Let $X=\left\{x_{1},x_{2},x_{3},.....,x_{N}\right\}$ be a sequence of N observations where each $x_{i}$ is either 1 or 0. Now we are trying to fit a first order Markov model with the transition probability \begin{bmatrix} p_{11} &1-p_{11} \\ 1-p_{22} & p_{22} \end{bmatrix} to the data. We calculate the Likelihood $f(X|\theta)$ where $\theta=(p_{11},p_{22})$ which comes out to be $p_{11}^{n_{11}}(1-p_{11})^{n_{12}}p_{22}^{n_{22}}(1-p_{22})^{n_{21}}$. Here $n_{ij}$ is the number of times the markov chain transitioned from state i to state j. Now suppose we are given a prior distribution on the parameters $\theta=(p_{11},p_{22})$. Suppose each of those two parameters come from independent uniform priors. Now we are supposed to calculate the posterior distribution which would be equal to $$p(\theta|X)=\frac{p(X|\theta)*p(\theta)}{p(X)}$$ where $p(X)=\int_{\theta}p(\theta)p(X|\theta)d\theta$. Now we are given that the two parameters come from independent uniform prior. Thus $p(\theta)=1$.
The difficulty I am facing is trying to make sense of the posterior distribution which effectively is $$p(\theta|X)=\frac{p(X|\theta)*1}{p(X)}$$
$$p(\theta|X)=\frac{p_{11}^{n_{11}}(1-p_{11})^{n_{12}}p_{22}^{n_{22}}(1-p_{22})^{n_{21}}}{\int_{\theta}p(\theta)p(X|\theta)d\theta}$$
What is the posterior distribution of the parameters $\theta$ ? And what are the parameters of that posterior distribution ? What does the integral in the denominator calculates to ?