In this paper, we show that the Chapman-Kolmogorov formula could be used as a recursive formula for computing the m-step-ahead conditional density of a Markov bilinear model. The stationary marginal probability density function of the model may be approximated by the m-step-ahead conditional density for sufficiently large m.

This paper studies in particular an aspect of the estimation of conditional probability distributions by maximum likelihood that seems to have been overlooked in the literature on Bayesian networks: The information conveyed by the conditioning event should be included in the likelihood function as well.

The concept of conditional probability has been employed for hundreds of years. Thomas Bayes used the expression "the probability that [B] on the supposition that [A]" in the statement of a basic law (1763, p. 378). Frank Ramsey, developing the application of probability to uncertain epistemic attitudes, the "logic of partial belief (1926, p. 166), wrote of your "degree of belief in [B] given [...