Weiss–Weinstein Bound on Multiple Change-Points Estimation

Abstract

In the context of multiple change-points estimation, performance analysis of estimators such as the maximum likelihood is often difficult to assess since the regularity assumptions are not met. Focusing on the estimators variance, one can, however, use lower bounds on the mean square error. In this paper, we derive the so-called Weiss-Weinstein bound (WWB) that is known to be an efficient tool in signal processing to obtain a fair overview of the estimation behavior. Contrary to several works about performance analysis in the change-point literature, our study is adapted to multiple changes. First, useful formulas are given for a general estimation problem whatever the considered distribution of the data. Second, closed-form expressions are given in the cases of Gaussian observations with changes in the mean and/or the variance, and changes in the mean rate of a Poisson distribution. Furthermore, a semidefinite programming formulation of the minimization procedure is given in order to compute the tightest WWB. Specifically, it consists of finding the unique minimum volume covering the set constituted by hyperellipsoid elements that are generated using the derived candidate WWB matrices w.r.t. the so-called Loewner partial ordering. Finally, simulation results are provided to show the good behavior of the proposed bound.

Prior Influence on Weiss-Weinstein Bounds for Multiple Change- Point Estimation

“…However, it has been noticed that in a number of applications (see [6] for instance), the values s a = s b = 1/2 lead to the tightest bound. It is the case for the multiple change-point estimation problem as well, as it has been noticed in [8] after extensive simulations. For this reason, and to simplify the exposition, these values have been set into (6).…”

“…In this paper, we propose to study this prior influence in the context of multiple change-points. A WWB for this problem was recently derived using a particular uniform random walk as the prior [8]. We extend this work here to a wider class of prior distribution for the change locations.…”

“…Let us now enumerate the various prior distributions considered throughout this paper, and give their expressions. Some them have been used in the literature [8]- [10] 1) Uniform on independent segments (UIS) distribution:…”

“…Note that for this problem, the parameter space T is discrete and finite (T ⊂ N Q ), thus the set H and the set of lower bounds W {W (H) | H ∈ H} are discrete and finite as well. Hence, as explained in [8], the supremum of W (that is the tightest WWB) can be obtained by finding the matrix W associated with the minimum volume centered hyper-ellipsoid that covers all the centered hyper-ellipsoids associated with matrices W ∈ W. Such a task can actually be set as a convex optimization problem [11], which can be solved efficiently using an appropriate CVX toolbox.…”

On Bayesian Fisher Information Maximization for Distributed Vector Estimation

“…Then W W B is the supremum of set W, where the supremum operation is taken with respect to Loewner partial ordering [36]. It is worth mentioning the difference between the maximum and the supremum of the set W. The largest element of W, if it exists, is defined as W W * , ∀W ∈ W. On the other hand, the supremum of W is a minimal-upper bound on W that is not necessarily contained in W. This implies that the largest element of W may not exist, but if it exists, it is also the supremum.…”