de Finetti’s theorem and exchangeability

This week, we will start to work on multivariate models, and non-independence. The first idea to discuss non-independence will be to use the concept ofexchangeability. A sequence of random variable http://freakonometrics.blog.free.fr/public/perso5/exch-06.gif is said to be exchangeable if for all http://freakonometrics.blog.free.fr/public/perso5/exch-05.gif

http://freakonometrics.blog.free.fr/public/perso5/exch-01.giffor any permutation http://freakonometrics.blog.free.fr/public/perso5/exch-02.gif of http://freakonometrics.blog.free.fr/public/perso5/exch-03.gif. A standard example is the case wherehttp://freakonometrics.blog.free.fr/public/perso5/exch-07.gif, with

http://freakonometrics.blog.free.fr/public/perso5/exch-08.gifand

http://freakonometrics.blog.free.fr/public/perso5/exch-09.gifSince http://freakonometrics.blog.free.fr/public/perso5/exch-19.gif, a necessary condition is that

http://freakonometrics.blog.free.fr/public/perso5/exch-11.gifi.e.

http://freakonometrics.blog.free.fr/public/perso5/exch-12.gif
Since this inequality should hold for all http://freakonometrics.blog.free.fr/public/perso5/exch-05.gif it comes that necessarily http://freakonometrics.blog.free.fr/public/perso5/exch-13.gif.
de Finetti (1931): Let http://freakonometrics.blog.free.fr/public/perso5/exch-06.gif be a sequence of random variables with values in http://freakonometrics.blog.free.fr/public/perso5/exch-14.gifhttp://freakonometrics.blog.free.fr/public/perso5/exch-06.gif is exchangeable if and only if there exists a distribution function http://freakonometrics.blog.free.fr/public/perso5/exch-15.gif on http://freakonometrics.blog.free.fr/public/perso5/exch-16.gif such that

http://freakonometrics.blog.free.fr/public/perso5/credit-04.gifwhere http://freakonometrics.blog.free.fr/public/perso5/exch-20.gif. Note that http://freakonometrics.blog.free.fr/public/perso5/exch-15.gif is the distribution function of random variable

http://freakonometrics.blog.free.fr/public/perso5/exch-22.gifA nice proof of that result can be found in Heath & Sudderth (1995) – see alsoSchervish (1995)Chow & Teicher (1997) or Durrett (2010) and also probably in several bayesian books because that result has a strong interpretation in bayesian inference (as far as I understood, see e.g. Jaynes (1982)).
From the exchangeability condition, for any permutation http://freakonometrics.blog.free.fr/public/perso5/defi02.gif of http://freakonometrics.blog.free.fr/public/perso5/defi03.gif,

http://freakonometrics.blog.free.fr/public/perso5/defi01b.gifthat can be inverted in

http://freakonometrics.blog.free.fr/public/perso5/defi05.gifThe idea is then to extend the size of the vector http://freakonometrics.blog.free.fr/public/perso5/defi09.gif, i.e. for all http://freakonometrics.blog.free.fr/public/perso5/defi07.gif, define

http://freakonometrics.blog.free.fr/public/perso5/defi10.gifso that, if we condition on http://freakonometrics.blog.free.fr/public/perso5/defi11.gif,

http://freakonometrics.blog.free.fr/public/perso5/defi08.gifbut since given the sum of components of http://freakonometrics.blog.free.fr/public/perso5/defi11.gif, all possible rearrangements of the ones among the http://freakonometrics.blog.free.fr/public/perso5/GPD11.gif elements are equally likely, we can write

http://freakonometrics.blog.free.fr/public/perso5/defi15.gifThe first idea is to work on the blue term, and to invocate a theorem of approximation of the hypergeometric distribution http://freakonometrics.blog.free.fr/public/perso5/defi17.gif to a binomial distribution http://freakonometrics.blog.free.fr/public/perso5/defi19.gif, when http://freakonometrics.blog.free.fr/public/perso5/defi50.gif becomes large. Then

http://freakonometrics.blog.free.fr/public/perso5/defi16.gifLet http://freakonometrics.blog.free.fr/public/perso5/defi20.gif and let http://freakonometrics.blog.free.fr/public/perso5/defi21.gif denote the cumulative distribution function of http://freakonometrics.blog.free.fr/public/perso5/defi28.gif.

http://freakonometrics.blog.free.fr/public/perso5/defi33.gifThe idea is then to write the sum as an integral, with respect to that distribution,

http://freakonometrics.blog.free.fr/public/perso5/defi30.gifThe theorem is then obtained since http://freakonometrics.blog.free.fr/public/perso5/defi31.gif, i.e.

http://freakonometrics.blog.free.fr/public/perso5/defi32.gifIn the case of non-binary sequences, there is an extension of the previous result,
Hewitt & Savage (1955): Let http://freakonometrics.blog.free.fr/public/perso5/exch-06.gif be a sequence of random variables with values in http://freakonometrics.blog.free.fr/public/perso5/exch-24.gif.  http://freakonometrics.blog.free.fr/public/perso5/exch-06.gif is exchangeable if and only if there exists a measure http://freakonometrics.blog.free.fr/public/perso5/exch-25.gif on http://freakonometrics.blog.free.fr/public/perso5/exch-26.gif such that

http://freakonometrics.blog.free.fr/public/perso5/exc99.gifwhere http://freakonometrics.blog.free.fr/public/perso5/exch-25.gif is the measure associated to the empirical measure

http://freakonometrics.blog.free.fr/public/perso5/exch-29.gifand

http://freakonometrics.blog.free.fr/public/perso5/exc98.gifFor instance, in the Gaussian case mentioned earlier, if

http://freakonometrics.blog.free.fr/public/perso5/exch-23.gifthen

http://freakonometrics.blog.free.fr/public/perso5/exch-30.gifwhere

http://freakonometrics.blog.free.fr/public/perso5/exch-31.gifi.e. conditionally on http://freakonometrics.blog.free.fr/public/perso5/exch-32.gif, the http://freakonometrics.blog.free.fr/public/perso5/exch-06.gif are conditionally independent, with distribution http://freakonometrics.blog.free.fr/public/perso5/exch-33.gif. The proof can be found in Kingman (1978) and is based on martingale arguments.
Note that in the Gaussian case, http://freakonometrics.blog.free.fr/public/perso5/excccc02.gif where http://freakonometrics.blog.free.fr/public/perso5/exccc03.gif 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),

 

Assuming a Beta distribution for the latent factor, we can derive the probability distribution of the sum

http://freakonometrics.blog.free.fr/public/perso5/credit-01.gifSince

http://freakonometrics.blog.free.fr/public/perso5/exch61.gifif we assume that – given the latent factor – http://freakonometrics.blog.free.fr/public/perso5/exch67.gif (either the company defaults, or not),

http://freakonometrics.blog.free.fr/public/perso5/exch66.gifi.e.

http://freakonometrics.blog.free.fr/public/perso5/exch63.gifThus, we can derive the (unconditional) distribution of the sum

http://freakonometrics.blog.free.fr/public/perso5/exch60.gifi.e.

http://freakonometrics.blog.free.fr/public/perso5/exch68.gif 

> 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 http://freakonometrics.blog.free.fr/public/perso5/credit-5.gif. 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)

http://freakonometrics.blog.free.fr/public/perso5/exchangeable-beta.gif

Those two theorems are extremely close,

De Finetti’s theorem: a random sequence http://freakonometrics.blog.free.fr/public/perso5/dfhs1.gif of http://freakonometrics.blog.free.fr/public/perso5/dfhs4.gif random variables is exchangeable if and only if http://freakonometrics.blog.free.fr/public/perso5/dfhs2.gif‘s are conditionnally independent, conditionnally on some random variable http://freakonometrics.blog.free.fr/public/perso5/dfhs3.gif.

Hewitt-Savage’s theorem: a random sequence http://freakonometrics.blog.free.fr/public/perso5/dfhs1.gif is exchangeable if and only if http://freakonometrics.blog.free.fr/public/perso5/dfhs2.gif‘s are conditionnally independent, conditionnally on some sigma-algebra http://freakonometrics.blog.free.fr/public/perso5/dfhs5.gif

Olshen (1974), proposed an interesting discussion about those theorems, see also in the Encyclopedia of Statistical Science,

http://freakonometrics.free.fr/copecran1.png

The subtle difference between those two theorem is also discussed in Freedman (1965)

http://freakonometrics.free.fr/copecran2.png


2 thoughts on “de Finetti’s theorem and exchangeability”

  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).

  2. 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).

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