Proving tautological versus trivial results in mathematics

There is something that might be fun in mathematics, which is the connexion between trivial, tautological and difficult questions. Sometimes, things are so intuitive, that they seem to be obvious. But mathematicians aren’t jedis, and they should not trust too much their intuition… I mean intuition is fine, but it is not a proof. It is like those standard results we learn in topology courses, e.g. “the closure of an open ball is not necessarily the closed ball”. The other thing is that after a while, you try to prove something, until someone makes you realize that it is the definition…

And this morning, while I was trying to make a coffee, @renaudjf came with a simple question (yes, it always starts like that). Consider the standard algorithm to generate a conditional random variable. Assume that  has a priori distribution , and that , given , has (conditional) distribution .

The standard idea is monte carlo simulation, to generate values of , is
  •  step 1: generate 
  •  step 2: given that generation of , generate 
“Can we prove that we actually generate from the (true, maybe hard to characterize) non-conditional distribution of  ? Or is it just trivial ?”. After having those previous philosophical questions, we came to the point that if it was trivial, then we should be able to prove it. A standard way of writing the algorithm is to use the quantile based technique
  •   with ,
  •   with ,
For instance, to generate negative binomial distribution
n=1
theta=rgama(n,3,3)
X=rpois(n,lambda=theta)
Thus, let  where  and  are two independent random variables with a uniform distribution on the unit interval. Let us try to derive its distribution, i.e.
so
if we consider the following change of variate 
which is exactly the non-conditional distribution of .
And then, you’re quite happy because you’ve been able to prove a trivial result ! But next time, I promise, we’ll try to derive an amazing theorem that will change humanity… but next time only, first, let us prove trivial results.


Cite this blog post
Arthur Charpentier (2012, April 27). Proving tautological versus trivial results in mathematics. Freakonometrics. Retrieved April 21, 2024, from https://doi.org/10.58079/oulb

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