Referring to this thread: How would you explain Markov Chain Monte Carlo (MCMC) to a layperson?.
I can see that it is a combination of Markov Chains and Monte Carlo: a Markov chain is created with the posterior as invariant limiting distribution and then Monte Carlo draws (dependent) are made from the limiting distribution (=our posterior).
Let's say (I know that I am simplifying here) that after $L$ steps we are at the limiting distribution $\Pi$ (*).
The Markov chain being a sequence of random variables, I get a sequence $X_1, X_2, \dots , X_L, \Pi, \Pi, \Pi, \dots \Pi$, where $X_i$ is a random variable and $\Pi$ is the limiting ''random variable'' from which we wish to sample.
The MCMC starts from an initial value, i.e. $X_1$ is a random variable with all mass at that one value $x_1$. If I use capital letters for random variables and small letters for realizations of a random variable, then the MCMC gives me a sequence $x_1,x_2,x_3, \dots x_L, \pi_1, \pi_2, \pi_3, ....\pi_n$. So the length of the MCMC chain is L+n.
[[*Note: the capital letters are random variables (i.e. a whole bunch of outcomes) and the small $x$ are outcomes, i.e. one particular value. * ]]
Obviously, only the $\pi_i$ belong to my ''posterior'' and for approximating the posterior ''well'' the value of $n$ should be ''large enough''.
If I summarise this then I have an MCMC chain $x_1,x_2,x_3, \dots x_L, \pi_1, \pi_2, \pi_3, ....\pi_n$ of length $N=L+n$, only $\pi_1,\pi_2,\dots, \pi_n$ are relevant for my posterior approximation, and $n$ should be large enough.
If I do include some of the $x_i$ (i.e. realizations before the invariant distribution is reached) in the computation of the approximation of the posterior, then it will be ''noisy''.
I know the length of the MCMC chain $N=L+n$, but without knowledge of the $L$, i.e. the step where I am sure to sample from the limiting distribution, I can not be sure that I did not include noise, nor can I be sure about $n=N-L$, the size of my sample from the limiting distribution, in particular, I can not be sure whether it is ''large enough''.
So, as far as I understood, this value of $L$ is of critical importance for the quality of approximation of the posterior (exclusion of noise and a large sample from it).
Are there any ways to find a reasonable estimate for $L$ when I apply MCMC ?
(*) I think that, in general, $L$ will depend on the initial value $x_1$.