The paper studies a probabilistic notion of causes in Markov chains that relies on the counterfactuality principle and probability-raising property. This is motivated by use for monitoring purposes where aim to detect faulty or undesired behaviours before they actually occur. A cause set finite executions system after which probability effect exceeds given threshold. We introduce multiple types...

We study the entropy rate of a hidden Markov process, defined by observing the output of a symmetric channel whose input is a first order Markov process. Although this definition is very simple, obtaining the exact amount of entropy rate in calculation is an open problem. We introduce some probability matrices based on Markov chain's and channel's parameters. Then, we try to obtain an estimate ...

The Metropolis-Hastings algorithm generates correlated samples from a target distribution by constructing a Markov chain which has as its stationary distribution the desired target distribution. One property of this algorithm is that it creates reversible Markov chains. As a result, reversible chains are often used in Monte Carlo simulations. Reversible Markov chains also have the added benefit...

A framework for finite-dimensional quantum Markov chains on Hilbert spaces is introduced. Quantum Markov chains generalize both classical Markov chains with possibly hidden states and existing models of quantum walks on finite graphs. Quantum Markov chains are based on Markov operations that may be applied to quantum systems and include quantum measurements, for example. It is proved that quant...