* Coherence ( philosophical gambling strategy ), analogous concept in Bayesian probability

Bayesian probability is one of the different interpretations of the concept of probability and belongs to the category of evidential probabilities.

The Bayesian interpretation of probability can be seen as an extension of logic that enables reasoning with propositions whose truth or falsity is uncertain.

To evaluate the probability of a hypothesis, the Bayesian probabilist specifies some prior probability, which is then updated in the light of new, relevant data.

Nevertheless, it was the French mathematician Pierre-Simon Laplace, who pioneered and popularised what is now called Bayesian probability.

Broadly speaking, there are two views on Bayesian probability that interpret the probability concept in different ways.

In the Bayesian view, a probability is assigned to a hypothesis, whereas under the frequentist view, a hypothesis is typically tested without being assigned a probability.

In Bayesian statistics, a probability can be assigned to a hypothesis that can differ from 0 or 1 if the truth value is uncertain.

Broadly speaking, there are two views on Bayesian probability that interpret the ' probability ' concept in different ways.

For objectivists, probability objectively measures the plausibility of propositions, i. e. the probability of a proposition corresponds to a reasonable belief everyone ( even a " robot ") sharing the same knowledge should share in accordance with the rules of Bayesian statistics, which can be justified by requirements of rationality and consistency.

The objective and subjective variants of Bayesian probability differ mainly in their interpretation and construction of the prior probability.

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 ).

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, which is itself regarded as Bayesian ).

Jaynes in the context of Bayesian probability

simple: Bayesian probability

Bayesian versus Frequentist interpretations of probability.

* Coherence ( philosophical gambling strategy ), a concept in Bayesian statistics

Broadly speaking, there are two views on Bayesian probability that interpret the probability concept in different ways.

The concept, as well as the term " conjugate prior ", were introduced by Howard Raiffa and Robert Schlaifer in their work on Bayesian decision theory.

To refine the equilibria generated by the Bayesian Nash solution concept or subgame perfection, one can apply the Perfect Bayesian equilibrium solution concept.

This is a form of Bayesian inference-the quantity is called the prior odds, and the posterior odds.

A Bayesian statistician often seeks the conditional probability distribution of a random quantity given the data.

Both frequentist and Bayesian statistical theory involve making a decision based on the expected value of the loss function: however this quantity is defined differently under the two paradigms.

In Bayesian statistical inference, a prior probability distribution, often called simply the prior, of an uncertain quantity p ( for example, suppose p is the proportion of voters who will vote for the politician named Smith in a future election ) is the probability distribution that would express one's uncertainty about p before the " data " ( for example, an opinion poll ) is taken into account.

Prediction intervals are used in both frequentist statistics and Bayesian statistics: a prediction interval bears the same relationship to a future observation that a frequentist confidence interval or Bayesian credible interval bears to an unobservable population parameter: prediction intervals predict the distribution of individual future points, whereas confidence intervals and credible intervals of parameters predict the distribution of estimates of the true population mean or other quantity of interest that cannot be observed.

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

However, since we know that most lossy compression techniques operate on data that will be perceived by human consumers ( listening to music, watching pictures and video ) the distortion measure should preferably be modeled on human perception and perhaps aesthetics: much like the use of probability in lossless compression, distortion measures can ultimately be identified with loss functions as used in Bayesian estimation and decision theory.

A Bayesian interpretation of the standard error is that although we do not know the " true " percentage, it is highly likely to be located within two standard errors of the estimated percentage ( 47 %).

Using this phylogenetic framework, we inferred the genus ' historical biogeography by using weighted ancestral-area analysis and dispersal-vicariance analysis in combination with a Bayesian relaxed molecular-clock approach and paleogeographical data.

In this and other cases, we can quantify a probability for our confidence in the conjecture itself and then apply a Bayesian analysis, with each experimental result shifting the probability either up or down.

As a logic of induction rather than a theory of belief, Bayesian inference does not determine which beliefs are a priori rational, but rather determines how we should rationally change the beliefs we have when presented with evidence.

We begin by committing to a prior probability for a hypothesis based on logic or previous experience, and when faced with evidence, we adjust the strength of our belief in that hypothesis in a precise manner using Bayesian logic.

From a Bayesian point of view, we would regard it as a prior distribution.

Regardless of the method used to perform the learning, SpamAssassin's Bayesian test will subsequently assign a higher score to e-mails that are similar to previously received spam ( or, more precisely, to those emails that are different from non-spam in ways similar to previously received spam e-mails ).

A Bayesian spam filter will eventually assign a higher probability based on the user's specific patterns.

By contrast, in a Bayesian approach to statistical inference, one would assign a probability distribution to p regardless of the non-existence of any such " frequency " interpretation, and one would construe the probabilities as degrees of belief that p is in any interval to which a probability is assigned.

To meet the needs of science and of human limitations, Bayesian statisticians have developed " objective " methods for specifying prior probabilities.

Indeed, methods for constructing " objective " ( alternatively, " default " or " ignorance ") priors have been developed by avowed subjective ( or " personal ") Bayesians like James Berger ( Duke University ) and José-Miguel Bernardo ( Universitat de València ), simply because such priors are needed for Bayesian practice, particularly in science.

Thus, the Bayesian statistician needs either to use informed priors ( using relevant expertise or previous data ) or to choose among the competing methods for constructing " objective " priors.

* Conjugate prior, in Bayesian statistics, a family of probability distributions that contains a prior and the posterior distributions for a particular likelihood function ( particularly for one-parameter exponential families )

Bayesian decision theory allows these failures of rationality to be described as part of a statistically optimized system for decision making.

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.

Similarly, if funding is withdrawn part way through an experiment, and the analyst must work with incomplete data, this is a possible source of bias for classical methods but not for Bayesian methods, which do not depend on the intended design of the experiment.

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

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.

Algorithms for cladograms include least squares, neighbor-joining, parsimony, maximum likelihood, and Bayesian inference.

In the Bayesian interpretation, it expresses how a subjective degree of belief should rationally change to account for evidence.

In a Bayesian inference step, the probability of evidence is constant for all models.

In statistics, Bayesian inference is a method of inference in which Bayes ' rule is used to update the probability estimate for a hypothesis 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 works as well as any competing method, for some cases.

" Bayesian probability provides a rational method for updating beliefs ; however, non-Bayesian updating rules are compatible with rationality, according to Ian Hacking and Bas van Fraassen.

Bayesian inference derives the posterior probability as a consequence of two antecedents, a prior probability and a " likelihood function " derived from a probability model for the data to be observed.