= Prior Probability Distribution =

A '''prior probability distribution''' is a probability distribution describing a random variable with uncertainty, which is also a random variable.

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== 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) = E,,π,,[p,,Θ,,(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 ''E,,π,,[p,,Θ,,(Y|X)]'', but the latter term '''''cannot''''' be rewritten as ''E,,π,,[ p,,Θ,,(X,Y) / p,,Θ,,(Y) ]''. Instead it should be expanded like:

{{attachment:expansion.svg}}



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