# Is having a burn-in time relevant when only trying to sample from a distribution?

I'm trying to simulate - via the Metropolis-Hastings algorithm - a sample $$X$$ of size 10000 from a density $$f$$ using a proposal distribution $$g$$.
The Markov chain $$X$$ obtained by this algorithm has the stationary distribution $$f$$, i.e: for every starting point $$x, y\in M$$ we have :
$$P_x(X_n = y) → f(y) \text{ as } n→\infty.$$
A classical step after generating my sample X is to discard the first thousand values or so, so I only have $$X_n$$ with $$n$$ big enough such that $$X_n$$ approximately follows $$f$$.
However, after some reading (here and here), I am under the impression that this is unnecessary if we start from a state $$x_0\in M$$ that should be reached with high probability.
While I think I get the point these texts are trying to make, starting at a large $$n$$ seems absolutely necessary to me so that $$X$$ starting from $$n$$ follows $$f$$.
So, should I skip a thousand values and only consider my chain from then on, or should I inspect the output values and start from the mode of $$f$$ ?

• I find that burn-ins are useful when you don't know much about the (possibly multiple) mode(s) of the distribution. So you start the chain somewhere that is simply convenient and let the burn-in move you to somewhere sensible to start the recorded samples. If you know the mode of $f$ then it's acceptable to start there without a burn-in. If you are worried about damaging the reliability of the results by choosing the first sample, you can always do a burn-in anyway, unless it's prohibitively expensive. – Alex Dec 22 '18 at 12:41