Our main goal for this chapter is to present Bayesian inferences for processes in continuous time. As in earlier chapters, Bayesian ways to estimate parameters, test hypotheses about those parameters, and predict future observations will be developed. There are many examples of Markov chains in continuous times, and the best known is the Poisson process. The Poisson process is a counting process that records many interesting events such as the number of accidents in a given stretch of highway over a selected period, the number of telephone calls at a switchboard, the arrival of customers at a counter, the number of visits to a website, earthquake occurrences in a particular region, etc.