a funny mistake

While watching the early morning activity in Tofino inlet from my rental desk, I was looking at a recent fivethirthyeight Riddle, which consisted in finding the probability of stopping a coin game which rule was to wait for the n consecutive heads if (n-1) consecutive heads had failed to happen when requested, which is

p+(1-p)p²+(1-p)(1-p²)p³+…

or

While the above can write as

or

hence suggesting

the answer is (obviously) false and the mistake in separating the series into a difference of series is that both terms are infinite. The correct answer is actually

which is Euler’s function. Maybe nonstandard analysis can apply to go directly from the difference of the infinite series to the answer!

asymptotics of M³C²L

In a recent arXival, Blazej Miasojedow, Wojciech Niemiro and Wojciech Rejchel establish the convergence of a maximum likelihood estimator based on an MCMC approximation of the likelihood function. As in intractable normalising constants. The main result in the paper is a Central Limit theorem for the M³C²L estimator that incorporates an additional asymptotic variance term for the Monte Carlo error. Where both the sample size n and the number m of simulations go to infinity. Independently so. However, I do not fully perceive the relevance of using an MCMC chain to target an importance function [which is used in the approximation of the normalising constant or otherwise for the intractable likelihood], relative to picking an importance function h(.) that can be directly simulated.

little people of Tofino [jatp]

optimal approximations for importance sampling

“…building such a zero variance estimator is most of the times not practical…”

As I was checking [while looking at Tofino inlet from my rental window] on optimal importance functions following a question on X validated, I came across this arXived note by Pantaleoni and Heitz, where they suggest using weighted sums of step functions to reach minimum variance. However, the difficulty with probability densities that are step functions is that they necessarily have a compact support, which thus make them unsuitable for targeted integrands with non-compact support. And making the purpose of the note and the derivation of the optimal weights moot. It points out its connection with the reference paper of Veach and Guibas (1995) as well as He and Owen (2014), a follow-up to the other reference paper by Owen and Zhou (2000).

sunrise stars [jatp]