Estimands

This package defines dedicated types to describe empirical Bayes estimands (that can be used for estimation or inference).

Empirikos.EBayesTarget — Type

Abstract type that describes Empirical Bayes estimands (which we want to estimate or conduct inference for).

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Example: PosteriorMean

Empirikos.PosteriorMean — Type

PosteriorMean(Z::EBayesSample) <: AbstractPosteriorTarget

Type representing the posterior mean, i.e.,

\[E_G[\mu_i \mid Z_i = z]\]

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A target::EBayesTarget, such as PosteriorMean, may be used as a callable on distributions (priors).

julia> G = Normal()
Normal{Float64}(μ=0.0, σ=1.0)

julia> postmean1 = PosteriorMean(StandardNormalSample(1.0))
PosteriorMean{StandardNormalSample{Float64}}(Z=     1.0 | σ=1.0  )
julia> postmean1(G)
0.5

julia> postmean2 = PosteriorMean(NormalSample(1.0, sqrt(3.0)))
PosteriorMean{NormalSample{Float64,Float64}}(Z=     1.0 | σ=1.732)
julia> postmean2(G)
0.25000000000000006

Posterior estimands

In addition to PosteriorMean, other implemented posterior estimands are the following:

Empirikos.PosteriorProbability — Type

PosteriorProbability(Z::EBayesSample, s) <: AbstractPosteriorTarget

Type representing the posterior probability, i.e.,

\[\Prob_G[\mu_i \in s \mid Z_i = z]\]

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Empirikos.PosteriorVariance — Type

PosteriorVariance(Z::EBayesSample) <: AbstractPosteriorTarget

Type representing the posterior variance, i.e.,

\[V_G[\mu_i \mid Z_i = z]\]

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Linear functionals

A special case of empirical Bayes estimands are linear functionals:

Empirikos.LinearEBayesTarget — Type

LinearEBayesTarget <: EBayesTarget

Abstract type that describes Empirical Bayes estimands that are linear functionals of the prior G.

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Currently available linear functionals:

Empirikos.PriorDensity — Type

PriorDensity(z::Float64) <: LinearEBayesTarget

Example call

julia> PriorDensity(2.0)
PriorDensity{Float64}(2.0)

Description

This is the evaluation functional of the density of $G$ at z, i.e., $L(G) = G'(z) = g(z)$ or in Julia code L(G) = pdf(G, z).

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Empirikos.MarginalDensity — Type

MarginalDensity(Z::EBayesSample) <: LinearEBayesTarget

Example call

MarginalDensity(StandardNormalSample(2.0))

Description

Describes the marginal density evaluated at $Z=z$ (e.g. $Z=2$ in the example above). In the example above the sample is drawn from the hierarchical model

\[\mu \sim G, Z \sim \mathcal{N}(0,1)\]

In other words, letting $\varphi$ the Standard Normal pdf

\[L(G) = \varhi \star dG(z)\]

Note that 2.0 has to be wrapped inside StandardNormalSample(2.0) since this target depends not only on G and the location, but also on the likelihood.

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Posterior estimands such as PosteriorMean can be typically decomposed into two linear functionals, a numerator' and adenominator':

Base.numerator — Method

Base.numerator(target::AbstractPosteriorTarget)

Suppose a posterior target $\theta_G(z)$, such as the posterior mean can be written as:

\[\theta_G(z) = \frac{ a_G(z)}{f_G(z)} = \frac{ \int h(\mu)dG(\mu)}{\int p(z \mid \mu)dG(\mu)}.\]

For example, for the posterior mean $h(\mu) = \mu \cdot p(z \mid \mu)$. Then Base.numerator returns the linear functional representing $G \mapsto a_G(z)$.

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Base.denominator — Method

Base.denominator(target::AbstractPosteriorTarget)

Suppose a posterior target $\theta_G(z)$, such as the posterior mean can be written as:

\[\theta_G(z) = \frac{ a_G(z)}{f_G(z)} = \frac{ \int h(\mu)dG(\mu)}{\int p(z \mid \mu)dG(\mu)}.\]

For example, for the posterior mean $h(\mu) = \mu \cdot p(z \mid \mu)$. Then Base.denominator returns the linear functional representing $G \mapsto f_G(z)$ (i.e., typically the marginal density). Also see Base.numerator(::AbstractPosteriorTarget).

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