Accelerated Inference for Partially Observed Markov Processes using Automatic Differentiation (2407.03085v1)

Abstract: Automatic differentiation (AD) has driven recent advances in machine learning, including deep neural networks and Hamiltonian Markov Chain Monte Carlo methods. Partially observed nonlinear stochastic dynamical systems have proved resistant to AD techniques because widely used particle filter algorithms yield an estimated likelihood function that is discontinuous as a function of the model parameters. We show how to embed two existing AD particle filter methods in a theoretical framework that provides an extension to a new class of algorithms. This new class permits a bias/variance tradeoff and hence a mean squared error substantially lower than the existing algorithms. We develop likelihood maximization algorithms suited to the Monte Carlo properties of the AD gradient estimate. Our algorithms require only a differentiable simulator for the latent dynamic system; by contrast, most previous approaches to AD likelihood maximization for particle filters require access to the system's transition probabilities. Numerical results indicate that a hybrid algorithm that uses AD to refine a coarse solution from an iterated filtering algorithm show substantial improvement on current state-of-the-art methods for a challenging scientific benchmark problem.

• The paper introduces a novel hybrid algorithm that integrates automatic differentiation with particle filtering to enhance maximum likelihood estimation in POMPs.
• It employs a discount factor α to optimize the bias-variance tradeoff, significantly lowering mean squared error compared to traditional methods.
• The method shows superior convergence and performance in real-world applications like the Dhaka cholera model, indicating broad interdisciplinary impact.

Accelerated Inference for Partially Observed Markov Processes using Automatic Differentiation

The paper presented by K. Tan, G. Hooker, and E. L. Ionides introduces a novel approach to inference in Partially Observed Markov Processes (POMP) through a hybrid algorithm, leveraging recent advances in automatic differentiation (AD). The core contribution lies in addressing the quantitative challenges of maximum likelihood estimation in POMPs without requiring access to transition probabilities. This work builds on statistical efficiency while maintaining the flexibility offered by a plug-and-play methodology in complex, nonlinear stochastic systems.

Summary of Methodology

The typical difficulty with POMPs is their intractability due to nonlinearities and stochastic dependencies, which render simple differentiation techniques ineffective. Traditional particle filters, crucial for such models, yield discontinuous likelihood functions concerning parameters, hence impeding straightforward gradient-based optimization.

The proposed model innovatively integrates two established AD methods within a theoretical framework to form a new class of algorithms, named Measurement Off-Parameter (MOP) particle filters. This class establishes a balance between bias and variance, reducing mean squared error significantly compared to existing techniques. Particularly, the introduction of a discount factor $\alpha$ within the MOP algorithm is critical. This factor ranges between 0 and 1, interpolating between biased gradients with low variance ($\alpha=0$) and unbiased gradients with high variance ($\alpha=1$).

Numerical Efficacy

Numerical results substantiate the efficacy of the proposed methods. A hybrid algorithm, referred to as Iterated Filtering with Automatic Differentiation (IFAD), shows superior performance compared to iterated filtering methods (e.g., IF2). IFAD begins with an iterated filtering solution and refines it through AD, achieving substantial improvements in the likelihood maximization task.

The paper demonstrates this through the practical application to the Dhaka cholera model, a widely recognized benchmark in epidemiology. IFAD achieves rapid convergence to higher likelihood values compared to the state-of-the-art IF2. The MOP-$\alpha$ variants, particularly with $\alpha$ values strictly between 0 and 1, exhibit reduced bias and variance, effectively optimizing complex models significantly faster and with greater stability in estimated parameters.

Theoretical Contributions

The theoretical underpinnings are robust. The authors prove that MOP-$\alpha$ targets the correct filtering distribution and provide conditions for the strong consistency of the gradient estimates. Key results include a strong law of large numbers for triangular arrays, demonstrating almost sure convergence under appropriate weight corrections. Importantly, the paper rigorously establishes that MOP-$\alpha$ encompasses previous gradient estimators, offering flexibility and an improved performance landscape.

Bias, Variance, and Convergence Analysis

A central analytical focus is on understanding the bias-variance tradeoff inherent in the MOP-$\alpha$ algorithm. The variance of MOP-$\alpha$ is shown to be $\tilde{O}(Np G'(\theta)^2(k+\psi(\alpha)))$, where $\psi(\alpha)$ characterizes the bias-variance landscape effectively, benefiting from the introduced discount factor. This contributes significantly to the field by analytically demonstrating how $\alpha<1$ transitions to a lower MSE region compared to MOP-$1$ ($\alpha=1$) or MOP-$0$ ($\alpha=0$).

Implications and Future Directions

The implications of this research extend broadly. Practically, the significant reduction in computational demand and improvement in efficiency can expedite the inferential processes in highly stochastic models within a range of disciplines such as epidemiology, ecology, and finance. Theoretically, the successful integration of AD within particle filters opens new avenues for complex model optimization, potentially minimizing the cumbersome need for problem-specific algorithm tuning and extensive computation.

Future work is slated to address limitations regarding discontinuous simulators and extending AD's applicability without requiring differentiability. This would further generalize the utility of AD in inference processes. Another promising direction is enhancing Bayesian inference methodologies, such as integrating the MOP-$\alpha$ gradient estimates into variational inference frameworks.

Overall, the paper presents a concrete advancement in the methodology for maximum likelihood estimation in POMPs, demonstrating substantial improvements in efficiency and accuracy in dealing with complex scientific models. The hybrid approach, validated both theoretically and numerically, shows promise for broad application and further development in the landscape of stochastic dynamical systems.