The term " Bayesian " refers to the 18th century mathematician and theologian Thomas Bayes, who provided the first mathematical treatment of a non-trivial problem of Bayesian inference.

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 ).

The use of Bayesian probabilities as the basis of Bayesian inference has been supported by several arguments, such as the Cox axioms, the Dutch book argument, arguments based on decision theory and de Finetti's theorem.

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

Experiments and computational models in Multimodal integration have shown that sensory input from different senses is integrated in a statistically optimal way, in addition, it appears that the kind of inferences used to infer single sources for multiple sensory inputs uses a Bayesian inference about the causal origin of the sensory stimuli.

As such, it appears neurobiologically plausible that the brain implements decision-making procedures that are close to optimal for Bayesian inference.

* Bayesian inference

In fact, Bayesian inference can be used to show that when the long-run proportion of different outcomes are unknown but exchangeable ( meaning that the random process from which they are generated may be biased but is equally likely to be biased in any direction ) previous observations demonstrate the likely direction of the bias, such that the outcome which has occurred the most in the observed data is the most likely to occur again.

The Inverse-Wishart distribution is important in Bayesian inference, for example in Bayesian multivariate linear regression.

Those who promote Bayesian inference view " frequentist statistics " as an approach to statistical inference that recognises only physical probabilities.

The most common ones are parsimony, maximum likelihood, and MCMC-based Bayesian inference.

However, certain phenetic methods, such as neighbor-joining, have found their way into cladistics, as a reasonable approximation of phylogeny when more advanced methods ( such as Bayesian inference ) are too computationally expensive.

* Bayesian inference

model selection, test set, minimum description length, Bayesian inference, etc.

" Bayesian approaches which make use of Carnap's theory of inductive inference include Humburg, Maher, and Fitelson et al.

* Bayesian inference

* Bayesian inference

* Bayesian inference in phylogeny

Each of these methods has been useful in Bayesian practice.

The fact that Bayesian and frequentist arguments differ on the subject of optional stopping has a major impact on the way that clinical trial data can be analysed.

The complexity penalty has a Bayesian interpretation as the negative log prior probability of,, in which case is the posterior probabability of.

The most important distinction between the frequentist and Bayesian paradigms, is that frequentist makes strong distinctions between probability, statistics, and decision-making, whereas Bayesians unify decision-making, statistics and probability under a single philosophically and mathematically consistent framework, unlike the frequentist paradigm which has been proven to be inconsistent, especially for real-world situations where experiments ( or " random events ") can not be repeated more than once.

Recent research has shown that Bayesian methods that involve a Poisson likelihood function and an appropriate prior ( e. g., a smoothing prior leading to total variation regularization or a Laplacian prior leading to-based regularization in a wavelet or other domain ) may yield superior performance to expectation-maximization-based methods which involve a Poisson likelihood function but do not involve such a prior.

In the Bayesian interpretation, Bayes ' theorem is fundamental to Bayesian statistics, and has applications in fields including science, engineering, economics ( particularly microeconomics ), game theory, medicine and law.

Bayesian inference has applications in artificial intelligence and expert systems.

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.

As applied to statistical classification, Bayesian inference has been used in recent years to develop algorithms for identifying e-mail spam.

Eric R. Bittner's group at the University of Houston has advanced a statistical variant of this approach that uses Bayesian sampling technique to sample the quantum density and compute the quantum potential on a structureless mesh of points.

Practical rationality has also a formal component, that reduces to Bayesian decision theory, and a material component, rooted in human nature ( lastly, in our genome ).

In 2004, analysis of the Bayesian classification problem has shown that there are some theoretical reasons for the apparently unreasonable efficacy of naive Bayes classifiers.

We can see this from the Bayesian update rule: letting U denote the unlikely outcome of the random process and M the proposition that the process has occurred many times before, we have

" Bayesian " has been used in this sense since about 1950.

:: In the above Chapter 20 covers confidence intervals, while Chapter 21 covers fiducial intervals and Bayesian intervals and has discussion comparing the three approaches.

Analysis is traditionally carried out with some form of multiple regression, but more recently the use of hierarchical Bayesian analysis has become widespread, enabling fairly robust statistical models of individual respondent decision behaviour to be developed.

The Bayesian view has a number of desirable features — one of them is that it embeds deductive ( certain ) logic as a subset ( this prompts some writers to call Bayesian probability " probability logic ", following E. T. Jaynes ).

This has led researchers such as David MacKay to view MDL as equivalent to Bayesian inference: code length of the model and code length of model and data together in MDL correspond to prior probability and marginal likelihood respectively in the Bayesian framework.

The Bayesian approach also fails to provide an answer that can be expressed as straightforward simple formulae, but modern computational methods of Bayesian analysis do allow essentially exact solutions to be found.

Although FDR's disease was earlier attributed to poliomyelitis, a 2003 peer-reviewed study found that six of eight Bayesian posterior probabilities favored a diagnosis of Guillain-Barré syndrome over poliomyelitis.

Bayesian also refers to the application of this probability theory to the functioning of the brain

The application of Bayes ' theorem to update beliefs is called Bayesian inference.

For more on the application of Bayes ' theorem under the Bayesian interpretation of probability, see Bayesian inference.

The critical point about Bayesian inference, then, is that it provides a principled way of combining new evidence with prior beliefs, through the application of Bayes ' rule.

* The scientific method is sometimes interpreted as an application of Bayesian inference.

For more detail on the application of Bayes ' rule under the Bayesian interpretation of probability, see Bayesian model selection.

He wrote extensively on statistical mechanics and on foundations of probability and statistical inference, initiating in 1957 the MaxEnt interpretation of thermodynamics, as being a particular application of more general Bayesian / information theory techniques ( although he argued this was already implicit in the works of Gibbs ).

Bayesian search theory is the application of Bayesian statistics to the search for lost objects.

It is closely related to the chi-squared distribution and its specific importance is that it arises in the application of Bayesian inference to the normal distribution, where it can be used as the prior and posterior distribution for an unknown variance.

His contributions to the theoretical development and practical use of Bayesian inference networks and language modelling for retrieval, and to their evaluation through extensive experiment and application, are particularly important.

The analysis step is an application of the Bayes theorem and the overall assimilation procedure is an example of Recursive Bayesian estimation.