Reverse Telescoping Sampler

• Reverse Telescoping Sampler is an algorithmic approach that inverts block decomposition to achieve exact Bayesian posterior sampling with improved computational efficiency.
• It applies to high-dimensional Gaussian graphical models with sparsity-inducing priors, reducing per-iteration complexity from O(p^4) to O(p^3) and ensuring global positive definiteness.
• Empirical studies in gene network inference demonstrate its capability to produce accurate posterior estimates with 5-7x speed improvements over traditional sampling methods.

A reverse telescoping sampler is an algorithmic scheme that leverages the reversal of a telescoping decomposition—frequently block-based or hierarchical in character—to achieve computational and inferential advantages in Bayesian sampling and, more broadly, in the efficient resolution of structurally complex sampling problems. The term principally arises in the context of high-dimensional Gaussian graphical models, but is informed by developments in symbolic computation, lattice models, and creative telescoping in combinatorics. The distinguishing feature is that the sampler inverts a telescoping transformation (originally used to collapse structure for marginal or likelihood calculations) to attain efficient, fully Bayesian posterior sampling with provable correctness and substantially improved scaling in computational complexity.

1. Conceptual and Mathematical Foundations

Reverse telescoping sampling originates from block decompositions of structured parameter spaces, notably the precision (inverse covariance) matrices of Gaussian graphical models (GGMs) endowed with element-wise sparsity-inducing priors (e.g., the graphical horseshoe). In the standard telescoping (or sweeping) block decomposition, the precision matrix $\Theta$ is partitioned blockwise, typically as

$$\Theta = \begin{bmatrix} \Theta_{[p-1]} & \Theta_{[\cdot\,p]} \ \Theta_{[\cdot\,p]}^{\mathrm{T}} & \theta_{pp} \end{bmatrix}$$

and recursed, facilitating marginal likelihood evaluations via Schur complements.

The reverse telescoping sampler repurposes this decomposition for posterior sampling: instead of sweeping structure away to reduce dimensions, it samples in a reparameterized domain (the "tilde" parameters) and sequentially reconstructs the full matrix in reverse order. The reverse mapping at each level $j$ "adds back" the block correction by inverting a Schur complement: $$\theta_j = \tilde{\theta}_j + f(\tilde{\theta}_{j+1}, \ldots, \tilde{\theta}_p)$$ where $f(\cdot)$ denotes the nonlinear correction implied by the recomposition. The mapping is guaranteed to be one-to-one and preserves the domain required for positive definiteness via stepwise enforcement of simple positivity constraints.

2. Algorithmic Implementation and Workflow

The reverse telescoping sampler operates as follows in GGMs with $p$ nodes:

1. Initialization: Start with a positive definite $\Theta$ or its forward block-decomposition $\tilde{\Theta}$.
2. Blockwise Sampling: For $j = p, p-1, \ldots, 1$, update block $\tilde{\theta}_j$:
   • Sample diagonal (precision) and off-diagonal elements conditionally, leveraging conjugacy where possible (often drawing from gamma or normal distributions).
   • The positivity constraints on $\tilde{\theta}_j$ directly enforce global positive definiteness of $\Theta$ after reversal.
3. Reverse Mapping: Sequentially reconstruct the original $\Theta$ via the reverse telescoping mapping.
4. Hyperparameter Updates: Update element-wise or global scale/shrinkage parameters where applicable.
5. Iteration: Repeat until posterior mixing and convergence criteria are satisfied.

A critical feature is that updates are performed on the sample data matrix (not just the empirical scatter matrix), retaining compatibility with a broad class of non-conjugate priors.

3. Computational Complexity and Scaling Advantages

Whereas classical block Gibbs samplers for element-wise priors in GGMs require $O(p^4)$ per iteration (due to $O(p)$ block updates, each necessitating inversion of a $(p-1)\times(p-1)$ submatrix), the reverse telescoping sampler achieves $O(p^3)$ per iteration. This is made possible by:

• Sequential block updates that avoid redundant full-matrix inversions,
• Propagation of conditional independence via the reverse decomposition such that only local (block) operations are needed at each step,
• Reuse of submatrix factorizations in recursive updates.

For high-dimensional settings $p \gg n$, this one-order-of-magnitude improvement brings non-conjugate, element-wise Bayesian approaches on par with the computational cost of the classical Bartlett decomposition (applicable only to conjugate Wishart priors).

4. Empirical Evaluation and Practical Applications

Extensive simulation studies confirm that the reverse telescoping sampler targets the exact Bayesian posterior in GGMs with element-wise priors. In comparative studies (notably under the graphical horseshoe prior), posterior mean estimates (e.g., in Frobenius norm) and inferred sparsity patterns match those from traditional cyclic samplers, but the reverse telescoping sampler is dramatically faster—offering up to one-order-of-magnitude speedup at large $p$.

A notable application is gene network inference from high-dimensional RNA-sequencing data (breast cancer datasets), where the reverse telescoping sampler produces network topologies and edge strengths closely concordant with standard approaches while achieving a $5$--$7\times$ runtime reduction. This improvement enables fully Bayesian uncertainty quantification in regimes previously limited to approximate or pseudolikelihood-based methods.

The method does not employ variational techniques or ABC relaxations: all draws are from the true posterior, retaining exactness.

5. Broader Context, Connections, and Theoretical Considerations

Reverse telescoping sampling is tightly connected to the broader class of creative telescoping methods in symbolic computation and lattice model theory:

• In creative telescoping of sums or integrals, one seeks to find recurrences (i.e., to "collapse" structure). A reverse telescoping sampler, by contrast, could be viewed as an algorithmic re-expansion: given a suitable recurrence (or block decomposition), one reconstructs or samples the original objects, carefully managing boundary contributions and invariants.
• Submodule approaches to creative telescoping (Hoeij, 16 Jan 2024) further illustrate efficient decomposition, recombination, and the role of automorphisms in symbolic structure recovery, offering a metaphor for reverse telescoping in probabilistic inference as well.

In Bayesian nonparametric settings, the conceptual "reverse" of telescoping samplers has been considered: collapsing infinite- or large-component models to finite, parsimonious representations (Frühwirth-Schnatter et al., 2020). However, maintaining coverage and detailed balance without costly reversible jump proposals presents ongoing challenges.

6. Implications and Future Prospects

The reverse telescoping sampler resolves a major computational bottleneck in high-dimensional Bayesian inference with non-conjugate priors, expanding feasible application domains to genomics, proteomics, and any high-dimensional precision matrix estimation problem requiring sparse, structure-adaptive priors. Its methodology, targeting the full posterior distribution, aligns with critical demands for valid uncertainty quantification and network structure recovery.

Further extensions could include:

• Adaptation to alternative priors (e.g., spike-and-slab, hierarchical shrinkage) beyond the graphical horseshoe,
• Application to time-varying or dynamic graphical models, evidence estimation, and integrative analyses across multi-modal data,
• Distributed or parallelized versions for very large-scale inference.

A plausible implication is that similar reverse telescoping paradigms could inform other computationally constrained Bayesian models, especially wherever hierarchical marginalization or telescoping decomposition can be reversed to obtain efficient, exact sampling schemes (Gao et al., 30 Sep 2025).