Conditional Intensities and Likelihoods
The ideas to be discussed in this chapter have been the subject of intensive development during the last two decades, as much by engineers as by mathematicians and statisticians. The underlying theme is the need for a theory of estimation, prediction, and control for point processes. In the late 1960s and early 1970s engineers, in particular, began to exploit a remarkable analogy between point processes and diffusion processes, with the Poisson process playing a role analogous to that of Brownian motion. Early papers by Yashin (1970) in the Soviet Union and Snyder (1972) and Rubin (1972) in the United States explored the analogy between filtering and detection problems for point processes and the Kaiman filtering techniques for signal-from-noise problems in the Gaussian context; the analogy is closest for doubly stochastic (i.e., Cox) processes. The paper by Gaver (1963) may be regarded as some kind of precursor of these developments. These papers were followed by more systematic studies in the theses by Brémaud (1972) and van Schuppen (1973), and papers by Boel, Varaiya, and Wong (1975), Kailath and Segall (1975), and Davis (1976), to mention only a few. On the probabilistic side, the possibility of a powerful link with martingale theory was noted as early as 1964 by Watanabe (1964) who gave a martingale characterization of the Poisson process; the martingale theory was developed further in Kunita and Watanabe (1967). A synthesis of these approaches was presented by Kabanov, Liptser, and Shiryayev (1975) and incorporated in Volume II of Liptser and Shiryayev (1978). Further important reviews are found in Brémaud and Jacod (1977), Brémaud (1981), Shiryayev (1981), and Jacobsen (1982).
KeywordsPoisson Process Point Process Generalize Entropy Predictable Process Conditional Intensity
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