MAT8886 de Finetti's theorem and exchangeability
By arthur charpentier on Friday, February 10 2012, 00:53 - MAT8886 copulas and extremes - Permalink
is said to be exchangeable if for all
of 
. A standard example is the case where 
, with
, a necessary condition is that 
Since this inequality should hold for all
it comes that necessarily 
.de Finetti (1931): Let
be a sequence of random variables with values in 
. is exchangeable if and only if there exists a distribution function 
on 
such that
. Note that is the distribution function of random variable
From the exchangeability condition, for any permutation
of 
,
, i.e. for all 
, define
,
, all possible rearrangements of the ones among the 
elements are equally likely, we can write
to a binomial distribution 
, when 
becomes large. Then
and let 
denote the cumulative distribution function of 
. 
, i.e. 
Hewitt & Savage (1955): Let
be a sequence of random variables with values in 
. is exchangeable if and only if there exists a measure 
on 
such that
is the measure associated to the empirical measure
, the are conditionally independent, with distribution 
. The proof can be found in Kingman (1978) and is based on martingale arguments. Note that in the Gaussian case,
where 
are i.i.d. random variables. To go further on exchangeability and related topics, see Aldous (1985) (see also here).This construction can be used in credit risk, to model defaults in an homogeneous portfolio, see e.g. Frey (2001),
Since
if we assume that - given the latent factor - 
(either the company defaults, or not),
i.e.
Thus, we can derive the (unconditional) distribution of the sum
i.e.
> proba=function(s,a,m,n){ + b=a/m-a + choose(n,s)*integrate(function(t){t^s*(1-t)^(n-s)* + dbeta(t,a,b)},lower=0,upper=1)$value + }
Based on that function, it is possible to plot the probability distribution over 
. In the upper corner is plotted the density of the Beta distribution.
> a=2 > m=.2 + n=10 + V=rep(NA,n+1) + for(i in 0:n){ + V[i+1]=proba(i,a,m,n)} > barplot(V,names.arg=0:10)
Those two theorems are extremely close,
De Finetti's theorem: a random sequence 
of 
random variables is exchangeable if and only if 
's are conditionnally independent, conditionnally on some random variable 
.
Hewitt-Savage's theorem: a random sequence is exchangeable if and only if 's are conditionnally independent, conditionnally on some sigma-algebra 
Olshen (1974), proposed an interesting discussion about those theorems, see also in the Encyclopedia of Statistical Science,
The subtle difference between those two theorem is also discussed in Freedman (1965)
Comments
For the Bayesian paradigm justification: de Finetti's representation theorem proves that under the exchangeability assumption for the observations, there exist a model and a prior distribution on its parameter, of which the observations are a random sample.
It's interesting to see that an infinite exchangeable sequence necessarily has a positive covariance (not only a Gaussian sequence, as anyway the covariance is constant), and for a finite sequence, it is bounded below by -\sigma^2/(n-1).
thanks Julyan for the comment on the Bayesian paradigm... about the last comment, indeed, exchangeability (infinite exchangeability) is a strong concept of positive dependence. Moshe Shaked introduced the PDM concept (positive dependent by mixture) in 1977, and it is one of the strongest concept of positive dependence... But I don't think I'll have time to discuss too much concepts of (multivariate) stochastic orderings, even if I plan to discuss definitions of positive dependence (and perhaps, also, negative dependence).