Archive for completion

Typo in MCSM

Posted in Books, Statistics with tags , , , , on February 20, 2010 by xi'an

I received the following email from Liaosa Xu this morning about Monte Carlo Statistical Methods:

I am reading your text to learn MCMC. However I feel confused for Example 5.14.  I don’t understand how to derive Likelihood of y by the integration of product completed data’s likelihood and conditional density of z given y. Since conditional density of z is just f(z-theta)/(1-F(a-theta)), then we actually integrate f(z-a)^2 over [a, +infinity). How to finally get the likelihood of y? I feel confused about this problem. Could you help me out. Thanks a lot for your answer.

and there is indeed a mistake in the example. The integration of the complete likelihood is

L(\theta|\mathbf{y}) = \int L^p(\theta|\mathbf{y}) f(\mathbf{z}|\mathbf{y},\theta) \text{d}\mathbf{z}

where L^p (\theta|\mathbf{y}) is the part of the likelihood only involving y (this is how it should appear on page 175 of Monte Carlo Statistical Methods). In the current printing, we somehow got confused with the EM completion scheme this example illustrates and wrote

L(\theta|\mathbf{y}) = \mathbb{E}[L(\theta|\mathbf{y,Z})]

and

L(\theta|\mathbf{y}) = \mathbb{E}[L(\theta|\mathbf{y,Z})]= \int L^c(\theta|\mathbf{y},\mathbf{z}) f(\mathbf{z}|\mathbf{y},\theta) \text{d}\mathbf{z}

which is plain wrong, as pointed above by Liaosa Xu.

Ps-If you see the above LaTeX formula for the correct decomposition twice, it is because I wrote it twice in the HTML code. (If you don’t, forget about the following.) For some incomprehensible reason, including the first formula for the complete likelihood erases the two following lines, at least on my output. I tried to play with the entries in the LaTeX formula but did not find a clear culprit!

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