Archive for asymptotic variance

Fisher’s lost information

Posted in Books, Kids, pictures, Statistics, Travel with tags , , , , , , , on February 11, 2019 by xi'an

After a post on X validated and a good discussion at work, I came to the conclusion [after many years of sweeping the puzzle under the carpet] that the (a?) Fisher information obtained for the Uniform distribution U(0,θ) as θ⁻¹ is meaningless. Indeed, there are many arguments:

  1. The lack of derivability of the indicator function for x=θ is a non-issue since the derivative is defined almost everywhere.
  2. In many textbooks, the Fisher information θ⁻² is derived from the Fréchet-Darmois-Cramèr-Rao inequality, which does not apply for the Uniform U(0,θ) distribution.
  3. One connected argument for the expression of the Fisher information as the expectation of the squared score is that it is the variance of the score, since its expectation is zero. Except that it is not zero for the Uniform U(0,θ) distribution.
  4. For the same reason, the opposite of the second derivative of the log-likelihood is not equal to the expectation of the squared score. It is actually -θ⁻²!
  5. Looking at the Taylor expansion justification of the (observed) Fisher information, expanding the log-likelihood around the maximum likelihood estimator does not work since the maximum likelihood estimator does not cancel the score.
  6. When computing the Fisher information for an n-sample rather than a 1-sample, the information is n²θ⁻², rather than nθ⁻².
  7. Since the speed of convergence of the maximum likelihood estimator is of order n⁻², the central limit theorem does not apply and the limiting variance of the maximum likelihood estimator is not the Fisher information.

revisiting the Gelman-Rubin diagnostic

Posted in Books, pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , , on January 23, 2019 by xi'an

Just before Xmas, Dootika Vats (Warwick) and Christina Knudson arXived a paper on a re-evaluation of the ultra-popular 1992 Gelman and Rubin MCMC convergence diagnostic. Which compares within-variance and between-variance on parallel chains started from hopefully dispersed initial values. Or equivalently an under-estimating and an over-estimating estimate of the MCMC average. In this paper, the authors take advantage of the variance estimators developed by Galin Jones, James Flegal, Dootika Vats and co-authors, which are batch mean estimators consistently estimating the asymptotic variance. They also discuss the choice of a cut-off on the ratio R of variance estimates, i.e., how close to one need it be? By relating R to the effective sample size (for which we also have reservations), which gives another way of calibrating the cut-off. The main conclusion of the study is that the recommended 1.1 bound is too large for a reasonable proximity to the true value of the Bayes estimator (Disclaimer: The above ABCruise header is unrelated with the paper, apart from its use of the Titanic dataset!)

In fact, I have other difficulties than setting the cut-off point with the original scheme as a way to assess MCMC convergence or lack thereof, among which

  1. its dependence on the parameterisation of the chain and on the estimation of a specific target function
  2. its dependence on the starting distribution which makes the time to convergence not absolutely meaningful
  3. the confusion between getting to stationarity and exploring the whole target
  4. its missing the option to resort to subsampling schemes to attain pseudo-independence or scale time to convergence (albeit see 3. above)
  5. a potential bias brought by the stopping rule.

single variable transformation approach to MCMC

Posted in Books, Statistics, Travel with tags , , , , on September 9, 2014 by xi'an

I read the newly arXived paper “On Single Variable Transformation Approach to Markov Chain Monte Carlo” by Dey and Bhattacharya on the pleasant train ride between Bristol and Coventry last weekend. The paper actually follows several earlier papers by the authors that I have not read in detail. The notion of single variable transform is to add plus or minus the same random noise to all components of the current value of the Markov chain, instead of the standard d-dimensional random walk proposal of the reference Metropolis-Hastings algorithm, namely all proposals are of the form

x_i'=x_i\pm \epsilon\ i=1,\cdots,d

meaning the chain proceeds [after acceptance] along one and only one of the d diagonals. The authors’ arguments are that (a) the proposal is cheaper and (b) the acceptance rate is higher. What I find questionable in this argument is that this does not directly matter in the evaluation of the performances of the algorithm. For instance, higher acceptance in a Metropolis-Hasting algorithm does not imply faster convergence and smaller asymptotic variance. (This goes without mentioning the fact that the comparative Figure 1 is so variable with the dimension as to be of limited worth. Figure 1 and 2 are also found in an earlier arXived paper of the authors.) For instance, restricting the moves along the diagonals of the Euclidean space implies that there is a positive probability to make two successive proposals along the same diagonal, which is a waste of time. When considering the two-dimensional case, joining two arbitrary points using an everywhere positive density g upon ε means generating two successive values from g, which is equivalent cost-wise to generating a single noise from a two-dimensional proposal. Without the intermediate step of checking the one-dimensional move along one diagonal. So much for a gain. In fine, the proposal found in this paper sums up as being a one-at-a-time version of a standard random walk Metropolis-Hastings algorithm.