# EBayes.jl

A package for empirical Bayes (EB) estimation in Julia. Currently the main functionality is estimation in the Gaussian compound decision problem in which we observe:

$Z_i$ is observed, $\sigma_i^2$ is assumed to be known and $\mu_i$ is unknown. The goal is to use all $Z_i$ and potential covariates $X_i$ to design estimators $\hat{\mu}_i$ of $\mu_i$ such that $\sum_i \mathbb{E}\left[(\mu_i - \hat{\mu}_i)^2\right]$ is small.

A longer term goal of this package is to provide a unified interface for EB method development and applications.

# Getting started

## SURE EB estimator

Let us generate toy data:

```
using Random
Random.seed!(1)
n = 10000
σs = fill(1.0, n)
μs = randn(n) .* σs
Zs = μs .+ randn(n)
```

We first check the mean squared error if we try to estimate $\mu_i$ by $Z_i$:

```
using StatsBase
mean( (μs - Zs).^2 )
```

Instead let us use the Normal SURE method of Xie, Kou, and Brown (2012), which has been implemented in this package. To do this, we will first need to wrap the `Zs`

and `σs`

as `NormalSamples`

.

```
using EBayes
ss = NormalSamples(Zs, σs)
sure_fit = fit(Normal(), SURE(), ss)
sure_pred = predict(sure_fit)
mean( (μs - sure_pred).^2 )
```

0.5131564861710659

## EB with covariates: EBayesCrossFit

### Regression estimators

Now let us consider the case that we also have contextual side information about each unit $i$, i.e., we have covariates $X_i$ for each $i$ that may be informative about the $\mu_i$. The difference compared to $Z_i$ is that we do not make any probabilistic assumptions about how $X_i$ is related to $\mu_i$ (while we know that $Z_i$ is unbiased for $\mu_i$).

If the $X_i$'s capture $\mu_i$ perfectly, then a powerful machine learning method will be able to learn the relationship if we regress $Z_i \sim X_i$. For this task we can use MLJ.jl, which provides a unified framework for working with machine learning methods.

Continuing with our toy problem, let us simulate two-dimensional covariates; the first dimension is informative for $\mu_i$, while the second is not.

```
using MLJ, MLJBase
Xs1 = μs .+ randn(n)
Xs2 = randn(n)
Xs = table([Xs1 Xs2])
```

[ Info: Model metadata loaded from registry. WARNING: using MLJ.predict in module ex-run_ebcf conflicts with an existing identifier. WARNING: using MLJBase.fit in module ex-run_ebcf conflicts with an existing identifier. WARNING: using MLJBase.predict in module ex-run_ebcf conflicts with an existing identifier.

We now use MLJ to fit a model (here just ordinary least squares linear regression) and then use the model to predict the $\mu_i$'s:

```
lin_mlj = MLJ.@load LinearRegressor pkg=GLM
lin_mach = fit!(machine(lin_mlj, Xs, Zs), verbosity=0)
lin_preds = predict_mean(lin_mach)
mean((lin_preds .- μs).^2)
```

0.5103593047734509

In this example, the regression has almost the same mean squared as the SURE method.

### EBayesCrossFit

We have seen two strong baselines: The empirical Bayes (SURE) method that utilizes the information in the $Z_i$'s by accounting for their known sampling distributions and the regression (ML) method that utilizes the information in the $X_i$'s. The `EBayesCrossFit`

method (Ignatiadis and Wager, 2019) synthesizes both sources of information through the empirical Bayes principle and by simultaneously leveraging any (black-box) ML method.

Let us apply the `EBayesCrossFit`

method in conjuction with the linear regression we used previously:

```
ebcf_lin = EBayesCrossFit(lin_mlj) # EBCF with linear regression
ebcf_fit = fit(ebcf_lin, Xs, ss) # fit EBCF
ebcf_preds = predict(ebcf_fit) # gather EBCF predictions
mean((ebcf_preds .- μs).^2) # evaluate EBCF
```

0.34409618164824574

We see that indeed the `EBayesCrossFit`

method outperforms both baselines.