= Bayesian Notation =

Because the Bayesian approach to probability differs in meaningful ways from classical statistics, slightly different [[Statistics/ProbabilityNotation|notation]] is typically used to express the even more precise intention of certain words.

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== Priors ==

The prior probability function of a random variable ''X'' ([[Statistics/ProbabilityNotation#Probability_mass_functions|PMF]] or [[Statistics/ProbabilityNotation#Probability_density_functions|PDF]] depending on what ''X'' represents) is notated ''p(X|θ)'' to indicate that it reflects priors about ''X'' captured in an uncertainty (''θ'') term. This is sometimes instead notated as ''p,,θ,,(X)''.

The prior uncertainty term (''θ'') itself is a random variable with a PDF notated as ''π(θ)''.

The expected value for an ''X'' with an uncertainty term ''θ'' 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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== Posteriors ==

Uncertainty is updated given ''X''; this is notated with the probability function ''p(θ|X)''. Here ''X'' is the observed data, which typically is a vector, rather than a random variable. To differentiate the meaning, sometimes the function is notated ''p(θ|D)''.

The posterior uncertainty probability function is now notated as ''π|X(θ)'' (or ''π|D(θ)'').



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