Estimands

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

`EBayesTarget`

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

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Example: PosteriorMean

`PosteriorMean`

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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))
julia> postmean1(G)
0.5

julia> postmean2 = PosteriorMean(NormalSample(1.0, sqrt(3.0)))
julia> postmean2(G)
0.25000000000000006

Posterior estimands

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

`PosteriorProbability`

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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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`PosteriorVariance`

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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:

`LinearEBayesTarget`

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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:

`PriorDensity`

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PriorDensity(z::Float64) <: LinearEBayesTarget

Example call {.unnumbered}

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

Description {.unnumbered}

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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`MarginalDensity`

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MarginalDensity(Z::EBayesSample) <: LinearEBayesTarget

Example call {.unnumbered}

MarginalDensity(StandardNormalSample(2.0))

Description {.unnumbered}

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

$L(G) = \varphi \star dG(z)$

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 a denominator:

`numerator`

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numerator(x)

Numerator of the rational representation of x.

Examples {.unnumbered}

julia> numerator(2//3)
2

julia> numerator(4)
4

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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`denominator`

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denominator(x)

Denominator of the rational representation of x.

Examples {.unnumbered}

julia> denominator(2//3)
3

julia> denominator(4)
1

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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