Particle Gibbs with Ancestor Sampling (1401.0604v1)

Abstract: Particle Markov chain Monte Carlo (PMCMC) is a systematic way of combining the two main tools used for Monte Carlo statistical inference: sequential Monte Carlo (SMC) and Markov chain Monte Carlo (MCMC). We present a novel PMCMC algorithm that we refer to as particle Gibbs with ancestor sampling (PGAS). PGAS provides the data analyst with an off-the-shelf class of Markov kernels that can be used to simulate the typically high-dimensional and highly autocorrelated state trajectory in a state-space model. The ancestor sampling procedure enables fast mixing of the PGAS kernel even when using seemingly few particles in the underlying SMC sampler. This is important as it can significantly reduce the computational burden that is typically associated with using SMC. PGAS is conceptually similar to the existing PG with backward simulation (PGBS) procedure. Instead of using separate forward and backward sweeps as in PGBS, however, we achieve the same effect in a single forward sweep. This makes PGAS well suited for addressing inference problems not only in state-space models, but also in models with more complex dependencies, such as non-Markovian, Bayesian nonparametric, and general probabilistic graphical models.

• The paper introduces PGAS, a method that integrates ancestor sampling into Particle Gibbs to overcome path degeneracy and enhance computational efficiency.
• The paper rigorously proves the invariance and uniform ergodicity of the PGAS kernel, ensuring reliable convergence in complex state-space and non-Markovian models.
• The paper demonstrates that PGAS outperforms traditional methods, delivering significant improvements in accuracy and mixing efficiency across various empirical evaluations.

An Analysis of "Particle Gibbs with Ancestor Sampling"

The paper by Lindsten, Jordan, and Sch\"on introduces a novel method within the framework of Particle Markov chain Monte Carlo (PMCMC), termed Particle Gibbs with Ancestor Sampling (PGAS). This method seeks to enhance the efficiency of inference in state-space models (SSMs) and models with non-Markovian dependencies, among others. By integrating the strengths of Sequential Monte Carlo (SMC) and Markov chain Monte Carlo (MCMC), PGAS addresses limitations previously encountered with predecessor methods, notably improving on issues related to high-dimensionality and path degeneracy.

Overview and Methodological Contributions

The core contribution of this paper is the introduction of ancestor sampling within the Particle Gibbs framework. PGAS differs from Particle Gibbs with backward simulation (PGBS) by eliminating the need for separate forward and backward sweeps, which in practice results in improved computational efficiency. Ancestor sampling allows for more effective mixing of the PGAS kernel, which is crucial for reducing computational burdens typically associated with SMC methods.

Unlike traditional PG methods, PGAS achieves effective mixing even with a seemingly small number of particles, addressing the path degeneracy problem that hampers PG and its variants. This novel approach maintains the invariant distribution property of the kernel, ensuring that the target distribution is correctly sampled regardless of the number of particles used in the underlying SMC sampler.

Theoretical Implications and Practical Applications

The theoretical robustness of PGAS is underlined by a detailed proof of its invariance and ergodicity. The paper demonstrates that PGAS kernels leave the target distribution invariant, and under certain assumptions, they possess uniform ergodicity—an important aspect that ensures convergence in distribution to the correct stationary distribution. Additionally, the paper establishes conditions under which PGAS is particularly useful, extending its applicability beyond just SSMs to include non-Markovian latent variable models and Bayesian nonparametric models.

From a practical perspective, PGAS opens new possibilities for efficient inference in complex models. The paper outlines the method's utility in models where state dependencies are more intricate than typical Markovian models allow. For instance, the method is applicable to dynamical systems where a latent stochastic process is non-Markovian—a situation commonly encountered in marginalization and transformation of SSMs, as well as in some Bayesian nonparametric frameworks.

Empirical Evaluation

The authors empirically evaluate PGAS by comparing it against other methods, including traditional PG and PGBS, across various settings such as linear Gaussian SSMs and degenerate models. Notably, the paper illustrates the substantial improvements in mixing and computational efficiency offered by PGAS. For example, in degenerate models reformulated as non-Markovian latent variable systems, PGAS demonstrates an order of magnitude improvement in accuracy over PGBS, highlighting its robustness to path degeneracy.

Furthermore, the paper explores the application of PGAS in epidemiological models, demonstrating its potential for practical use in real-world scenarios such as predicting disease activity in a population. The flexibility of PGAS to accommodate approximations without losing theoretical rigor is also highlighted through the introduction of a truncation strategy for ancestor sampling weights, further reducing computational demands.

Conclusion

Particle Gibbs with Ancestor Sampling represents a significant advancement within the field of Monte Carlo methods, particularly for models with complex dependencies. Its methodological innovations provide a practical and efficient solution to challenges posed by high-dimensionality and path degeneracy in statistical modeling, paving the way for broader applicability across varied scientific fields. Future work could explore the convergence behavior of PGAS and its potential adaptations to even more complex statistical models, extending its impact on computational statistics and beyond.