# Maximum Likelihood With a Time Varying Parameter

## TL;DR

This paper introduces a recursive stochastic gradient descent method for tracking a time-varying parameter in probabilistic models, proving convergence and illustrating its effectiveness with exponential family distributions.

## Contribution

It proposes a novel recursive scheme using the log-likelihood as a gain function for tracking time-varying parameters, with proven convergence properties.

## Key findings

- Convergence in mean-square error near the true parameter
- Effective tracking demonstrated with exponential family distributions
- The method adapts to changing parameters in real-time

## Abstract

We consider the problem of tracking an unknown time varying parameter that characterizes the probabilistic evolution of a sequence of independent observations. To this aim, we propose a stochastic gradient descent-based recursive scheme in which the log-likelihood of the observations acts as time varying gain function. We prove convergence in mean-square error in a suitable neighbourhood of the unknown time varying parameter and illustrate the details of our findings in the case where data are generated from distributions belonging to the exponential family.