Efficient Computation of Probability Distributions of Pattern Statistics Through Auxiliary Markov Chains - Some Considerations

This paper considers the computation of probability distributions of pattern statistics using an auxiliary Markov chain, and discusses several aspects of the method’s use. A way is given to increase the method’s applicability by minimizing the number of states in the auxiliary Markov chain’s state space. Whereas minimizing state spaces allows computation of pattern distributions that are not feasible otherwise, an example is given where a reduced state space that is not minimal can be more efficient in terms of computation time than first obtaining the minimal state space. We also show, in the context of computing probabilities of accepting a unit in a start-up demonstration test with varied input probabilities, that simple recursive equations may be developed, and computation through these equations is more efficient than using Markov chains in terms of computation time. The bottom line is that while Markov chains can be very effective for computing pattern distributions, wisdom is needed when considering the method’s use.