Bayesian Inference via Approximation of Log-likelihood for Priors in Exponential Family
This work addresses computational challenges in Bayesian inference for specific prior distributions, representing an incremental improvement in approximation methods for filtering and estimation tasks.
The paper tackles the problem of Bayesian inference with priors in the exponential family by developing a Taylor series approximation of the log-likelihood to linearize it with respect to the prior's sufficient statistic, resulting in a posterior that retains the same exponential family form as the prior, and applies this to derive an extended target measurement update for random matrix models with inverse Wishart distributions.
In this paper, a Bayesian inference technique based on Taylor series approximation of the logarithm of the likelihood function is presented. The proposed approximation is devised for the case, where the prior distribution belongs to the exponential family of distributions. The logarithm of the likelihood function is linearized with respect to the sufficient statistic of the prior distribution in exponential family such that the posterior obtains the same exponential family form as the prior. Similarities between the proposed method and the extended Kalman filter for nonlinear filtering are illustrated. Furthermore, an extended target measurement update for target models where the target extent is represented by a random matrix having an inverse Wishart distribution is derived. The approximate update covers the important case where the spread of measurement is due to the target extent as well as the measurement noise in the sensor.