I'm reading Performance Modeling and Design of Computer Systems which contains some analysis of Markov chains. In particular, it emphasises various analytical methods for finding the stationary distribution of a Markov chain.
Here's what confuses me: when I have formally studied computer-intensive statistics (more specifically, Monte Carlo integration) in the past, a core theme has been about constructing Markov chains that have stationary distributions that approximate the distribution you want to draw from, and then running those Markov chains to get draws from the desired distribution.
So here we have one source preferring to analytically derive the distribution implied by a Markov chain, and another that prefers to imply a Markov chain and then run it to get draws from a distribution.
It seems much easier to me to just run the chain -- so why is it that is not preferred by some statisticians?
I get the argument for analytic solutions in some edge cases where running the chain might have extreme computational cost -- but that has not been the case in the examples I've seen.
Will I make a mistake in practical terms if I choose to run the chain instead of derive probabilities analytically?