## Archive for the Kids Category

Posted in Kids, pictures, Statistics, University life with tags , , , , , , , , , , on October 29, 2020 by xi'an

## inverse Gaussian trick [or treat?]

Posted in Books, Kids, R, Statistics, University life with tags , , , , , , , , , , , , , , on October 29, 2020 by xi'an

When preparing my mid-term exam for my undergrad mathematical statistics course, I wanted to use the inverse Gaussian distribution IG(μ,λ) as an example of exponential family and include a random generator question. As shown above by a Fortran computer code from Michael, Schucany and Haas, a simple version can be based on simulating a χ²(1) variate and solving in x the following second degree polynomial equation

$\dfrac{\lambda(x-\mu)^2}{\mu^2 x} = v$

since the left-hand side transform is distributed as a χ²(1) random variable. The smallest root x¹, less than μ, is then chosen with probability μ/(μ+x¹) and the largest one, x²=μ²/x¹ with probability x¹/(μ+x¹). A relatively easy question then, except when one considers asking for the proof of the χ²(1) result, which proved itself to be a harder cookie than expected! The paper usually referred to for the result, Schuster (1968), is quite cryptic on the matter, essentially stating that the above can be expressed as the (bijective) transform of Y=min(X,μ²/X) and that V~χ²(1) follows immediately. I eventually worked out a proof by the “law of the unconscious statistician” [a name I do not find particularly amusing!], but did not include the question in the exam. But I found it fairly interesting that the inverse Gaussian can be generating by “inverting” the above equation, i.e. going from a (squared) Gaussian variate V to the inverse Gaussian variate X. (Even though the name stems from the two cumulant generating functions being inverses of one another.)

## artificial EM

Posted in Books, Kids, R, Statistics, University life with tags , , , , , , on October 28, 2020 by xi'an

When addressing an X validated question on the use of the EM algorithm when estimating a Normal mean, my first comment was that it was inappropriate since there is no missing data structure to anchor by (right preposition?). However I then reflected upon the infinite number of ways to demarginalise the normal density into a joint density

$$∫ f(x,z;μ)dz = φ(x–μ)$$

from the (slice sampler) call to an indicator function for $$f(x,z;μ)$$ to a joint Normal distribution with an arbitrary correlation. While the joint Normal representation produces a sequence converging to the MLE, the slice representation utterly fails as the indicator functions make any starting value of $$μ$$ a fixed point for EM.

Incidentally, when quoting from Wikipedia on the purpose of the EM algorithm, the following passage

Finding a maximum likelihood solution typically requires taking the derivatives of the likelihood function with respect to all the unknown values, the parameters and the latent variables, and simultaneously solving the resulting equations.

struck me as confusing and possibly wrong since it seems to suggest to seek a maximum in both the parameter and the latent variables. Which does not produce the same value as the observed likelihood maximisation.

## help for Cox’s Bazar

Posted in Kids, pictures, Travel with tags , , , , , , , on October 25, 2020 by xi'an

## Nous continuerons, Professeur.

Posted in Kids, pictures with tags , , , , , , , , on October 24, 2020 by xi'an

“Nous continuerons, Professeur. Avec tous les instituteurs et professeurs de France, nous enseignerons l’histoire, ses gloires comme ses vicissitudes. Nous ferons découvrir la littérature, la musique, toutes les œuvres de l’âme et de l’esprit. Nous aimerons de toutes nos forces le débat, les arguments raisonnables, les persuasions aimables. Nous aimerons la science et ses controverses. Comme vous, nous cultiverons la tolérance. Comme vous, nous chercherons à comprendre, sans relâche, et à comprendre encore davantage cela qu’on voudrait éloigner de nous. Nous apprendrons l’humour, la distance. Nous rappellerons que nos libertés ne tiennent que par la fin de la haine et de la violence, par le respect de l’autre.”

Emmanuel Macron, 21 October 2020