Properties of Total Positivity and an Application to Job Search under Uncertainty

The total positivity of order two is a fundamental property to investigate the sequential decision problem, and it also plays an important role in the Bayesian learning procedure for a partially observable Markov process. For this process, we also deals with a job search, and observe the probability densities on the state space after some additional transitions by employing the optimal policy. This problem is considered as an extension of a job search in a dynamic economy discussed in Lippman and MacCall [8], and we will investigate a problem where the state changes according to a partially observable Markov process. Associated to each state of the process, the wages of a job is a random variable, and information about the unobservable state is obtained through it. All information are summarized by probability distributions on the state space, and we employ the Bayes’ theorem as a learning procedure. By using a property called a total positive of order two, some relationships among information, the optimal policy and the probability density on the state space after some additional transitions are obtained.