## a [Gregorian] calendar riddle

Posted in R with tags , , , , , on April 17, 2018 by xi'an

A simple riddle express this week on The Riddler, about finding the years between 2001 and 2099 with the most cases when day x month = year [all entries with two digits]. For instance, this works for 1 January, 2001 since 01=01 x 01. The only difficulty in writing an R code for this question is to figure out the number of days in a given month of a given year (in order to include leap years).

The solution was however quickly found on Stack Overflow and the resulting code is

```#safer beta quantile
numOD <- function(date) {
m <- format(date, format="%m")
while (format(date, format="%m") == m) date <- date + 1
return(as.integer(format(date - 1, format="%d")))
}
dayz=matrix(31,12,99)
for (i in 2001:2099)
for (j in 2:11)
dayz[j,i-2000]=numOD(as.Date(
paste(i,"-",j,"-1",sep=""),"%Y-%m-%d"))

monz=rep(0,99)
for (i in 1:99){
for (j in 1:12)
if ((i==(i%/%j)*j)&((i%/%j)<=dayz[j,i]))
monz[i]=monz[i]+1}
```

The best year in this respect being 2024, with 7 occurrences of the year being the product of a month and a day…

## Le Monde puzzle [#1045]

Posted in Books, Kids with tags , , , on March 19, 2018 by xi'an

An arithmetic Le Monde mathematical puzzle of limited proportions (also found on Stack Exchange):

1. If x,y,z are distinct positive integers such that x+y+z=19 and xyz=p, what is the value of p that has several ordered antecedents?
2.  If the sum is now x+y+z=22, is there a value of p=xyz for which there are several ordered antecedents?
3. If the sum is now larger than 100, is there a value of p with this property?

The first question is dead easy to code

```entz=NULL
for (y in 1:5) #y<z<x
for (z in (y+1):trunc((18-y)/2))
if (19-y-z>z) entz=c(entz,y*z*(19-y-z))
```

and return p=144 as the only solution (with ordered antecedents 2 8 9 and 3 4 12). The second question shows no such case. And the last one requires more than brute force exploration! Or the direct argument that a multiple by κ of a non-unique triplet produces a sum multiplied by κ and a product multiplied by κ³. Hence leads to another non-unique triplet with an arbitrary large sum.

## golden Bayesian!

Posted in Statistics with tags , , , , , , , , , on November 11, 2017 by xi'an

## ghost [parameters] in the [Bayesian] shell

Posted in Books, Kids, Statistics with tags , , , , , , , on August 3, 2017 by xi'an

This question appeared on Stack Exchange (X Validated) two days ago. And the equalities indeed seem to suffer from several mathematical inconsistencies, as I pointed out in my Answer. However, what I find most crucial in this question is that the quantity on the left hand side is meaningless. Parameters for different models only make sense within their own model. Hence when comparing models parameters cannot co-exist across models. What I suspect [without direct access to Kruschke’s Doing Bayesian Data Analysis book and as was later confirmed by John] is that he is using pseudo-priors in order to apply Carlin and Chib (1995) resolution [by saturation of the parameter space] of simulating over a trans-dimensional space…

## simulation under zero measure constraints

Posted in Books, Kids, R, Statistics, University life with tags , , , , , , , on November 17, 2016 by xi'an

A theme that comes up fairly regularly on X validated is the production of a sample with given moments, either for calibration motives or from a misunderstanding of the difference between a distribution mean and a sample average. Here are some entries on that topic:

In most of those questions, the constraint in on the sum or mean of the sample, which allows for an easy resolution by a change of variables. It however gets somewhat harder when the constraint involves more moments or, worse, an implicit solution to an equation. A good example of the later is the quest for a sample with a given maximum likelihood estimate in the case this MLE cannot be derived analytically. As for instance with a location-scale t sample…

Actually, even when the constraint is solely on the sum, a relevant question is the production of an efficient simulation mechanism. Using a Gibbs sampler that changes one component of the sample at each iteration does not qualify, even though it eventually produces the proper sample. Except for small samples. As in this example

``````n=3;T=1e4
s0=.5 #fixed average
sampl=matrix(s0,T,n)
for (t in 2:T){
sampl[t,]=sampl[t-1,]
for (i in 1:(n-1)){
sampl[t,i]=runif(1,
min=max(0,n*s0-sum(sampl[t,c(-i,-n)])-1),
max=min(1,n*s0-sum(sampl[t,c(-i,-n)])))
sampl[t,n]=n*s0-sum(sampl[t,-n])}}``````

For very large samples, I figure that proposing from the unconstrained density can achieve a sufficient efficiency, but the in-between setting remains an interesting problem.