Memory and the efficient use of information

We consider the problem of how an animal's memory should be designed in order to cope with a stochastic and changing environment. In particular we consider the problem of forming the best estimate of an unknown and possibly changing environmental parameter. Under the simple model we consider, the effect of an observation is to update this estimate using a linear operator. Two models of a changing environment are analysed. For each model we show how estimates change as a function of time elapsed and observations taken. The effect of a regular sequence of observations is also considered, and it is shown that an exponential weighting of past observations is a sufficient statistic on which to base decisions. The weighting factors are different in the two model environments considered, but each is shown to be a function of the rate at which the environment is changing.