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"How much evidence is enough?." Likely qualification to a claim is, depending on the evidence for or against it. The technique is called Bayesian reasoning It is in fact a quantitative way of calculating what most people do informally when deciding whether or not something is likely, by adding up the evidence for or against a particular claim. Scientific inquiry is an iterative process of integrating accumulating information. Investigators assess the current state of knowledge regarding the issue of interest, gather new data to address remaining questions, and then update and refine their understanding to incorporate both new and old data. Bayesian inference provides a logical, quantitative framework for this process. Scientific hypotheses typically are expressed through probability distributions for observable scientific data. These probability distributions depend on unknown quantities called parameters. In the Bayesian paradigm, current knowledge about the model parameters is expressed by placing a probability distribution on the parameters, called the "prior distribution", often written as
When new data y become available, the information they contain regarding the model parameters is expressed in the "likelihood," which is proportional to the distribution of the observed data given the model parameters, written as
This information is then combined with the prior to produce an updated probability distribution called the "posterior distribution," on which all Bayesian inference is based. Bayes' Theorem, an elementary identity in probability theory, states how the update is done mathematically: the posterior is proportional to the prior times the likelihood, or more precisely,
In theory, the posterior distribution is always available, but in realistically complex models, the required analytic computations often are intractable. Over several years, in the late 1980s and early 1990s, it was realized that methods for drawing samples from the posterior distribution could be very widely applicable. The probability that the hypothesis (Hi) is
true, given both the newly observed data (D) and pre-existing
information (I) The probability that the hypothesis is true given only the
previously known information The probability of observing such a particular outcome
(D) assuming our previous knowledge (I) and also
assuming that the hypothesis is true (Hi) The probability that such a piece of data (D) could be expected to be observed given ONLY the previously existing information (I) BibliographyInternational Society for Bayesian Analysis (ISBA)
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