Empirical Bayes Jackknife Regression

• Empirical Bayes Jackknife Regression is a framework that generates synthetic replicates via data-fission or jackknife-resampling to recast posterior mean estimation as a regression task.
• It utilizes regression models—ranging from linear to tree-based methods—to approximate the optimal posterior mean, ensuring consistency and asymptotic efficiency.
• The approach extends to high-dimensional covariance estimation, providing competitive performance compared to established shrinkage and nonparametric techniques.

Empirical Bayes Jackknife Regression is a general framework for empirical Bayes (EB) estimation in both univariate and high-dimensional settings, leveraging synthetic or pseudo-replicates to recast posterior mean estimation as a regression problem. This approach encompasses the "Aurora" methodology for the classical one-sample EB case via data-fission, as well as EB strategies for covariance matrix estimation via jackknife-resampling and regression. The methodology fundamentally circumvents the need for multiple independent data replicates or explicit nonparametric likelihood estimation by generating pseudo-replicates—through data-fission or jackknife—enabling regression-based recovery of posterior means.

1. Data-Fission and Synthetic Replicates

Data-fission is a technique to generate synthetic replicates from a single data observation in classical hierarchical models. Consider independent observations $\theta_i \sim H$, $X_i|\theta_i\sim p(\cdot|\theta_i)$ for $i=1, ..., n$, with the goal of estimating the Bayes rule $m_B(x)=E_H[\theta|X=x]$ in the absence of knowledge about $H$. If $K\geq 2$ independent replicates were available, direct regression of one on the remaining $K-1$ would recover the posterior mean (see "Aurora" of Ignatiadis–Sun). For the single-replicate regime, data-fission produces two synthetic replicates $f_\tau(X)$ and $g_\tau(X)$ constructed such that

$$E_H[g_\tau(X)\,|\,f_\tau(X)] = E_H[\theta\,|\,f_\tau(X)],$$

making regression of $g_\tau(X)$ on $f_\tau(X)$ a valid EB estimator.

Canonical examples:

• Gaussian noise: For $X|\theta \sim N(\theta, \sigma^2)$, let $Z \sim N(0, \sigma^2)$ be independent, and set

$$f_\tau(X) = X + \tau Z,\;\; g_\tau(X) = X - Z/\tau\quad(\tau > 0).$$

• Poisson noise: For $X|\theta \sim \mathrm{Poisson}(\theta)$, let $Z|X \sim \mathrm{Binomial}(X, 1-\tau)$, then

$$f_\tau(X) = Z/(1-\tau),\;\; g_\tau(X) = (X - Z)/\tau.$$

In both constructions, $f_\tau(X)$ introduces noise (holding back randomness) so $g_\tau(X)$ is conditionally unbiased for $\theta$. As $\tau\to0$, $f_\tau(X)\to X$ in distribution, yielding asymptotic matching to the Bayes estimator (Ignatiadis et al., 15 Oct 2024).

2. Regression Formulation of Empirical Bayes Estimation

Empirical Bayes Jackknife Regression reframes the estimation of $m_B(x)=E[\theta|X=x]$ as a regression task. The practitioner generates pairs $(f_i,g_i)$, $i=1,...,n$, and solves

$$\hat m = \arg\min_{m \in M} \sum_{i=1}^n (g_i - m(f_i))^2,$$

where $M$ is a class of regression functions (e.g., linear, spline, random forest). The estimator for $\theta_i$ is then $\hat\theta_i = \hat m(f_\tau(X_i))$. Repeating data-fission $B$ times and averaging the resultant $\hat\theta_i^{(b)}$ further stabilizes the estimator. This generalizes to univariate or multivariate EB by appropriate choices of regression function (Ignatiadis et al., 15 Oct 2024).

In the high-dimensional setting (notably, covariance estimation), the jackknife regression approach generates pseudo-replicates by block-wise sample splitting and leverages them to recover analogs of the oracle posterior mean via regression (Xin et al., 19 Jun 2024).

3. Algorithmic Workflow

The following table summarizes the steps in Empirical Bayes Jackknife Regression for univariate EB and covariance estimation:

Step	Aurora (Data-Fission)	Covariance (Jackknife Regression)
1	Pick $\tau$, select regression class $M$	Partition data into $M$ blocks
2	Generate $(f_i,g_i)$ pairs from $X_i$	Compute block covariances $s_{jk}^{(m)}$
3	Regress $g$ on $f$ over all $i$	Regress $s_{jk}^{(m)}$ on features from $(-m)$
4	Evaluate $\hat m(f_\tau(X_i))$ for each $i$	Average predictions over splits/samples

The regression step may utilize linear, clustered-linear, kNN, or tree regressors, with tuning of model complexity and split parameters by cross-validation. The outputs are EB posterior mean estimates ($\hat\theta_i$ or $\hat\Sigma$) (Ignatiadis et al., 15 Oct 2024, Xin et al., 19 Jun 2024).

In covariance estimation, the responses are pseudo-covariances $s_{jk}^{(m)}$, and features are the remaining block covariances, mimicking leave-one-block-out resampling. Algorithmic projections to the nearest positive-definite matrix are applied as necessary.

4. Key Theoretical Properties

Empirical Bayes Jackknife Regression admits consistency and asymptotic optimality guarantees under mild regularity. Specifically, provided the regression class is rich enough to approximate the target (posterior mean) function and the noise parameter $\tau$ (or number of jackknife blocks $M$) is tuned appropriately with increasing sample size, the procedure achieves vanishing mean squared error (MSE):

$\sup_i E[(\hat\theta_i - \theta_i)^2]\to 0$

as $n\to\infty$ (Ignatiadis et al., 15 Oct 2024).

For covariance estimation, it is proven that the jackknife regression estimator attains

$$R(\hat\Sigma, \Sigma) - \inf_{\delta\in\mathcal{D}}R(\delta, \Sigma) \to 0$$

where $\mathcal{D}$ is the class of generalized-separable rules, and $R$ is the Frobenius risk. A finite-sample error bound holds when the regression function is uniformly close to the optimal posterior mean mapping (Xin et al., 19 Jun 2024).

The bias-variance tradeoff in data-fission and block-size selection in jackknife regression must be balanced with sample size to achieve theoretical guarantees.

5. Comparative Methodological Context

Empirical Bayes Jackknife Regression relates to a spectrum of EB and shrinkage techniques:

• Data-fission generalizes data splitting and plug-in regression for single-observation EB problems.
• For covariance estimation, the jackknife regression method requires no structural assumptions (such as sparsity or low rank) and is competitive with state-of-the-art procedures, including:
  • Linear shrinkage (Ledoit–Wolf)
  • Nonlinear shrinkage (QIS)
  • Eigen-regularized estimators (NERCOME)
  • Adaptive thresholding (Cai–Liu)
  • Nonparametric $g$-modeling (MSGCor)

Empirical evaluations demonstrate superiority or parity of jackknife regression in challenging covariance scenarios (orthogonal, spiked) and across Gaussian and non-Gaussian designs, with robustness to violations of parametric assumptions (Xin et al., 19 Jun 2024).

6. Implementation, Tuning, and Empirical Results

Implementation requires choices of regression function, number of pseudo-replicate generations (fissions or splits), and tuning parameters (e.g., $\tau$, block count $M$, local model complexity $K_c$ in clustered regression, $k$ in kNN). Regularization or complexity penalties (ridge, tree-size) may be applied as appropriate.

Computational cost scales as $O(np^2)$ in covariance problems (feature construction), with regression steps determined by chosen algorithm. Typical settings require $T=5$–$10$ repetitions and $M=5$–$10$ blocks for stability and computational efficiency.

Empirical Bayes Jackknife Regression exhibits strong empirical performance in canonical simulations (e.g., Gaussian–Gaussian with closed-form Bayes rule $x/2$), as well as in applied genomics data (mouse brain RNA-seq, $p=200$ genes), outperforming alternatives in both estimation error and biologically plausible network reconstruction (Ignatiadis et al., 15 Oct 2024, Xin et al., 19 Jun 2024).

7. Numerical Example

For illustration, consider the Gaussian–Gaussian model. Let $\theta_i \sim N(0,1)$ and $X_i|\theta_i \sim N(\theta_i,1)$, $i=1,\dots,n$, with true Bayes estimator $m_B(x)=x/2$. Aurora (data-fission) is run with $B=1$, $\tau=0.5$, and linear regression:

n <- 500 tau <- 0.5 theta <- rnorm(n,0,1) X <- theta + rnorm(n,0,1) Z <- rnorm(n,0,1) f <- X + tau*Z g <- X - Z/tau fit <- lm(g ~ f) a_hat <- coef(fit)["f"] b_hat <- coef(fit)["(Intercept)"] theta_hat <- a_hat * (X + tau*Z) + b_hat mse_aurora <- mean((theta_hat - theta)^2) mse_oracle <- mean((X/2 - theta)^2)

Empirically, Aurora achieves $\text{mse}_\text{aurora}\approx0.49$ versus the oracle $\text{mse}_\text{oracle}\approx0.50$, rapidly converging to the optimal slope of $1/2$ (Ignatiadis et al., 15 Oct 2024).

• Leiner, Duan, Wasserman & Ramdas (2023), “Data fission: splitting a single data point”.
• Ignatiadis & Sun (2023), “Empirical Bayes with multiple replicates via regression (Aurora)”.
• Brown, Johnstone & MacGibbon (2013), Poisson EB via data splitting.
• Efron (2019), Empirical Bayes methods.
• “Empirical Bayes estimation via data fission” (Ignatiadis et al., 15 Oct 2024).
• “An Empirical Bayes Jackknife Regression Framework for Covariance Matrix Estimation” (Xin et al., 19 Jun 2024).