mathcomp7

For a simple user of SO you could only care about de quality of answers, but for someone hiring it can be a very useful information. I agree with you that special care should be taken to avoid SO take a bag stance about new users to keep power.

I have updated the post. In SO I think such a scheme could be implemented for sharing prestige in case people care enough.

On one hand, mathematically speaking, the problem of sampling a finite state system when the probabilities of each state are known is a trivial one, as you know very well since you have shown that method in your posts (computing cumulative probabilities ...) On the other side, your blog is about computation and math, and in the realm of computation the complexity of algorithms is a key factor. Mixing time and burning time of markov chains are also computationally important.

The curse of dimensionality is a cause of Monte Carlo methods to be used instead of other numerical methods in high dimensional problems.

You should communicate to your audience that the advances in stochastic sampling and its application to science has revolutionize the field not because a breathrough in maths but because great key insights that allow us to explore multidimensional problems effectively with the help of computers.

Another important factor is that MCMC is used extensively in Bayesian statistics, but to explain that main application of MCMC in your blog it is necessary to introduce concepts like Bayer's theorem, likelihood, posterior density function, and prior probabilities.

There are some blogs in which all those concepts are illustrated and code in R or python is provided. Perhaps MCMC requires more than one post to be adequately explained.

To sum up, I am a follower of your posts and I appreciate the effort you take to show the main ingredients and the python code. My only desire is for you to continue improving the content of your blog for us to follow and enjoy.

Merry Christmas to you and your family from a follower in Spain.

Let X be the random variable that takes values in natural numbers n>1 with p(X=k)=A/k^3 with constant A such that sum p(X=k)=1, you can't apply MCMC with ypur explanation since you don't have an oracle for p(x=k). To apply MCMC you only need an oracle for the quotient of probabilies, here the transition probabilities are k^3/j^3 and you don't need to compute the unknown constant A.

I recommend the example in the book Doing Bayesian Data Analysis, in this book to motivate MCMC : A politicean want to visit all of US states and each with frequency proportional to the total population of the state. He takes decisions to visit a new state comparing the population of the current and the proposed state, He don't know in advance the total population of every state, but applying MCMC he archieves his goal. Here you may think that computing the total population of the US is a simple detail, but in other problem the computation involves a very difficult multivariable integral with thousand of variables, that in practice is impossible to calculate.

Your article is interesting. But I see a problem in your exposition, in MCMC we don't know the probabilities of individual events. For example let p_n proportional to 1/n^3. We know that 1/n^3 converges but we don't know the sum. The key factor here is that we can compute the transition probabilities without computing the sum of the series. In the continuous variable setting we can compute the transition probabilities without computing difficult integrals. I hope you can incorporate this key factor to improve the quality of your article.