Efficient Neural Likelihood-free Bayesian Inference of Implicit Hidden Markov Models

Core Concepts

A novel, sample-efficient likelihood-free method for estimating the high-dimensional hidden states of an implicit Hidden Markov Model, by learning the intractable posterior distribution of the hidden states using an autoregressive-flow.

Abstract

The content presents a novel technique for Bayesian inference of implicit Hidden Markov Models (HMMs), where the likelihood function is analytically intractable.
The key highlights are:

Existing neural likelihood-free inference (NLFI) methods can efficiently estimate the marginal posterior distribution of the model parameters, but fail to accurately estimate the joint posterior distribution of the parameters and the high-dimensional hidden states. This leads to an inaccurate assessment of the goodness-of-fit.

The authors propose a novel, sample-efficient method to estimate the posterior distribution of the hidden states. The method learns an approximation of the posterior distribution of the hidden states using an autoregressive-flow neural density estimator.

The proposed approach, when combined with any off-the-shelf NLFI method for estimating the parameter posterior, can perform full Bayesian inference of an implicit HMM in a sample-efficient manner, as an alternative to the computationally expensive Approximate Bayesian Computation (ABC) algorithms.

The method was evaluated on a nonlinear Gaussian state-space model with a tractable approximate factor, as well as two implicit biological HMM models. The results show that the proposed approach can produce estimates of the hidden states and the posterior predictive distribution that are comparable or better than what can be achieved using a much more computationally expensive Sequential Monte Carlo (SMC) algorithm, but with significantly fewer model simulations.

Stats

The content does not provide any specific numerical data or statistics. The evaluation is based on qualitative metrics such as mean squared error, 90% empirical coverage, and coefficient of variation.

Quotes

"Naive application of these methods to a HMM, ignoring the inference of this joint posterior distribution, will thus produce an inaccurate estimate of the posterior predictive distribution, in turn hampering the assessment of goodness-of-fit."
"To rectify this problem, we propose a novel, sample-efficient likelihood-free method for estimating the high-dimensional hidden states of an implicit HMM."
"Upon evaluating our approach on some implicit HMMs, we found that the quality of the estimates retrieved using our method is comparable to what can be achieved using a much more computationally expensive SMC algorithm."

Key Insights Distilled From

Sample-efficient neural likelihood-free Bayesian inference of implicit HMMs

by Sanmitra Gho... at arxiv.org 05-06-2024

https://arxiv.org/pdf/2405.01737.pdf

Deeper Inquiries

How can the proposed method be extended to handle model misspecification, where the observed data may not exactly follow the assumed implicit HMM structure

To handle model misspecification in the context of implicit HMMs, where the observed data may deviate from the assumed structure, the proposed method can be extended by incorporating a mechanism for model adaptation or flexibility. This can involve introducing additional parameters or latent variables that allow the model to adapt to variations in the data. For example, one approach could be to include a mechanism for model selection or model averaging, where multiple candidate models are considered, and the one that best fits the data is chosen. This can help account for uncertainties in the model structure and improve the robustness of the inference process in the presence of model misspecification.

Can the autoregressive-flow based density estimation approach be further improved to better capture the complex dependencies in the hidden state dynamics

The autoregressive-flow based density estimation approach can be further improved to better capture the complex dependencies in the hidden state dynamics by enhancing the flexibility and expressiveness of the neural network architecture. This can be achieved by using more sophisticated neural network models, such as deep neural networks or convolutional neural networks, that can capture nonlinear relationships and dependencies in the data more effectively. Additionally, incorporating attention mechanisms or recurrent neural networks can help model long-range dependencies in the hidden state dynamics. Furthermore, exploring advanced techniques like normalizing flows with more complex transformations can also enhance the modeling capabilities of the autoregressive-flow approach.

What are the potential applications of the proposed method beyond Bayesian inference of implicit HMMs, where efficient estimation of high-dimensional latent variables is crucial

The proposed method for efficient estimation of high-dimensional latent variables using autoregressive-flow based density estimation has potential applications beyond Bayesian inference of implicit HMMs. Some of the potential applications include:

Financial Modeling: The method can be applied to financial time series data to estimate latent variables in complex financial models, such as volatility modeling or asset price prediction.
Healthcare: In healthcare, the method can be used for patient monitoring and disease progression modeling by estimating latent variables from patient data to improve diagnosis and treatment planning.
Natural Language Processing: The approach can be utilized in natural language processing tasks, such as language modeling or text generation, by estimating latent variables in language models to capture semantic relationships and context.
Climate Modeling: The method can be applied to climate data analysis to estimate hidden variables in climate models and improve predictions of weather patterns and climate change effects.
Image and Video Analysis: In computer vision and video analysis, the method can help estimate latent variables in image and video data to extract meaningful features and patterns for tasks like object recognition and motion tracking.