How can I compute $P(X_1 = 1|X_0 = 1)$ from the given data?

Question

I know how to mathematically calculate the probability of various Markov Properties.

But, how can I calculate Markov probability from data?

Suppose, I have a Markov Chain as follows:

$S=\{1, 2\}$

$ \alpha = \begin{bmatrix} 0.5&0.5\end{bmatrix} $

$ P = \begin{bmatrix} 0.5&0.5\\ 0&1 \end{bmatrix} $

And, the following data regarding 5 steps of Markov Chain taken 12 times:

 Steps    1    2    3    4    5
 --------------------------------
   1      2    2    1    1    1
   2      1    1    1    1    1
   3      1    1    1    1    1
   4      1    1    1    1    1
   5      2    2    2    1    1
   6      2    2    2    1    1
   7      1    1    1    1    1
   8      1    1    1    1    1
   9      1    1    1    1    1
  10      2    2    2    2    2
  11      1    1    1    1    1
  12      1    1    1    1    1

How can I calculate $P(X_1 = 1|X_0 = 1)$ from this table?

Kindly, explain your answer.

Answer 1

Your question here is at cross-purposes because you have already specified the transition probabilities in your Markov chain as a known quantity, which means that the observed data have no relevance. If you are proceeding under those values then the transition probability in the Markov chain is simply the value you have specified ---i.e., $\mathbb{P}(X_{t+1} = 1|X_t = 1) = P_{1,1} = 0.5$. In this case the data have no input into the calculation, since you have already assumed a known matrix of transition probabilities.

Data enters into consideration only when there is some unknown aspect of the underlying probability mechanism and we are using data to make inferences about this unknown thing. If you want the data to become relevant then you will need to weaken your assumed form so that the transition probability of interest is unknown. In that case, you have a statistical inference problem, so you don't "calculate" the probability from the data, you estimate the unknown probability from the data.