MCMC with using MAP as starting value let $X$ be a random variable from my target distribution $\pi(x)$, which I know up to a normalizing constant, and I want to calculate $Ef(X)$ for some know function $f$. The dimensions of $X$ are around 30, and I wanted to use MCMC to draw samples from $\pi(x)$.
At first, I started with a random starting point with an acceptance rate somewhere between 0.2 and 0.3. Then, I calculate my object of interests
$$\hat{E}f(X) = \frac{1}{N}\sum_{i}^Nf(x_i), $$
where $N$ is the number of draws after burn-in. Then, the issue was that the sample standard deviation of $\hat{E}f(X)$ is too large. I want to reduce that.
To do reduce my sample standard deviation, I thought that maybe starting at MAP solves the problem. So What I did is as follows. I first find MAP
$$x^* = \arg\max_x \log \pi(x).$$
Then, I use transition matrix $q(x'|x_t) = q(x'|x^*)$ where $q(x|y)$ is the normal pdf of $x$ with mean $y$. Then draw a samples using MCMC while keeping acceptance rate around 0.25 and 0.3.
By doing this, I get a significantly smaller sample standard deviation. My guess is that this is because I am only looking around MAP. I am not sure if I can do this and claim that I have a reasonable approximation to my object of interests, especially when dimensions are around 30 and the target $\pi(x)$ may have multi modals.
 A: The dimensionality is not a problem in itself (30 is not that high), but multimodality might very well pose a problem if you have significantly peaked and separate modes.
When you propose new states from the same distribution every time it is typically called independence MCMC, as there is no past dependence on the trajectory of the chain. The problem is for this to work the proposal distribution has to have meaningful support across every region where $\pi$ has mass. It is very likely that unless you significantly increase your standard deviation for your proposal distribution you will only ever see a very small part of the space where $\pi$ is defined. This will clearly give you a very small standard deviation as every value will be close to the MAP.
In general, your idea is not very useful as there is little benefit from localising yourself completely. I'd rather use gradients, or if your problem is significantly multimodal, a tempered SMC sampler (depending on the cost of evaluating your density, as you will need many particles.)
