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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. <<TableOfContents>> |
Bayesian probabilities use a few non-standard notations. |
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| == Random Variables == Bayesians strictly notate random variables using capital letters, i.e. ''X''. |
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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 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)''. |
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| The prior uncertainty term (''θ'') itself is a random variable with a PDF notated as ''π(θ)''. | ''θ'' itself is a random variable, with a p.f. notated as ''π(θ)''. |
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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 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: |
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| Uncertainty is updated given ''x''; this is notated with the probability function ''p(θ|x)''. | 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)''. |
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| The posterior uncertainty probability function is now notated as ''π|x(θ)''. | The posterior uncertainty probability function is now notated as ''π|X(θ)'' (or ''π|D(θ)''). |
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 pθ(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) = 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:
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)).