Bayesian Threshold Estimation

Bayesian estimation of a threshold time (hereafter simply threshold) for the receipt of impulse signals is accomplished given the following: 1) data, consisting of the number of impulses received in a time interval from zero to one and the time of the largest time impulse; 2) a model, consisting of a uniform probability density of impulse time from zero to the threshold, and constituting the probability density from which the data impulses are drawn randomly and independently; and 3) a prior probability density of threshold which is linear from a positive initial threshold to one and that is zero otherwise, or which is uniform from zero to one. The posterior probability density of threshold is found with the number of impulses, the time of the largest time impulse, and the initial threshold as parameters, and with the first two parameters fixed, the initial threshold is found that maximizes the entropy of the posterior probability density. It is shown that for some values of initial threshold, including the value that maximizes entropy, the following three estimates of threshold from the posterior probability density are all different: the maximum (maximum a posteriori or MAP estimate), the mean (mean squared error or MMSE estimate), and the maximum for a uniform prior probability density (maximum likelihood or ML estimate). This easily understood and practical Bayesian threshold estimation problem has two aspects of pedagogical interest: 1) the problem is unusual in that the MAP, MMSE, and ML estimates are generally different-in typical problems with Gaussian or uniform models and prior probability densities, at least two of these three estimates are equal; 2) the problem constitutes a clear illustration of how an unknown parameter in the prior probability density (here, initial threshold) may be specified so as to achieve a desired property for the posterior probability density (here, maximum entropy).