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← Revision 7 as of 2026-07-28 02:37:38 ⇥
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| = Bayesian Notation = | = Prior Probability Distribution = |
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| Because the Bayesian approach to probability differs in meaningful ways from classical statistics, slightly different [[Statistics/FunctionNotation|notation]] is typically used to express the even more precise intention of certain words. | A '''prior probability distribution''' is a probability distribution describing a random variable with uncertainty, which is also a random variable. |
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| == Priors == | == Description == |
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| The prior probability function of a random variable ''x'' ([[Statistics/FunctionNotation#Probability_mass_functions|PMF]] or [[Statistics/FunctionNotation#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 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 prior uncertainty term (''θ'') itself is a random variable with a PDF notated as ''π(θ)''. | 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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| 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)''. The posterior uncertainty probability function is now notated as ''π|x(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:
