rd ed.). Open Court Publishing Company. ISBN 978-0812695786.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Phillips, L.D.; Edwards, W. (2008). "Chapter 6: Conservatism in a simple probability inference task (Journal of Experimental Psychology (1966) 72: 346-354)". In Jie W. Weiss and David J. Weiss (ed.). A Science of Decision Making:The Legacy of Ward Edwards. Oxford University Press. p. 536. ISBN 9780195322989. {{cite book}}: More than one of |author= and |last1= specified (help); Unknown parameter |month= ignored (help)

Further reading

Elementary

The following books are listed in ascending order of probabilistic sophistication:

• Colin Howson and Peter Urbach (2005). Scientific Reasoning: the Bayesian Approach (3rd ed.). Open Court Publishing Company. ISBN 978-0812695786.
• Berry, Donald A. (1996). Statistics: A Bayesian Perspective. Duxbury. ISBN 0534234763.
• Morris H. DeGroot and Mark J. Schervish (2002). Probability and Statistics (third ed.). Addison-Wesley. ISBN 9780201524888.
• Bolstad, William M. (2007) Introduction to Bayesian Statistics: Second Edition, John Wiley ISBN 0-471-27020-2
• Winkler, Robert L, Introduction to Bayesian Inference and Decision, 2nd Edition (2003) ISBN 0-9647938-4-9
• Lee, Peter M. Bayesian Statistics: An Introduction. Second Edition. (1997). ISBN 0-340-67785-6.
• Carlin, Bradley P. and Louis, Thomas A. (2008). Bayesian Methods for Data Analysis, Third Edition. Boca Raton, FL: Chapman and Hall/CRC. ISBN 1-584-88697-8.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Gelman, Andrew; Carlin, John B.; Stern, Hal S.; Rubin, Donald B. (2003). Bayesian Data Analysis, Second Edition. Boca Raton, FL: Chapman and Hall/CRC. ISBN 1-584-88388-X.

Intermediate or advanced

• Berger, James O (1985). Statistical Decision Theory and Bayesian Analysis. Springer Series in Statistics (Second ed.). Springer-Verlag. ISBN 0-387-96098-8.
• Bayesian Theory. Wiley. 1994.
• DeGroot, Morris H., Optimal Statistical Decisions. Wiley Classics Library. 2004. (Originally published (1970) by McGraw-Hill.) ISBN 0-471-68029-X.
• Jaynes, E.T. (1998) Probability Theory: The Logic of Science.
• O'Hagan, A. and Forster, J. (2003) Kendall's Advanced Theory of Statistics, Volume 2B: Bayesian Inference. Arnold, New York. ISBN 0-340-52922-9.
• Robert, Christian P (2001). The Bayesian Choice – A Decision-Theoretic Motivation (second ed.). Springer. ISBN 0387942963.
• Glenn Shafer and Pearl, Judea, eds. (1988) Probabilistic Reasoning in Intelligent Systems, San Mateo, CA: Morgan Kaufmann.

External links

• Bayesian Statistics from Scholarpedia.
• Introduction to Bayesian probability from Queen Mary University of London
• Mathematical notes on Bayesian statistics and Markov chain Monte Carlo
• Bayesian reading list, categorized and annotated by Tom Griffiths.
• Stanford Encyclopedia of Philosophy: "Inductive Logic"
• Bayesian Confirmation Theory
• What is Bayesian Learning?