# Online Algorithms For Parameter Mean And Variance Estimation In Dynamic Regression Models

We study the problem of estimating the parameters of a regression model from a set of observations, each consisting of a response and a predictor. The response is assumed to be related to the predictor via a regression model of unknown parameters. Often, in such models the parameters to be estimated are assumed to be constant. Here we consider the more general scenario where the parameters are allowed to evolve over time, a more natural assumption for many applications. We model these dynamics via a linear update equation with additive noise that is often used in a wide range of engineering applications, particularly in the well-known and widely used Kalman filter (where the system state it seeks to estimate maps to the parameter values here). We derive an approximate algorithm to estimate both the mean and the variance of the parameter estimates in an online fashion for a generic regression model. This algorithm turns out to be equivalent to the extended Kalman filter. We specialize our algorithm to the multivariate exponential family distribution to obtain a generalization of the generalized linear model (GLM). Because the common regression models encountered in practice such as logistic, exponential and multinomial all have observations modeled through an exponential family distribution, our results are used to easily obtain algorithms for online mean and variance parameter estimation for all these regression models in the context of time-dependent parameters. Lastly, we propose to use these algorithms in the contextual multi-armed bandit scenario, where so far model parameters are assumed static and observations univariate and Gaussian or Bernoulli. Both of these restrictions can be relaxed using the algorithms described here, which we combine with Thompson sampling to show the resulting performance on a simulation.

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