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1) I make confusions about what we call a "sampler". From what I understand, a sampler allows to generate a distribution of points that follows a known PDF (probability distribution function), doesn't it ?

2) On the other side, it exits Monte-Carlo method which allows for example to compute the number Pi by generating random values (x_i,y_i) and see if x_i^2+y_i^2 < R^2. The ratio between accepted points and total points generated will an estimation of Pi.

3) Moreover, I have used previously the Metropolis-Hasting in this simple form, i.e to generate a distribution of points with a known PDF. But I have also seen that we can use it to do estimation of parameters of a model : at which level can we distinguish the "Sampler" Metroplis-Hasting from "Estimation of parameters" method.

4) For example, there is also the acceptance method (called also Von Neumann method), very used in Nuclear physics, which generates also a distribution from a known PDF : can it be qualified also of "sampler" ?

5) Finally, the Markov chain coupled with Monte Carlo (MCMC) is a pure method to estimate the parameters of a model given the data: what is the respective role of Monte-Carlo and the Markov chain in this method.

To summarize, I show you below the problematic in which I am: it is about Forecast in astrophysics. In this post, I am talking about "Inverse problem" in Physics, i.e, we don't predict the data from a very accurate theoretical model but I want to estimate the parameters of my theoretical model given the data I have from experiment or from simulated data (what's we call fake data). The using of Bayes theorem is very practical in this kind of approach since we have a relation of proportionality between the posterior (probability of parameters given the data) and the likelihood (product of PDF taken at data values given the parameters model).

6) Fisher formalism is very useful for estimation of the standard deviation compared to the fiducial values but we need to know these fiducial values before and second point, we have to assume that posterior distribution is always Gaussian, haven't we ? (or that likelihood is Gaussian, I don't remember ... if someone could indicate this assumption).

So as you have seen, there are a lot of concept to integrate for me and I would like to convert this mess into ordered things.

The most important : I would like to make the difference between a "sampler" and an estimator method. After, any remark is welcome to clarify my confusions.

Any help is welcome, sorry for those who find all these questions boring. I think that I am going to start a bounty to clarify all these points.

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  • $\begingroup$ I am having a tough time understanding what you are asking for in the bounty statement. It seems to me like you might be better off asking an entirely new question. $\endgroup$
    – knrumsey
    Commented Jun 26, 2020 at 16:09
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    $\begingroup$ @knrumsey ok, you are surely right, I think that I am going to post a content similar to my UPDATE 1. For wanting multiple precisions or details in explanations at all costs, this may create confusion for the reader. Nevertheless, I let the bounty active in case ... Thanks $\endgroup$
    – user226073
    Commented Jun 26, 2020 at 18:37
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    $\begingroup$ I feel like I dropped 100 for nothing $\endgroup$
    – user226073
    Commented Jul 3, 2020 at 2:50

2 Answers 2

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1.

A sampler (or sampling algorithm) is any procedure which is designed to generate draws from a target distribution $\pi(\cdot)$.

2.

Your understanding seems correct to me. Monte Carlo essentially leverages the Law of Large Numbers. Suppose that $X$ is a distributed according to a distribution $\pi(x)$ and $\theta$ is a scalar quantity $\theta = E(g(X))$ which you would like to estimate.

\begin{align*} \theta &= E(g(X)) \\[1.2ex] &= \int g(x)\pi(x) dx \\[1.2ex] &\approx \frac{1}{M}\sum_{i=1}^Mg(x_i) && \text{(the MC estimator)} \end{align*} where $x_1, x_2, \cdots x_M$ are independent draws from the target distribution $\pi(x)$. Note that Monte Carlo, which is an estimation procedure, always requires that a sampler already exists for a target distribution.

3.

This seems to be where your confusion stems from. The Metropolis-Hastings algorithm (which is an MCMC method) is "just a sampler" which is commonly used for parameter inference in Bayesian statistics. The common use-case may be what's confusing you, so focus on the facts

  • the MH algorithm is used to sample from a target distribution $\pi(x)$, $x \in \mathbb R^d$.
  • Unlike most of the other "samplers" that you mention, the MH algorithm does NOT generate independent draws from the target distribution. Regardless, as the number of samples increase, each draw (in theory) is distributed according to $\pi(x)$. This allows us to estimate $\theta$ in the same way as above (i.e. question 2.).

Due to its many advantages (the target density need not be "normalized", easy to choose a fast "proposal distribution", works well in high dimensions), the MH algorithm is often used to sample from a posterior distribution $\pi(\theta|x)$. These samples from the posterior can then be used for inference, such as parameter estimation. The MH algorithm itself, however, refers to the sampler.

4.

Yes, the accept-reject algorithm is a sampler.

5.

Hopefully this has been mostly answered in the response to question 3. When using an MCMC algorithm to sample from a distribution (usually a posterior), each "sample" depends on the sample before it. That is, the generated samples are not independent, but can be viewed as a Markov Chain. Still, assuming the MCMC sampler has "converged" these draws can be used in the usual Monte Carlo way.

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Samplers are algorithms used to generate observations from a probability density (or distribution) function. Two examples are algorithms that rely on the Inverse Transform Method and Accept-Reject methods.

On the other hand, an estimator is an approximation of an often unknown quantity. Monte Carlo methods refer to a family of algorithms used to obtain these estimations. Monte Carlo methods have the characteristic that they rely on samples from probability distributions to obtain these approximations. This is where the two concepts connect.

Markov Chain Monte Carlo (MCMC) methods combine these two ideas to generate samples and estimate quantities of interest with these samples. Metropolis-Hastings is one of many MCMC algorithms.

For example, if your quantity of interest is the mean of a posterior distribution, this usually means you have to solve an integral. In higher dimensions, solving the integral is often very difficult or even impossible to solve analytically. The idea of MCMC methods is to simulate a sample from the posterior distribution, and then estimate the integral needed to calculate the mean using the average of the sample.

For a friendly introduction to these concepts, I think Introducing Monte Carlo Methods with R by Robert & Casella is a great reference.

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