# A Modularized Efficient Framework for Non-Markov Time Series Estimation

## TL;DR

This paper introduces a modular, efficient framework for estimating non-Markovian and non-Gaussian time series using a consensus ADMM approach, applicable to various complex signal processing problems.

## Contribution

It proposes a novel, modular ADMM-based method for MAP estimation that handles non-Markov dynamics and non-Gaussian measurements with broad applicability.

## Key findings

- Framework successfully estimates complex time series models.
- Convergence guaranteed under broad convexity assumptions.
- Demonstrated on neuroscience data with non-Gaussian and non-Markovian models.

## Abstract

We present a compartmentalized approach to finding the maximum a-posteriori (MAP) estimate of a latent time series that obeys a dynamic stochastic model and is observed through noisy measurements. We specifically consider modern signal processing problems with non-Markov signal dynamics (e.g. group sparsity) and/or non-Gaussian measurement models (e.g. point process observation models used in neuroscience). Through the use of auxiliary variables in the MAP estimation problem, we show that a consensus formulation of the alternating direction method of multipliers (ADMM) enables iteratively computing separate estimates based on the likelihood and prior and subsequently "averaging" them in an appropriate sense using a Kalman smoother. As such, this can be applied to a broad class of problem settings and only requires modular adjustments when interchanging various aspects of the statistical model. Under broad log-concavity assumptions, we show that the separate estimation problems are convex optimization problems and that the iterative algorithm converges to the MAP estimate. As such, this framework can capture non-Markov latent time series models and non-Gaussian measurement models. We provide example applications involving (i) group-sparsity priors, within the context of electrophysiologic specrotemporal estimation, and (ii) non-Gaussian measurement models, within the context of dynamic analyses of learning with neural spiking and behavioral observations.