Posterior Risk Calculation Method

• Posterior risk calculation methods use Bayesian updates to minimize expected loss, with applications in high-dimensional sparse estimation.
• The horseshoe estimator, with theoretically optimal or empirically estimated τ, contracts to the true parameter at the minimax rate balancing bias and variance.
• Empirical Bayes approaches adaptively tune τ for accurate uncertainty quantification and improved credible intervals compared to one-component priors.

A posterior risk calculation method refers to the set of probabilistic, algorithmic, or decision-theoretic procedures that leverage data-conditioned (posterior) distributions to evaluate, bound, or optimize risk-related metrics. In mathematical and applied statistics, such methods play a central role in high-dimensional estimation, uncertainty quantification, model selection, and sequential decision making. Posterior risk calculation connects the Bayesian machinery of posterior updates with the explicit aim of minimizing or evaluating expected loss, often under sparsity or other structural assumptions. The horseshoe estimator, particularly in the context of nearly black vectors, provides a highly influential approach rooted in these principles (Pas et al., 2014).

1. Sparse Mean Estimation and Posterior Concentration

The context is the high-dimensional Gaussian mean model: $$Y = \theta_0 + \varepsilon, \quad \varepsilon \sim \mathcal{N}(0, \sigma^2 I_n)$$ where the true mean vector θ₀ ∈ ℝⁿ is assumed to be "nearly black": only $p_n = o(n)$ components are nonzero, and $p_n$ may grow but much more slowly than $n$.

The horseshoe prior, designed for such sparse scenarios, induces a posterior distribution π_τ(θ|Y) that exhibits strong concentration around θ₀ or its Bayes estimate (the horseshoe estimator $T_τ(Y)$). For global shrinkage parameter τ chosen as τ ∼ (pₙ/n)^α, α ≥ 1, Theorem 3.3 demonstrates that the posterior contracts to θ₀ at the minimax rate: $$\Pi_τ\left(\{\theta: \|\theta-\theta_0\|^2 > M_n p_n \log(n/p_n)\}\mid Y \right) \to 0$$ for any $M_n \to \infty$, uniformly over all nearly black mean vectors.

This means the full posterior measure becomes sharply localized in shrinking $\ell_2$ balls of the optimal radius, matching the lower bound for minimax risk.

2. Attainment of the Minimax ℓ₂ Risk

The paper establishes sharp risk bounds for the horseshoe estimator in sparse settings. For a known $p_n$, choosing τ appropriately ensures that the mean squared error (MSE) of $T_τ(Y)$ matches the minimax lower bound: $$\inf_{\hat{\theta}}\, \sup_{\theta_0 \in \ell_0[p_n]} \mathbb{E} \|\hat{\theta} - \theta_0\|^2 \geq 2\sigma^2 p_n \log(n/p_n)(1 + o(1))$$ and

$$\sup_{\theta_0 \in \ell_0[p_n]} \mathbb{E}_{\theta_0} \|T_τ(Y) - \theta_0\|^2 \lesssim p_n \log(1/\tau) + (n - p_n) \tau \sqrt{\log(1/\tau)}$$

Thus, by setting τ ≍ (pₙ/n)^α, α ≥ 1, the horseshoe estimator achieves the minimax rate up to constants, with sharper matching possible by further refinement τ = (pₙ/n)√{log(n/pₙ)}. The risk bound naturally decomposes into a bias component from nonzero means and a variance component from zero means, influenced by τ.

3. Empirical Bayes Estimation of Sparsity

In practice, $p_n$ is unknown. The methodology therefore introduces a plug-in, empirical Bayes estimator for τ: $$\hat{\tau} = \frac{\#\{i: |y_i| \geq \sqrt{c_1 \sigma^2 \log n}\}}{c_2 n}$$ with constants c₁ > 2, c₂ > 1. Theorems in Section 5 show that, provided the estimator is not overly biased (e.g., its overestimation probability is controlled), the resulting empirical Bayes horseshoe estimator

$T_{\hat{\tau}}(Y)$

retains the minimax risk rate: $$\sup_{\theta_0 \in \ell_0[p_n]} \mathbb{E}_{\theta_0} \|T_{\hat{\tau}}(Y) - \theta_0\|^2 \sim p_n \log(n/p_n)$$ If only the upper bound on overestimation is satisfied, a log-factor in the risk may be lost.

Empirical Bayes thus offers a tractable, data-driven tuning of global shrinkage that retains the desirable risk properties of the theoretically optimal fixed-τ approach.

4. Posterior Variance and Uncertainty Quantification

A distinctive contribution is the analysis of the posterior variance. The posterior variance of each θᵢ—given yᵢ and τ—admits an explicit integral form (involving confluent hypergeometric functions) and, more importantly, the sum across all coordinates is tightly bounded: $$\sup_{\theta_0 \in \ell_0[p_n]} \mathbb{E}_{\theta_0} \sum_i \mathrm{Var}(\theta_i|Y_i) \lesssim p_n \log(1/\tau) + (n - p_n) \tau \sqrt{\log(1/\tau)}$$ with a matching lower bound (for true zeros): $$\inf_{\theta_0 \in \ell_0[p_n]} \mathbb{E}_{\theta_0} \sum_{i: \theta_0=0} \mathrm{Var}(\theta_i|Y_i) \gtrsim (n - p_n) \tau \sqrt{\log(1/\tau)}$$ Therefore, with optimal τ = (pₙ/n)√{log(n/pₙ)}, the posterior variance is neither collapsed nor overly dispersed, aligning with the minimax estimation error and ensuring credible intervals reflect true uncertainty.

5. Comparison to One-Component Shrinkage Priors

The horseshoe prior is contrasted with one-component shrinkage priors such as the Laplace prior of the Bayesian Lasso. One-component priors may yield estimators at or near the minimax risk, but their posteriors generally contract too slowly, leading to suboptimal or misleading uncertainty quantification.

The horseshoe prior possesses both infinite density at zero ("pole at zero") and heavy Cauchy tails. This results in adaptive shrinkage: very strong shrinkage for noise (nearly-zero coordinates) and negligible shrinkage for large signals. The full horseshoe posterior contracts at the optimal rate—its posterior mean squared radius matches the minimax risk—yielding informative and accurate credible sets. This is in contrast to the Bayesian Lasso, whose posterior is known to contract too slowly for the purpose of uncertainty quantification.

6. Application Guidance and Theoretical Implications

The theoretical results indicate that for high-dimensional, sparse estimation problems, the horseshoe estimator—augmented by either optimally set or empirically estimated τ—offers a posterior risk calculation that is fully adaptive and achieves minimax optimality, both for point estimation and for uncertainty quantification. Practical strategies involve:

• Adopting τ ≍ (pₙ/n) or τ ≍ (pₙ/n)√{log(n/pₙ)} when $p_n$ is known
• Employing empirical Bayes thresholding to estimate τ robustly
• Reporting posterior credible intervals based on the full horseshoe posterior, which are supported theoretically to have correct coverage
• Choosing the horseshoe over the Bayesian Lasso or similar one-component priors when valid uncertainty quantification is required

This approach is particularly well-suited for problems in signal processing, genomics, and other domains where high-dimensional sparsity is a dominant structural attribute. The risk properties ensure not only reliable estimation but also valid inference for uncertainty—subtle but critical in modern applications where both effect identification and reliability assessment are needed.

Table: Summary of Key Risk Results

Quantity	Horseshoe Estimator Bound	Condition/Parameterization
Posterior Contraction Radius ($\ell_2$)	$p_n \log(n/p_n)$	τ ≍ (pₙ/n)^α, α ≥ 1
Squared Estimation Risk (MSE)	$$p_n \log(1/\tau) + (n-p_n) \tau \sqrt{\log(1/\tau)}$$	τ ≍ (pₙ/n)^α, α ≥ 1
Posterior Variance (Sum, Upper Bound)	$$p_n \log(1/\tau) + (n-p_n) \tau \sqrt{\log(1/\tau)}$$
Posterior Variance (Zero Coords, Lower)	$(n-p_n) \tau \sqrt{\log(1/\tau)}$
Empirical Bayes Plug-in Estimator	Risk ≍ $p_n \log(n/p_n)$	Plug-in τ̂, suitable over/underestimation control

This theoretical framework underpins the accurate and robust application of the horseshoe estimator for nearly black vectors, ensuring that both point estimates and full uncertainty quantification (e.g., credible balls) are minimax-optimal and reflective of the true sparse signal structure.