Prior Probability Distribution
A prior probability distribution is a probability distribution describing a random variable with uncertainty, which is also a random variable.
Description
The Bayesian workflow begins with an uncertainty term θ, which is a random variable distributed according to π(θ). This term captures prior beliefs about some other random variable X.
The probability distribution of X is notated p(X|θ), indicating that it is conditioned on the priors. Therefore it is a prior probability distribution.
The expected value of X is expressed as p(X) = Eπ[pΘ(X)]. Note the capitalized Θ here, which reflects the expected value of the uncertainty term θ. This embedded expectation creates subtle limitations on computation. For example, p(Y|X) is equivalent to Eπ[pΘ(Y|X)], but the latter term cannot be rewritten as Eπ[ pΘ(X,Y) / pΘ(Y) ]. Instead it should be expanded like:
