  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.
