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A General Framework for the Parametrization of Hierarchical Models
Omiros Papaspiliopoulos, Gareth O. Roberts, Martin Sköld
TL;DR
Hierarchical-model MCMC can be highly sensitive to parametrization, while general guidance is limited for nonlinear models and latent stochastic processes. The paper develops centered, noncentered, and partially noncentered strategies, together with convergence theory and construction recipes, and finds that their complementary use can provide robust inference across contexts.
Problem
General parametrization strategies are limited for nonlinear hierarchical models, particularly those involving latent stochastic processes, even though parametrization strongly affects MCMC performance.
Method
The paper analyzes centered and noncentered Gibbs-sampler parametrizations, develops convergence-time theory, and gives construction recipes including state-space expansion and partial noncentering.
Results
Centered and noncentered parametrizations have complementary strengths, and their combined use is presented as an attractive, relatively robust option for hierarchical-model MCMC.
Takeaways & Limitations
Parametrization choice can be guided by model structure, while combining centered and noncentered approaches can balance convergence advantages against computational costs such as loss of conjugacy.
Takeaways & Limitations
Neither centering nor noncentering is uniformly effective, and centered samplers can deteriorate or become reducible as latent-state imputation is refined.
Abstract
from arXiv · showhide
In this paper, we describe centering and noncentering methodology as complementary techniques for use in parametrization of broad classes of hierarchical models, with a view to the construction of effective MCMC algorithms for exploring posterior distributions from these models. We give a clear qualitative understanding as to when centering and noncentering work well, and introduce theory concerning the convergence time complexity of Gibbs samplers using centered and noncentered parametrizations. We give general recipes for the construction of noncentered parametrizations, including an auxiliary variable technique called the state-space expansion technique. We also describe partially noncentered methods, and demonstrate their use in constructing robust Gibbs sampler algorithms whose convergence properties are not overly sensitive to the data.
1. INTRODUCTION
The paper addresses the lack of general parametrization strategies for nonlinear hierarchical models, especially those with latent stochastic processes, by developing a broad centered/noncentered framework for MCMC design.
- Research gap: Analytic convergence rates exist for Gaussian Gibbs samplers, but comparable general parametrization strategies for nonlinear hierarchical models remain underdeveloped.The gap is especially pronounced for models involving unobserved latent stochastic processes.
- Motivation: Increasingly complex hierarchical models create computational challenges across econometrics, geostatistics, and genetics.Existing high-performance MCMC strategies are often ad hoc and limited to particular model classes.
- Approach: The paper develops a general strategy based on Gibbs sampling or component-wise Metropolis–Hastings with centered and noncentered parametrizations.The framework targets a wide range of statistical contexts rather than a single model class.
- Core contribution: Centered and noncentered parametrizations are presented as complementary because one can converge slowly when the other converges much faster.The paper also argues that model structure can indicate the preferable parametrization before computation begins.
- Design goal: The framework prioritizes robustness across contexts when achieving optimality would require sacrificing generality.This emphasis responds to the breadth and specialization of the existing MCMC toolkit.
2. AUGMENTATION SCHEMES AND PARAMETRIZATIONS
The paper frames augmentation and parametrization as separate choices for MCMC over joint posteriors, focusing on centered and noncentered parametrizations as broadly applicable, complementary alternatives. Their relative performance depends on posterior dependence between latent variables and parameters, while convergence analysis identifies important failure modes and practical boundaries.
- Augmentation and parametrization: Augmentation introduces latent variables for analytical, scientific, or computational reasons, but the paper focuses on choosing their parametrization for joint-posterior MCMC.The augmentation scheme and parametrization are distinct choices; reparametrization aims solely to improve Monte Carlo efficiency.
- Augmentation and parametrization: A practical reparametrization combines a joint prior for transformed latent variables and parameters with a transformation whose conditional parameter distribution is tractable.The transformation need not be one-to-one, provided P(Θ | X∗) and therefore P(Θ | X∗,Y) are available up to normalization.
- Centered parametrization: The centered parametrization is the natural hierarchical form in which data are conditionally independent of parameters given imputed latent data, and it appears across diverse applications.The section situates this construction in common hierarchical models and examples including spatial and stochastic-volatility settings.
- Complementarity and convergence: Centered Gibbs sampling can worsen as sample size grows, become nongeometrically ergodic, or become reducible under some latent-model and augmentation choices.The convergence-rate intuition is that stronger posterior dependence between updated components produces slower convergence.
- Noncentered parametrization: Noncentered parametrizations make transformed latent variables and parameters a priori independent, often improving efficiency when the latent variables are weakly identified by the data.When the latent variables are well identified, the transformation can instead induce strong posterior dependence between transformed latents and parameters.
- Complementarity and convergence: The centered and noncentered algorithms are complementary general-purpose choices rather than universally optimal parametrizations, with performance often predictable from the model structure.Their main advantages are robustness, ease of extension to high-dimensional spaces and complicated data structures, and broad applicability.
- Complementarity and convergence: The paper’s convergence-rate analysis does not directly apply when component updates use Metropolis–Hastings, although posterior dependence remains a useful heuristic for such algorithms.This limits direct theoretical transfer from the Gibbs analysis to more general update schemes.
3. CASE STUDIES
The case studies show that centered and noncentered parametrizations have complementary convergence behavior, depending on sample size, tail behavior, and the amount of latent-variable imputation. They also motivate noncentering when finer augmentation creates strong dependence or reducibility.
- Overview: The examples are simplified illustrations of situations where centering or noncentering can be crucial for practical hierarchical models.The cases are organized by dependence on sample size, link tails, and amount of imputation.
- Efficiency and sample size: For a Gaussian random-effects model, τc = O(1/log n) while τnc = O(n), so centered sampling improves and noncentered sampling deteriorates with sample size.The Gaussian structure makes the convergence analysis tractable through linear functionals.
- Efficiency and sample size: For a hidden Markov model with moderate serial dependence, both centered and noncentered algorithms have O(1) convergence time complexity.With independent hidden states, the presence of data tying down the states is important for centered performance; without data, noncentering is O(1).
- Efficiency and sample size: Nonregular models can reverse the usual preference: one example has τc ≥ O(n) and τnc = O(1), while an asymmetric-error model has τc = O(1) and τnc ≥ O(n).The contrasting rates arise from how conditional variances scale under the two parametrizations.
- Efficiency and sample size: In Bayesian classification, perfect separation favors centering, whereas unrelated covariates favor noncentering, with the other parametrization having convergence worse than O(n) in each case.Under perfect classification, centered updates are independent; when f0 = f1, noncentered updates are independent.
- Efficiency and link tails: Posterior dependence can vary across the state space: centering may work near the mode but noncentering in the tails, while complementary heavy-tailed models reverse this behavior.In the heavy-tailed HMM example, centered sampling is not geometrically ergodic whereas noncentered sampling is uniformly ergodic; the complementary model reverses these properties.
- Efficiency and amount of imputation: When increasingly fine or infinite-dimensional imputation strongly links X and Θ, centered algorithms can become reducible or have O(n) convergence, whereas noncentering can remain O(1).This occurs because Θ may be determined by the imputed path in the limiting representation; the noncentered alternative is presented as a simpler efficient option.
4. CONSTRUCTING NCP’S
The paper constructs noncentered parametrizations by transforming simple, parameter-independent random inputs, including higher-dimensional and noninvertible state-space expansions. These constructions support practical MCMC updates for latent stochastic processes and complex hierarchical models.
- General recipes: Standard simulation devices—location, scale, inverse-CDF, and recursive transformations—provide general recipes for constructing noncentered parametrizations.For Markov chains, inverse-CDF transformations use independent uniforms recursively; Gaussian-field constructions use independent standard normals and a covariance factor.
- General recipes: When prior simulation is available, inspecting the simulation algorithm often suggests an NCP, although no simple universal construction exists for arbitrary latent structures.Representation theorems can also provide constructions for stochastic processes.
- General recipes: An NCP can use a latent representation ˜X with much higher dimension than X, and its transformation h can be noninvertible.This flexibility is central to applying noncentering to complex latent structures.
- State-space expansion: For Gibbs updates, ˜X may be sampled conditionally rather than deterministically inverted, allowing many-to-one transformations and partial auxiliary-state updates.This is useful when ˜X is infinite-dimensional but X remains finite-dimensional, including latent Poisson and hidden Markov jump-process settings.
- State-space expansion: State-space expansion constructs Poisson-process NCPs from unit-rate Poisson processes on either [0,1] × [0,∞) or [0,∞), followed by a parameter-dependent transformation.The resulting observed process depends on only finitely many auxiliary points for fixed Θ, so partial information can be stored during updates.
5. ROBUSTNESS AND DATA-BASED MODIFICATIONS
Centered and noncentered parametrizations can respond differently to the data and even within different posterior regions. Data-based and partially noncentered modifications therefore combine observed and unobserved components to improve robustness.
- Robustness: Centered and noncentered performance can vary across datasets and posterior regions: centered parametrization may work near the mode while noncentering is needed in the tails.This motivates methods whose behavior is less sensitive to how informative the data are about X.
- Robustness: Alternating centered and noncentered algorithms can greatly improve robustness without a practically relevant increase in computational time.The paper presents combined use as an alternative to choosing one parametrization globally.
- Data-based modifications: Because centered and noncentered parametrizations are constructed from priors alone, efficient posterior parametrization may require incorporating data information.The paper develops simple data-based modifications for settings where standard parametrizations are necessary to improve performance.
- Data-based modifications: For discretely observed diffusions, centered augmentation has O(n) mixing time, while the previously successful noncentered parametrization cannot update Θ without violating the data constraint.The failure occurs because Y is a deterministic function of Θ and ˜X.
- Data-based modifications: In partially observed Gaussian fields, centering observed values and noncentering unobserved values addresses the contrasting information content of augmented and observed data.When n ≪ m, augmented data can contain much more information about Θ than the observations, degrading the centered algorithm.
6. DISCUSSION
The discussion presents centering and noncentering as broadly applicable, complementary tools rather than uniformly optimal parametrizations. Their combination and partial variants offer robustness, while computational overhead and limited theory remain open concerns.
- Conclusions: Neither centered nor noncentered methods is uniformly effective, but their complementary strengths make combined use an attractive and relatively robust option.The paper gives qualitative and, in some cases, quantitative guidance about when noncentering is likely to work.
- Conclusions: Centered methods can exploit hierarchical conditional independence and conjugate parameter updates, whereas noncentering can offset these computational costs through convergence advantages.The relative benefit depends on the model and the parametrization used.
- Conclusions: State-space expansion makes noncentered parametrizations broadly practical because P(Θ | ˜X) = P(Θ) is available up to a normalizing constant.The construction permits complex transformations without requiring posterior-dependent Jacobian calculations.
- Conclusions: Parametrizations between centered and noncentered forms, or data-specific constructions, can outperform both standard strategies, although partially noncentered methods may be difficult to construct.Their appeal is robustness across datasets and data types.
- Open limitations: Applications of partial noncentering remain uncommon, and further work is needed to assess computing overheads and extend theory beyond the covered examples.This is the paper’s stated scope boundary for evidence about effectiveness.