An obscure integral

Posted on April 7, 2010 by xi'an in R bloggers | 0 Comments

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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