Posterior Distribution

In Bayesian analysis, the posterior distribution, or posterior, is the distribution of a set of unknown parameters, latent variables, or otherwise missing variables of interest, conditional on the current data. The posterior distribution uses the current data to update previous knowledge, called a prior, about that parameter. A posterior distribution, p(θ|x), is derived using Bayes’s theorem

$$p\left(\theta |x\right)=\frac{p\left(x|\theta \right)p\left(\theta \right)}{p\left(x\right)}=\frac{p\left(x|\theta \right)p\left(\theta \right)}{\int p\left(x|\theta \right)p\left(\theta \right)d\theta },$$

where θ is the unknown parameter(s) and x is the current data. The probability of the data given the parameter p(x|θ) is the likelihood L(θ|x). The prior distribution, p(θ), is user specified to represent prior knowledge about the unknown parameter(s). The last piece of Bayes’s theorem, the marginal [Page 1274]distribution of data, p(x), is computed using the likelihood and the prior. The distribution of the posterior is determined by the ...