Bayesian Notation

Bayesian probabilities use a few non-standard notations.

Random Variables

Bayesians strictly notate random variables using capital letters, i.e. X.

Priors

The prior p.f. of a random variable X is notated p(X|θ) to indicate that it reflects priors about X captured in an uncertainty term θ. This is sometimes instead notated as p_θ(X).

θ itself is a random variable, with a p.f. 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:

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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