MCMC sampling is often used to produce samples from Bayesian posterior distributions. However, the MCMC method in general associates with computational difficulty and lack of transparency. Specialized computer programs are needed to implement MCMC sampling and the convergence of MCMC calculations needs to be assessed.
A numerical method known as “probability domain simulation (PDS)” (Huang and Fergen 1997) might be an effective alternative to MCMC sampling. A two-dimensional PDS can be easily implemented with Excel spreadsheets (Huang 2020). It outputs the joint posterior distribution of the two unknown parameters in the form of an m×n matrix, from which the marginal posterior distribution of each parameter can be readily obtained. PDS guarantees that the calculation is convergent. Further study of comparing PDS with MCMC is warranted to evaluate the potential of PDS as a general numerical procedure for Bayesian methods.
Huang H 2020 A new Bayesian method for measurement uncertainty analysis and the unification of frequentist and Bayesian inference, preprint, Preprint A new modified Bayesian method for measurement uncertainty a...
Huang H and Fergen R E 1995 Probability-domain simulation - A new probabilistic method for water quality modeling. WEF Specialty Conference "Toxic Substances in Water Environments: Assessment and Control" (Cincinnati, Ohio, May 14-17, 1995), Conference Paper Probability-domain simulation - A new probabilistic method f...
It's the same thing! MCMC produces samples from a ``probability domain''.
The computational difficulties are due to the choice of the function used in defining the accept/reject step.
The novelty in the method in question isn't in the MCMC part; but in the choice of the function used in defining the accept/reject step.
Stam Nicolis
Thank you for your comments. To my knowledge, Markov Chain Monte Carlo (MCMC) is a combination of Markov Chain and Monte Carlo. Since Monte Carlo simulation is a "sampling domain simulation", I think MCMC is also a "sampling domain simulation". Monte Carlo simulation is a counterpart of "probability domain simulation (PDS)". For a dimensional problem (i.e. having two unknown parameters), the 2-dimensional PDS can be implemented with an Excel spreadsheet so it is transparent. I don't think the MCMC sampling for a 2-dimensional problem can be implemented with an Excel spreadsheet. It requires a specialized computer program that may not be as transparent as a Excel spreadsheet. It might be interesting to compare MCMC and PDS for a simple problem.
The Excel spreadsheet isn't transparent-it's exactly the opposite, since it provides the result, without showing how it's obtained. There's no problem of principle in programming a Monte Carlo sampling using Excel; just that the code won't be efficient, since an Excel speadsheet isn't designed for such calculations, when they really get interesting.
What is relevant is that Monte Carlo sampling is sampling of a probability domain.
Cf. Article Introduction To Monte Carlo Algorithms
MCMC in MrBayescan be perfectly transparent as it lists likelihood scores for two simutaneous runs each completed four chains from the beginning of the search through the end. Convergence is not a problem for statistically informative data. Failure to converge is often simply a matter of noisy or insufficient data to estimate the many parameters that form a realistic model -for phylogenies.
As a statistician I rarely encounter any problem with only two parameters. Most phenomena are quite multivariate.
Stam Nicolis
In a Monte Carlo simulation (MCS), we randomly draw (sample) a number from each distribution of input quantities (variables). There is no probability attached to this randomly sampled number. The calculated value of the output quantity does not have a probability either. We repeat the sampling to generate a large sample, from which the distribution of the output quantity can be generated. Therefore, MCS is a sampling domain simulation. In contrast, in a probability domain simulation, the range of an input quantity (or parameter) is divided into intervals. Each interval has a probability attached to it. The calculated value of the output quantity also has a probability. Therefore, the distribution of the output quantity is "directly" generated. This is why this procedure is called "probability domain simulation."
The point of my comment is that the claim about the procedure described in the followup statement being an alternative to Markov Chain Monte Carlo sampling is incorrect, that's all.
Using different words to describe the same procedure doesn't change the procedure.
Aside from the rather confusing statement about transparency, there is also a dangerous assumption here (although indirectly) that is too common in MCMC users:
The software is (usually) not hard to code - the mathematics of convergence and detailed balance is.
Believing you (always) need highly specialised software leads to misplaced trust in the (often insufficient) default setups that are out there. It also stops people from trying to write or understand codes to perform MCMC, which leads to a lot of erroneous conclusions and misuse of the algorithms. Of course, not everyone is a developer, but you should at least understand how to develop something simple - even if it's on paper, or excel - before using it.
There are codes that are difficult (I'm developing one right now that MCMCs over algebras rather than quantities and is of a reversible-jump type), but most of the practical quantitative sampling is a straightforward exercise in your favourite programming language.
I compared PDS with Monte Carlo simulation in a previous study, but did not compare with MCMC.
- Badges
- Science topic
- Similar topics
- Mathematics
- Algebra
- Matrix