An obscure integral

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! On the other hand, the top of page 75 is missing an x(1-x) (and so does the R code below… See this post.)

2 Responses to “An obscure integral”

  1. Edward Kao Says:

    The middle part of p. 75 (two lines or R-code) does not work.
    Error message: Error in a[, 1] : incorrect number of dimesions

  2. Hi,
    The readility of blog is very low!!! Not able to see equations properly ..

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