SMC^2: an efficient algorithm for sequential analysis of state-space models (1101.1528v3)

Abstract: We consider the generic problem of performing sequential Bayesian inference in a state-space model with observation process y, state process x and fixed parameter theta. An idealized approach would be to apply the iterated batch importance sampling (IBIS) algorithm of Chopin (2002). This is a sequential Monte Carlo algorithm in the theta-dimension, that samples values of theta, reweights iteratively these values using the likelihood increments p(y_t|y_1:t-1, theta), and rejuvenates the theta-particles through a resampling step and a MCMC update step. In state-space models these likelihood increments are intractable in most cases, but they may be unbiasedly estimated by a particle filter in the x-dimension, for any fixed theta. This motivates the SMC^2 algorithm proposed in this article: a sequential Monte Carlo algorithm, defined in the theta-dimension, which propagates and resamples many particle filters in the x-dimension. The filters in the x-dimension are an example of the random weight particle filter as in Fearnhead et al. (2010). On the other hand, the particle Markov chain Monte Carlo (PMCMC) framework developed in Andrieu et al. (2010) allows us to design appropriate MCMC rejuvenation steps. Thus, the theta-particles target the correct posterior distribution at each iteration t, despite the intractability of the likelihood increments. We explore the applicability of our algorithm in both sequential and non-sequential applications and consider various degrees of freedom, as for example increasing dynamically the number of x-particles. We contrast our approach to various competing methods, both conceptually and empirically through a detailed simulation study, included here and in a supplement, and based on particularly challenging examples.

• The paper introduces SMC^2, an algorithm for sequential Bayesian inference in state-space models with parameter and state uncertainty, by combining Sequential Monte Carlo (SMC) and Particle Markov Chain Monte Carlo (PMCMC).
• Key algorithmic innovations include using nested particle filters for state sequences conditional on parameter particles and employing PMCMC moves to effectively rejuvenate parameter particles.
• Empirical results demonstrate that SMC^2 is robust and computationally efficient, allowing for stable approximation of the posterior distribution and managing computational costs that increase linearly over time.

Overview of the SMC$^2$ Algorithm for Sequential Bayesian Inference in State-Space Models

The paper introduces the SMC$^2$ algorithm, a novel approach for conducting sequential Bayesian inference in state-space models, where both the parameter and state processes are subject to uncertainty. State-space models are widely used to model dynamical systems across various fields, and they require efficient computational algorithms to perform inference. The SMC$^2$ algorithm stands out by integrating Sequential Monte Carlo (SMC) with particle Markov chain Monte Carlo (PMCMC) techniques to handle the challenges associated with such models.

Sequential Monte Carlo in Parameter Inference

The algorithm addresses the recursive exploration of posterior distributions in state-space models:

$$\pi_{0}(\theta)=p(\theta),\quad\pi_{t}(\theta,x_{1:t})=p(\theta,x_{1:t}|y_{1:t}),\quad t\geq1\,.$$

SMC$^2$ leverages SMC methods not just for the filtering problem—where parameters are assumed known—but across the parameter space, $\theta$. The core innovation lies in processing multiple particle filters simultaneously, each propagating throughout the state space $x$, conditional on the parameter space $\theta$. The algorithm efficiently manages the intractable likelihood increments that characterize state-space models, employing unbiased particle filtering estimates to enable sequential learning of $\theta$.

Algorithmic Innovations

Key components of the SMC$^2$ algorithm include:

• Nested Particle Filters: For each parameter particle, the algorithm associates a separate particle filter on the state sequence, effectively tackling the dual uncertainties in both state and parameters.
• Use of PMCMC: The PMCMC framework enriches the SMC$^2$ algorithm by incorporating MCMC moves that preserve the ergodic properties of samples in the parameter space, thereby aiding in rejuvenating parameter particles during resampling stages.
• Dynamic Particle Count: The algorithm dynamically adjusts the number of state particles, $N_x$, according to the required computational accuracy, effectively balancing computational load and inference fidelity.

Empirical Performance and Theoretical Implications

The paper presents strong empirical results showcasing SMC$^2$ in challenging environments, such as financial volatility modeling and extreme value analysis in athletic records. The results indicate that SMC$^2$ remains robust over time, maintaining accuracy and handling state-space model complexity effectively compared with other methods. The proposal outlines how its sequential capabilities make it suitable even for vast datasets or scenarios demanding frequent updates.

Theoretical Implications: The SMC$^2$ algorithm provides significant computational advantages by allowing entire populations of particles to be managed and resampled with computational costs increasing linearly over time. The integration of unbiased likelihood estimates with PMCMC allows the method to stably approximate the posterior distribution despite intractable likelihood increments.

Conclusion and Future Directions

The SMC$^2$ represents a substantial contribution to the computational methods available for state-space modeling, promoting deeper understanding and applicability of these complex models. Future developments could extend the algorithm's capabilities to models with intractable observation processes by employing unbiased estimators in transition probabilities. Additionally, the potential for employing augmented SMC methods within a tempering framework offers an intriguing extension for exploring target distributions beyond the sequential domain.

In summary, the SMC$^2$ algorithm serves as a powerful tool for sequential analysis in complex dynamical systems, offering insights not easily approachable through traditional SMC or MCMC methods alone. This work encourages further exploration in adaptive methodologies and applications in real-time large-scale data environments.