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Bayesian Inference

Bayesian inference is a process which combines both common-sense and observed evidence in assigning a probability to a statement or belief, even in the absence of a random process. Bayesian inference is one of the two broad categories of statistical inference, along with frequentist inference. However, these two categories are largely different in terms of the fundamental aspects of probability.

Bayesian inference is linked to the concepts of Bayes’ Theorem and Bayesian probability. Bayesian probability allows for uncertainty to be modelled in terms of the outcomes of interest and involves a joining of observable evidence with common-sense knowledge. By combining these two aspects, Bayesian probability helps eliminate some of the complexity associated with the model, such as variables with a meaningless relationship towards the system or unknown relationship which is irrelevant1.

Bayes’ Theorem presents the relationship between two conditional probabilities and is the foundation of Bayesian inference2

Figure 1. This expression illustrates the conditional probability of A given B. The conditional probability in this case, is also referred to as the “posterior probability”. This is because the conditional probability of event A is being given after a new observation of event B. The prior probabilities of both A and B are being considered along with the conditional probability of B given A. 

The concepts: credible interval, prior distribution, posterior distribution, Bayes factors and the maximum a posterior estimator, are all related to Bayesian inference. Further information on these five concepts can be found on the BrainMass website.  



1. Cran-r-project. (2014). Bayesian Inference. Retrieved from cran.r-project.org
2. Statisticat, LLC. (2014). Bayes' Theorem. Retrieved from http://www.bayesian-inference.com/bayestheorem
Title Image Credit: Wikimedia Commons

Categories within Bayesian Inference

Credible Interval

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A credible interval, also referred to as a Bayesian probability interval, represents the probability domain around the posterior moment.

Prior Distribution

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The prior distribution, which commonly is referred to as the prior, represents the probability distribution for a particular uncertain quantity before the data has been taken into account.

Posterior Distribution

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Contrary to the prior, the posterior distribution represents the distribution for the unknown quantity or parameter of interest, after the observed or necessary data has been applied.

Bayes Factor

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Bayes factor is a method of model selection utilized for trying to determine which model better fits the data of interest when multiple models are being compared by hypothesis testing.

Maximum a Posteriori Estimator

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The Maximum a Posteriori Estimator (MAP) is based from the posterior distribution and is representative of a point estimate for an unobserved quantity.