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| ## page was renamed from Statistics/BayesianNotation = Bayesian Notation = |
= Prior Probability Distribution = |
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| Bayesian probabilities use a few non-standard notations. | A '''prior probability distribution''' is a probability distribution describing a random variable with uncertainty, which is also a random variable. <<TableOfContents>> |
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| == Random Variables == | == Description == |
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| Bayesians strictly notate random variables using capital letters, i.e. ''X''. | 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''. |
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| 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. | |
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== 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: |
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: |
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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)''). |
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:
