## Galton and Watson voluntarily skipping some generations

Posted in Books, Kids, R with tags , , , , , on June 2, 2023 by xi'an

A riddle on a form of a Galton-Watson process, starting from a single unit, where no one dies but rather, at each of 100 generations, Dog either opts for a Uniform number υ of additional units or increments a counter γ by this number υ, its goal being to optimise γ. The solution proposed by the Riddler does not establish his solution’s is the optimal strategy and considers anyway average gains. Solution that consists in always producing more units until the antepenultimate hour (ie incrementing only at the 99th and 100th generations),  I tried instead various logical (?) rules and compared outputs by bRute foRce, resulting in higher maxima (over numerous repeated calls) for the alternative principle

s<-function(p=.66){
G=0;K=1 for(t in 1:9){
i=sample(1:K,1)
K=K+i*(i>=K*p)
G=G+i*(i<K*p)}
return(c(G+sample(1:K,1),K))}

## go forth and X [or the reverse]

Posted in Books, Kids with tags , , , , on February 8, 2023 by xi'an

The New Year Riddle is about optimisation: starting with a single machine, between delivering one unit per machine – hour and delivering one new machine per machine every six days, what is the maximal number of units produced over 100 days?

Comparing the amounts produced by k machines after 6log2(k) days used to multiply the machines showed that 2¹⁵ -1 additional machines were first produced, to generate 7864320 items over the remaining 10 days. Which did not really require an R implementation (although I checked that intermediate solutions where only some of the machines were producing new machines were sub-optimal).

## master project?

Posted in Books, Kids, Statistics, University life with tags , , , , , , , on July 25, 2022 by xi'an

A potential master project for my students next year inspired by an X validated question: given a Gaussian mixture density

$f(x)\propto\sum_{i=1}^m \omega_i \sigma^{-1}\,\exp\{-(x-\mu_i)^2/2\sigma^2\}$

with m known, the weights summing up to one, and the (prior) information that all means are within (-C,C), derive the parameters of this mixture from a sufficiently large number of evaluations of f. Pay attention to the numerical issues associated with the resolution.  In a second stage, envision this problem from an exponential spline fitting perspective and optimise the approach if feasible.

## dice and sticks

Posted in Books, Kids, R with tags , , , , , , on November 19, 2021 by xi'an

A quick weekend riddle from the Riddler about the probability of getting a sequence of increasing numbers from dice with an increasing number of faces, eg 4-, 6-, and 8-face dice. Which happens to be 1/4. By sheer calculation (à la Gauss) or simple enumération (à la R):

> for(i in 1:4)for(j in (i+1):6)F=F+(8-j)
> F/4/6/8
[1] 0.25


The less-express riddle is an optimisation problem related with stick breaking: given a stick of length one, propose a fraction a and win (1-a) if a Uniform x is less than one. Since the gain is a(1-a) the maximal average gain is associated with a=½. Now, if the remaining stick (1-a) can be divided when x>a, what is the sequence of fractions one should use when the gain is the length of the remaining stick? With two attempts only, the optimal gain is still ¼. And a simulation experiment with three attempts again returns ¼.

## data assimilation and reduced modelling for high-D problems [CIRM]

Posted in Books, Kids, Mountains, pictures, Running, Statistics, University life with tags , , , , , , , , , , , , , , , , , on February 8, 2021 by xi'an

Next summer, from 19 July till 27 August, there will be a six week program at CIRM on the above theme, bringing together scientists from both the academic and industrial communities. The program includes a one-week summer school followed by 5 weeks of research sessions on projects proposed by academic and industrial partners.

Confirmed speakers of the summer school (Jul 19-23) are:

• Albert Cohen (Sorbonne University)
• Masoumeh Dashti (University of Sussex)
• Eric Moulines (Ecole Polytechnique)
• Anthony Nouy (Ecole Centrale de Nantes)
• Claudia Schillings (Mannheim University)

Junior participants may apply for fellowships to cover part or the whole stay. Registration and application to fellowships will be open soon.