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Prior Distribution

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. It is used to avoid having randomness associated with an unknown parameter before the posterior probability is created.

The uncertain quantity in the prior distribution can be either an unknown parameter or a latent variable. A latent variable is a quantity which is inferred from an observed variable, but is not necessarily observed itself.

For example, pretend there is an unknown variable “p”, which represents the proportion of days which will be above 25 degrees Celsius in the upcoming summer months. Thus the prior for p is the probability distribution for the uncertainty associated with p since the frequency is unknown for how many days will be above this temperature until the actual data is recorded.

Furthermore, a prior distribution refers to its parameters as hyperparameters. This is done so that the parameters of the prior distribution can be differentiated from those used in the model. For example, if a beta distribution is used for the prior distribution then β would be one of the hyperparameters.

Prior distributions are also used in estimating the posterior distribution. Basically, the posterior distribution is equal to the prior distribution multiplied by the likelihood function, which is then normalized, equalling the posterior distribution.

In essence, the prior distribution is utilized in Bayesian statistics to assess the uncertainty present before the actual data is collected. There are four categories of priors which are possible: informative priors, weakly informative priors, least informative priors and uninformative priors. The specific details of each category are beyond the scope of this discussion. However, in a basic sense, the category used is dependent upon the quality and specific information known about the unknown quantity of interest.