mcmc - 5 Markov Chain Monte Carlo 5.1 Comparing Two Poisson...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
5 Markov Chain Monte Carlo 5.1 Comparing Two Poisson Means Let’s revisit the problem of comparing the means from two independent Pois- son samples. Counts { y Ai } from the weekend days are assumed Poisson with mean λ A and counts { y Bj } from the weekday days are assumed Poisson with mean λ B . We are interested in learning about the ratio of means γ = λ B λ A . We showed that the likelihood function in terms of the first Poisson mean θ = λ A and γ is given by L ( θ, γ ) = exp( - n A θ ) θ s A exp( - n B ( θγ ))( θγ ) s B . Assuming that θ and γ are independent with θ Gamma ( a 0 , b 0 ) , γ Gamma ( a g , b g ) , Then the posterior density of ( θ, γ ) is given, up to a proportionality constant, by g ( θ, γ | data) exp( - n A θ ) θ s A exp( - n B ( θγ ))( θγ ) s B × θ a 0 - 1 exp( - b 0 θ ) γ a g - 1 exp( - b g γ ) By combining terms, we obtain the expression g ( θ, γ | data) exp ( - ( b 0 + n A + n B γ ) θ ) θ a 0 + s A + s B - 1 × exp( - b g γ ) γ a g + s B - 1 .
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
2 5 Markov Chain Monte Carlo Although this is a complicated joint density, the conditional posterior den- sities have familiar expressions. Suppose we fix a value of the first Poisson
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern