[This article was first published on Xi'an's Og » R, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)
Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.

Here is an email from Thomas I received yesterday about a computation in our book Introducing Monte Carlo Methods with R:

I’m currently reading your book “Introduction to Monte Carlo Methods with R” and I quite highly appreciate your work. I’m not able to see how the integral on page 74, that describes the marginal likelihood, simplifies to the fraction on the second line. If I’m not asking too much, could you confirm to me whether the fraction is as is given in the text.

Because the transform of the integral

$m(x) = int_{{mathbb R}^2_+} f(x|alpha,beta),pi(alpha,beta),text{d}alpha text{d}beta$

into the ratio of two integrals

$dfrac{int_{{mathbb R}^2_+} left{ frac{Gamma(alpha+beta)}{Gamma(alpha)Gamma(beta)} right}^{lambda+1}, [x x_0]^{alpha}[(1-x)y_0]^{beta} ,text{d}alpha text{d}beta}{x(1-x),int_{{mathbb R}^2_+} left{ frac{Gamma(alpha+beta)}{Gamma(alpha)Gamma(beta)} right}^{lambda}, x_0^{alpha} y_0^{beta} ,text{d}alpha text{d}beta}$

may sound curious (and possibly wrong) to many readers besides Thomas, let me explain that the bottom integral is the normalisation constant of the prior

$pi(alpha,beta)propto left{ frac{Gamma(alpha+beta)}{Gamma(alpha)Gamma(beta)} right}^lambda, x_0^{alpha}y_0^{beta}$

while the top integral is the product of the observation density:

$f(x|alpha,beta) = dfrac{Gamma(alpha+beta)}{Gamma(alpha)Gamma(beta)},dfrac{x^{alpha}(1-x)^{beta}}{x(1-x)}$

and of the prior (minus the normalisation constant). Nothing wrong then with the formula at the bottom of page 74, but this is a bit short on explanations!

Filed under: Books, R, Statistics Tagged: Introducing Monte Carlo Methods with R, normalising constant, typos

To leave a comment for the author, please follow the link and comment on their blog: Xi'an's Og » R.

R-bloggers.com offers daily e-mail updates about R news and tutorials about learning R and many other topics. Click here if you're looking to post or find an R/data-science job.
Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.

# Never miss an update! Subscribe to R-bloggers to receive e-mails with the latest R posts.(You will not see this message again.)

Click here to close (This popup will not appear again)