Confidence intervals

The basic function to construct confidence intervals is the following

StatsBase.confint — Method

StatsBase.confint(target::EBayesTarget, estimator, Zs; 
				  level = 0.9, clip = true)

Construct confidence intervals for target based on samples Zs with the estimator estimator. The nominal coverage is set to level. If clip=true (Default) and target is bounded (say in $[0,1]$), then the confidence intervals are clipped so as to satisfy these bounds.

Some of the estimators have already been fitted on the Zs, so that the following call is also possible:

StatsBase.confint(target::EBayesTarget, estimator; 
                  level = 0.9, clip = true)

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Note that when the estimator is a SteinMinimaxEstimator, it is typically assumed that Zs is derived from data that is separate from the bata used in the construction of the quasi-minimax estimator.

Typical workflow

A typical workflow looks as follows (see data analysis tutorial for two examples):

First, set up a MinimaxCalibratorOptions instance:

mceb_options = MinimaxCalibratorOptions(prior_class = ..., marginal_grid = ..., pilot_options = ...)

Next fit mceb_options to the data and thereby construct a MinimaxCalibratorSetup object. This contains for example the fitted KDEInfinityBand.

mceb_setup = fit(mceb_options, nbhood_Zs)

Next for a given empirical Bayes target, we compute its pilot estimator, as well as the minimax linear estimator for the calibrated target:

target = PosteriorMean(StandardNormalSample(1.0))
calibrated_estimator_fit = fit(mceb_setup, target)

Finally we can compute confidence interval as

confint(target, calibrated_estimator_fit)

Reference

MinimaxCalibratedEBayes.MinimaxCalibratorOptions — Type

MinimaxCalibratorOptions(split = :random
                         prior_class::ConvexPriorClass,
                         marginal_grid,
                         infinity_band_options,
                         pilot_options,
                         tuner = HalfCIWidth
                         cache_target = false)

Struct that wraps all options required to automatically create a MinimaxCalibratorSetup object. The options are as follows

split: Defaults to :random, i.e., a random 50-50 split of the data.
prior_class: A ConvexPriorClass in which the true $G$ is believed to lie.
marginal_grid: Marginal domain discretization grid.
infinity_band_options: Options for constructing the localization, cf. KDKDEInfinityBandOptions.
pilot_options: Options that describe how to derive the pilot estimators, for example ButuceaComteOptions.
tuner: Specifies how the bias-variance tradeoff (i.e., $\delta$) will be navigated, for example RMSE or HalfCIWidth.
cache_target: Defaults to false. Setting it to true will speed up computations when multiple similar targets are being estimated (e.g., PosteriorMeans along a grid).

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MinimaxCalibratedEBayes.MinimaxCalibratorSetup — Type

MinimaxCalibratorSetup(;Zs_test, 
                       Zs_train,
                       fkde_train,
                       prior_class,
                       Zs_test_discr,
                       delta_tuner,
                       pilot_method,
                       cache_target)

Description:

This creates the object that holds all preliminary calculations necessary to conduct inference for empirical Bayes estimands. It can be either constructed manually, or more typically through fit(opt::MinimaxCalibratorOptions, Zs) where Zs comprises all the data and opt is a MinimaxCalibratorOptions object. See data_analysis.html for examples of both approaches.

Arguments:

Zs_test: Samples in the second data fold.
Zs_train: Samples in the first data fold.
fkde_train: A KDEInfinityBand instance computed based on Zs_train or external data (but not Zs_test).
Zs_test_discr: A discretization of Zs_test into DiscretizedStandardNormalSamples.
delta_tuner: Specifies how the bias-variance tradeoff (i.e., $\delta$) will be navigated, for example RMSE or HalfCIWidth.
pilot_method: Method for computing the pilot estimator for the numerator and denominator of the empirical Bayes estimand. Should support the interface estimate(target::LinearEBayesTarget, pilot_method, Zs_train).
cache_target: Defaults to false. Setting it to true will speed up computations when multiple similar targets are being estimated (e.g., PosteriorMeans along a grid).

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