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By Andrew P. Sage and James L. Melsa (Eds.)

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MAXIMUM A POSTERIORI IDENTIFICATION 47 where du ( t ) is a Wiener process. I n a similar manner, results of Eqs. 1-13) should be obtained rigorously from the stochastic calculus (Sage and Melsa, 1971). T h e foregoing comments also apply to the limiting procedure used in the remainder of the chapter. , z(k,) are denoted by X(k,) and Z(k,), respectively. I n like manner, the continuous values of x(t) and z ( t ) in the interval [ t o ,tt] are denoted by X(t,) and Z(t,). p [ X ( k f )1 Z(k,)] and p [ X ( t , ) [ Z(t,)] denote the conditional probability density function of X given measurements Z .

Almost all of the existing search techniques (Wilde, 1964) have been applied at one time or another in learning model identification with varying degrees of success. ‘The two most popular and most successful have been random search and “gradient” techniques. T h e random search approach is very effective in discarding local minima and obtaining solutions on “bas” error surfaces. T h e gradient techniques] are relatively easy to program and can be quite effective although they seek only local minima.

We will now turn our attention to this problem and will consider nonlinear system dynamics. 2. 2. MAXIMUM A POSTERIORI IDENTIFICATION I n this section, we will examine the Bayes maximum a posteriori (MAP) approach to generalized estimation or system identification. T h e n we will show that many system identification problems may be cast in the framework of maximum a posteriori estimation. We will show that, with Gaussian a priori statistics, the maximum a posteriori estimate is equivalent to an appropriate least squares curve fit estimate.

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