A note on the differential equations of conditional probability density functions

Abstract

In practice we quite often have to deal with a noise-corrupted signal of the type V(t) = X(t) + b(l) which re resents an inaccurate observation p (or measurement) of the parameter X(t). Quite often s(t) is an additive wide-band noise and is usually described as “white.” The statistical properties of both X(t) and B(t) are known a priori. On observing q(t) we obtain more accurate information about the values of the parameter to be estimated, X(t). It is now described by the “posterior” probability density function (p.d.f.) which is conditional upon observation. The equation governing the evolution of the conditional p.d.f. contains the function of observation as a driving term and describes in mathematical terms the “updating” effect while observing the signal V(t). By choosing a criterion of optimality (rms, maximum posterior probability etc.) one can obtain another dynamic equation for the estimate of the parameter x(t). The optimal filter may be then designed on the basis of that equation. The differential equation of conditional p.d.f.‘s was first obtained for the case of a Markovian parameter and an additive Gaussian wide-band noise by R. L. Stratonovich [l, 21. A somewhat different equation has been obtained by H. J. Kushner in [3]. The latter paper contains one or two misleading statements upon the nature of contradictions between the results of [l, 21, on the one hand, and those of [3], on the other. It is therefore thought desirable to look more closely into both “conflicting” equations (96) of [l] and (2.8) of [3] and investigate if any “errors due to the omission of certain significant terms” have been committed in [l], as Kushner alleges.