# A comparison between the convergence of Metropolis-Hastings and INDEPENDENT Metropolis-Hastings algorithm

I am trying to explore the Metropolis-Hastings algorithm but I am wondering about the characteristics of this sampling method. It is known about the Metropolized independent sampling that if we designed everything well then for sure it will converge to the target when sample sizes grows to infinity according to this paper J. Liu 1995.

My question is:

Suppose we designed two Markov chains and sampled in the first time using the original M-H and in the second time using the independent M-H, which chain will converge quicker under the same conditions (i.e mean by same conditions: same sample size, and the same distributions...etc ). I think that the metropolized independent should converge quicker because we sampled independent objects although the whole chain is dependent since we are using MCMC which generate repeated samples Robert & Casella 2010.

I tried to find a reference to prove or reject this but I couldn't!!

I am so sorry if there is any ambiguity or anything wrong but as I said I am trying to understand it.

Any kind of help is appreciated.

Suppose we designed two Markov chains and sampled in the first time using the original M-H and in the second time using the independent M-H, which chain will converge quicker under the same conditions

The thing with Markov chain Monte Carlo is that the convergence rate for a sampler is most dependent on the target distribution. There is not natural ordering as to whether a usual M-H sampler converges faster or slower than an Independent M-H. In addition, it also depends on the choice of proposal distributions for both the samplers.

Here is a simple example. Suppose the target distribution is the normal distribution with mean 5 and variance 2, i.e, $N(5,2)$. We will use two different M-H proposals and two different independent M-H proposals. For the first we use

• M-H proposal: $N(\cdot, 9)$
• Independent M-H proposal: $N(0, 4)$.

For the second we use

• M-H proposal: $N(\cdot, .0025)$
• Independent M-H proposal: $N(4.8, 3)$.

I ran all four samplers for 100,000 iterations. Below are the density estimates and their comparison to the truth.

In the first plot, because the proposal distribution for Independent M-H is far from the truth, it only rarely proposes value that are good, and thus doesn't seem to converge fast.

In the second plot, the variance in the M-H proposal is so small, that it is not able to propose values far away and explore the space well, leading to slower convergence.

One could further play around with different target distributions and see how the samplers behave. It will end up being problem specific.

I think that the metropolized independent should converge quicker because we sampled independent objects although the whole chain is dependent

Both the regular M-H and independent M-H sample independent objects when they draw from the proposal. Generally, when independent M-H sampler works, it works really well. But it is very difficult to get it to work in real settings since you really need to know a low about the target distribution. You need to know where a lot of the mass is, so that you can use a proposal density centered around that mass. This can be very hard for higher dimensional problems.

• thanks for answer but I have two comments: The first one: I said previously that sampling done under the same conditions so we suppose the same target distribution for both methods? so what do you think will happen if we supposed in your previous example that the target distribution is N(5,2) and the proposal distribution is N(0,4)? which one will converge quicker? – Noah16 Jul 26 '17 at 12:25
• You said: "Both the regular M-H and independent M-H sample independent objects when they draw from the proposal" but this is not true in view of what is written in wikipedia "The samples are correlated. Even though over the long term they do correctly followP(x), a set of nearby samples will be correlated with each other and not correctly reflect the distribution."[en.wikipedia.org/wiki/Metropolis%E2%80%93Hastings_algorithm] – Noah16 Jul 26 '17 at 12:28
• @Noah16 First point: I don't quite understand what you mean. The target distribution is the same for all four samplers, it is $N(5, 2)$ for all samplers. A proposal of $N(0,4)$ is not dependent on the current position of the Markov chain, so it yields an independent M-H. I can't use the same proposal for a random walk metropolis, because that has to depend on the current position of the Markov chain, so I used $N(\cdot, 9)$ where $\cdot$ means the current position. I can change 9 to be 4, if that is what you're asking. – Greenparker Jul 26 '17 at 16:48
• @Noah16 Second: When the samplers draw from the proposal distributions, the sampling is independent. The resulting samples obtained after doing the Metropolis-Hastings ratio step, are indeed correlated. – Greenparker Jul 26 '17 at 16:50
• Tahnks a lot, I understand it now but didn't understand what do you mean by "the sampling is independent"? – Noah16 Jul 27 '17 at 10:41