1. Introduction. The present study was motivated by consideration of a" tracking" problem. Observations are taken on the successive positions of an object traveling on a path, and it is desired to estimate its current position. If the path is smooth, regression estimates seem appropriate. If, however, the path is subject to occasional changes in direction, regression will give misleading results long after a naive observer would have made corrections. Our objective is to arrive at a simple formula which implicitly accounts for possible changes in direction and discounts observations taken before the latest change. To develop insight into the nature of a reasonable procedure, we study a simpler problem. In this problem successive observations are taken on n independently and normally distributed random variables X1, X2,..., X. with means ui,, UX** U and variance 1. Each mean, 4i is equal to the preceding mean, 4i-i except when an occasional change takes place. The object is to estimate the current mean jv

We shall study this problem from a Bayesian point of view. First, we assume that the time points of change obey an arbitrary specified a priori probability distribution appropriate to the special case being studied. Second, we assume that the amounts of change in the means, when changes take place, are independently and normally distributed random variables, with mean 0 and variance a2. Third, we assume that the current mean gU is a normally distributed random variable with mean 0 and variance 2. Letting r2 approach infinity, we derive, according to these assumptions, a Bayes estimator of g, U for an a priori uniform distribution on the whole real line and a quadratic loss function. This estimator is invariant under translations of Xi. The minimum variance linear unbiased estimator (MVLU) of Un is also derived. The MVLU estimator is considerably simpler than the Bayes estimator. However, when the expected number of changes in the means is neither zero nor n-1 the Bayes estimator is more efficient than the MVLU one. Generally, the Bayes estimator is very difficult for applications, since it requires many involved computations. A considerable simplification is attainable in the formula for the general Bayes estimator by letting the a-priori variance of the changes,. 2 approach zero. This simplified estimator might not be an efficient one in cases of large changes. As an alternative, we consider the problem where the a priori distribution of time points of change