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Suppose the only random number generator that you have is the one for log-normal distributions lnN(x|$\mu,\sigma^2$). Propose an MCMC algorithm for estimating the following integral

\begin{equation} I=\int_0^\infty e^{-x^4-x^6-x^8}\frac{e^x}{\alpha}dx, \end{equation}

where $\alpha=\int_0^\infty e^{-x^4-x^6-x^8}dx$ (is unknown). Describe the algorithm.

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    You might want to google something about monte carlo integration with importance sampling – 2017-01-03
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    What type of MCMC algorithms for integrals do you know? Do you know for instance Metropolis-Hastings? – 2017-01-03
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    This is also a bit underspecified because you can hack together other generators from your log-normal generator. For example, you could make a Metropolis algorithm as follows: given $x_k$, generate $p_k$ as $x_k+ \ln(L_k)$ and $L_k$ is drawn independently from a lognormal distribution with parameters to be determined later. Then use Metropolis update against the pdf proportional to $e^{-x^4-x^6-x^8} 1_{[0,\infty)}(x)$ (which of course does not require you to know $\alpha$). But is it "illegal" to "cheat" by using the logarithm to turn your lognormal variables into normal variables? – 2017-01-03
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    I took the class a few years ago. Now I'm trying to remember it by reviewing some sets of problems. I think it's ok to use logarithm to turn lognormal variables into normal variables, since it's not specified as you mentioned. Thanks for your help! – 2017-01-04
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    The main reason I ask is because there is basically no point in saying you *only* have a lognormal generator if you are allowed to cheat however you want, because taking the logarithm and then using the "norminv" function (i.e. the inverse of the standard normal CDF) gets you uniform random numbers, which is what you usually use for this sort of thing. Yet *that* seems against the spirit of the question. I'm less certain about whether my idea is against the spirit of the question. Perhaps any "cheating" should be only through elementary functions? – 2017-01-04

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