Proposed transition matrix for MCMC in two-state Markov Chain

Question

Suppose we would like to model the weather (either sunny $S$ or cloudy $C$) using a two-state Markov Chain, given a set of data collected from 10000 days: $$CCCSSSSSSCCCSSSSSSCCCC...$$

We can use the metropolis-hastings algorithm to get a target transition matrix $P$ whose invariant distribution is given by $$\pi = \left(\frac{\text{# of $S$'s from the data}}{10000}, \frac{\text{# of $C$'s from the data}}{10000}\right)$$ from the data.

Now how do we choose the proposed transition matrix $Q$? Naturally, we don't want to choose $Q$ to be $$Q = \begin{pmatrix} 1/2 & 1/2 \\ 1/2 & 1/2 \\ \end{pmatrix}$$ because then a typical sample from this $P$ would look like $$CCSCSSCSCSCCSCSCCCSCCS...$$ (which is unrealistic for modeling the weather) instead of our data $$CCCSSSSSSCCCSSSSSSCCCC...$$

Is there a systematic way of picking the $Q$ given the data set like above? I know the intuition would be say let
$$Q = \begin{pmatrix} 0.9 & 0.1 \\ 0.1 & 0.9 \\ \end{pmatrix}$$ but the $0.9$ and $0.1$ are still guesses.

Answer 1

There seems to be a major confusion behind the question: MCMC techniques are simulation tools that aim at reproducing generations from a known distribution, e.g., a posterior distribution in a Bayesian setting. They are not inference methods.

In the current setting, if the transition matrix $P$ is the quantity of interest and is inferred from the data at hand, an MCMC method such as Metropolis-Hastings would operate on the space of such transitions matrices, provided a prior distribution on the $P$'s and a sampling distribution on the data, indexed by $P$, are given .