**F**ollowing a question on X validated as to why the mean of the log of a uniform distribution is not log(0.5), I replied with the obvious link to Jensen’s inequality and the more general if equally obvious remark that expectation was rarely invariant under transforms and ended up with an high number of up-votes on that answer. Which bemuses me given the basic question and equally basic answer..!

## Archive for Jensen’s inequality

## 45 votes for Jensen’s inequality

Posted in Books, Statistics with tags cross validated, Jensen's inequality, uniform distribution, vote on November 27, 2019 by xi'an## more typos in Monte Carlo statistical methods

Posted in Books, Statistics, University life with tags capture-recapture, EM algorithm, frequentist inference, integer set, Jensen's inequality, missing data, Monte Carlo Statistical Methods, optimisation, typos, UNC on October 28, 2011 by xi'an**J**an Hanning kindly sent me this email about several difficulties with Chapters 3, *Monte Carlo Integration*, and 5, *Monte Carlo Optimization*, when teaching out of our book *Monte Carlo Statistical Methods**[my replies in italics between square brackets, apologies for the late reply and posting, as well as for the confusion thus created. Of course, the additional typos will soon be included in the typo lists on my book webpage.]*:

- I seem to be unable to reproduce
on page 88 – especially the chi-square column does not look quite right.*Table 3.3**[No, they definitely are not right: the true χ² quantiles should be 2.70, 3.84, and 6.63, at the levels 0.1, 0.05, and 0.01, respectively. I actually fail to understand how we got this table*that*wrong…]* - The second question I have is the choice of the U(0,1) in this
. It feels to me that a choice of Beta(23.5,18.5) for*Example 3.6**p*and Beta(36.5,5.5) for_{1}*p*might give a better representation based on the data we have. Any comments?_{2}*[I am plainly uncertain about this… Yours is the choice based on the posterior Beta coefficient distributions associated with Jeffreys prior, hence making the best use of the data. I wonder whether or not we should remove this example altogether… It is certainly “better” than the uniform. However, in my opinion, there is no proper choice for the distribution of the**p*]_{i}‘s because we are mixing there a likelihood-ratio solution with a Bayesian perspective on the predictive distribution of the likelihood-ratio. If anything, this exposes the shortcomings of a classical approach, but it is likely to confuse the students! Anyway, this is a very interesting problem. - My students discovered that
has the following typos, copying from their e-mail: “x_x” should be “x_i”*Problem 5.19**[sure!]*. There are a few “( )”s missing here and there*[yes!]*. Most importantly, the likelihood/density seems incorrect. The normalizing constant should be the reciprocal of the one showed in the book*[oh dear, indeed, the constant in the exponential density did not get to the denominator…]*. As a result, all the formulas would differ except the ones in part (a).*[they clearly need to be rewritten, sorry about this mess!]* - I am unsure about the
*if and only if*part of the**Theorem 5.15***[namely that the likelihood sequence is stationary*if and only if*the Q function in the E step has reached a stationary point]*. It appears to me that a condition for the “if part” is missing*[the “only if” part is a direct consequence of Jensen’s inequality]*. Indeed Theorem 1 of Dempster et al 1977 has an extra condition [*note that the original proof for convergence of EM has a flaw, as discussed here]*. Am I missing something obvious?*[maybe: it seems to me that, once Q reaches a fixed point, the likelihood L does not change… It is thus tautological, not a proof of convergence! But the theorem says a wee more, so this needs investigating. As Jan remarked, there is no symmetry in the Q function…]* - Should there be a (n-m) in the last term of formula
?*(5.17)**[yes, indeed!, multiply the last term by (n-m)]* - Finally, I am a bit confused about the likelihood in
*Example 5.22**[which is a capture-recapture model]*. Assume that H_{ij}=k*[meaning the animal i is in state k at time j]*. Do you assume that you observed X_{ijr}*[which is the capture indicator for animal i at time j in zone k: it is equal to 1 for at most one k]*as a Binomial B(n,p_{r}) even for r≠k?*[no, we observe all X*The nature of the problem seems to suggest that the answer is no_{ijr}‘s with r≠k equal to zero]*[for other indices,**X*If that is the case I do not see where the power on top of (1-p_{ijr}is always zero, indeed]_{k}) in the middle of the page 185 comes from*[when the capture indices are zero, they do not contribute to the sum, which explains for this condensed formula. Therefore, I do not think there is anything wrong with this over-parameterised representation of the missing variables.]* - In Section 5.3.4, there seems to be a missing minus sign in the approximation formula for the variance [
*indeed, shame on us for missing the minus in the observed information matrix!]* - I could not find the definition of in Theorem 6.15. Is it all natural numbers or all integers? May be it would help to include it in Appendix B. [
*Surprising! This is the set of all positive integers, I thought this was a standard math notation…]* - In Definition 6.27, you probably want to say covering of
*A*and not*X*.*[Yes, we were already thinking of the next theorem, most likely!]* - In Proposition 6.33 – all x in A instead of all x in X.
*[Yes, again! As shown in the proof. Even though it also holds for all x in X]*

Thanks a ton to Jan and to his UNC students (and apologies for leading them astray with those typos!!!)