In a recent project, I needed to generate random messages according to a Poisson distribution, where the probability that a message occurs time 't' after the previous one is lambda*e^(-lambda*t). So, given a random number generator which draws numbers from a uniform distribution, I needed to map those to the above distribution. A method that I read about was to find the CDF of the target distribution and then find the inverse into which the random numbers could be plugged in. Although this works for exponential distributions, it is not possible for every case (don't have examples off hand) -- how would the Monte Carlo methods work then?

Essentially, Monte Carlo methods employ standard distribution like Poisson of Gaussian, from which samples can be generated easily as analytic formulations of these standard distributions that employ uniform random numbers are available. Then the problem is converted to finding the relationship between the non-standar function and the standard function in a numeric sense, for example by comparing the ratios of function value and the employed standard function value at a particular point to decide whether it will be a suitable sample. I will be going over these during the presentation.

I was interested in the physical-system analogy and use of the Hamiltonian to explore probability space. Since hamiltonian systems are conservative (no dissipative losses), they often fall into simple, periodic orbits. Sometimes though they have quite complex, or even chaotic orbits, depending on initial conditions. I presume one would restart this several times, using a variety of initial conditions? This seems like a neat idea for exploring a manifold! I wonder if it could be adapted for other purposes....?

Is Gibbs Sampling affected by the order in which the conditional probabilities are calculated? As given, first x_1,(t+1) is calculated from P( x_1,t | x_j,t ) where j = 2:K; then x_2,t conditioned on x_1,(t+1) and x_j,t for j = 3:K and so on.

Would things change much (convergence rate, accuracy etc) if the order was changed, say starting with x_K,(t+1)?

Does detailed balance have any bearing on this question?

How useful or reliable are Monte-Carlo methods for estimating definite integrals that are hard to solve analytically?