Last week, during the workshop on Statistical Methods for Meteorology and Climate Change (here), I discovered how powerful bayesian techniques could be, and that there were more and more bayesian statisticians. So, if I was to fully understand applied statisticians in conferences and workshops, I really have to understand basics of bayesian statistics. I have published some time ago some posts on bayesian statistics applied to actuarial problems (here or there), but so far, I always thought that bayesian was a synonym for magician. To be honest, I am a Muggle, and I have not been trained as a bayesian. But I can be an opportunist…
So I decided to publish some posts on bayesian techniques, in order to prove that it is actually not that difficult to implement.
As far as I understand it, in bayesian statistics, the parameter is considered as a random variable (which is also the case, in classical mathematical statistics). But here, here assume that this parameter does have a parametric distribution….
Consider a classical statistical problem: assume we have a sample 
i.i.d. with distribution 
. Here we note
since parameter 
is a random variable. The idea is to assume that has a (so called a priori) distribution, e.g.
So far it was simple. The idea is then to consider the posterior distribution of , given the observations 
. Thus, we need to compute the distribution of 
which is here extremely simple (due to properties of the Gaussian distribution), i.e.
where
And them, it becomes extremely natural to consider 
as an estimator of given our sample data (and thus, we also have a confidence interval since we know the distribution of given the observations ).
In order to be sure that we understood, consider now a heads and tails problem, i.e. 
. Note, first, that \theta has support 
. So we need a distribution on that support. Why not a beta distribution ? E.g.
Thus,
and
From Bayes formula,
and we get easily
which is the density of a Beta distribution, i.e.
The estimator proposed is then the expected value of that conditional distribution,
Note that
Further, it is possible to derive confidence intervals using quantiles of the posterior distribution.
On the graphs below, we consider the following heads/tails sample
A first idea is to consider a uniform prior distribution.
A second idea is to consider an asymmetric beta distribution. First, with an asymmetry on the left,
or on the right
Finally a third idea is simply to get back to the standard Gaussian approximation,
If we compare the four models, we obtain (the plain black line is the Gaussian approximated distribution for the empirical mean), and red lines are obtained from prior beta distributions
The code to generate those graphs is the following
a1=1; b1=1
D1[1,]=dbeta(u,a,b)
a2=4; b2=2
D2[1,]=dbeta(u,a,b)
a3=2; b3=4
D3[1,]=dbeta(u,a,b)
setseed(1)
S=sample(0:1,size=100,replace=TRUE)
COULEUR=rev(rainbow(120))
D1=D2=D3=D4=matrix(NA,101,length(u))
for(s in 1:100){
y=sum(S[1:s])
D1[s+1,]=dbeta(u,a1+y,s-y+b1)
D2[s+1,]=dbeta(u,a2+y,s-y+b2)
D3[s+1,]=dbeta(u,a3+y,s-y+b3)
D4[s+1,]=dnorm(u,y/s,sqrt(y/s*(1-y/s)/s))
plot(u,D1[1,],col="black",type="l",ylim=c(0,8),
xlab="",ylab="")
for(i in 1:s){lines(u,D1[1+i,],col=COULEUR[i])}
points(y/s,0,pch=3,cex=2)
plot(u,D2[1,],col="black",type="l",ylim=c(0,8),
xlab="",ylab="")
for(i in 1:s){lines(u,D2[1+i,],col=COULEUR[i])}
points(y/s,0,pch=3,cex=2)
plot(u,D3[1,],col="black",type="l",ylim=c(0,8),
xlab="",ylab="")
for(i in 1:s){lines(u,D3[1+i,],col=COULEUR[i])}
points(y/s,0,pch=3,cex=2)
plot(u,D4[1,],col="white",type="l",ylim=c(0,8),
xlab="",ylab="")
for(i in 1:s){lines(u,D4[1+i,],col=COULEUR[i])}
points(y/s,0,pch=3,cex=2)
plot(u,D4[s+1,],col="black",lwd=2,type="l",
ylim=c(0,8),xlab="",ylab="")
lines(u,D1[1+i,],col="blue")
lines(u,D2[1+i,],col="red")
lines(u,D3[1+i,],col="purple")
points(y/s,0,pch=3,cex=2)
}
Here, we can see that computations are simple if the prior distribution has a distribution which is the conjugate of the observations’ distribution (see here for the list of prior and posterior standard distributions).
So far, I have two questions that naturally show up
- is it possible to start with a neutral prior distribution, non informative ?
- what if we are no longer working with conjugate distributions ?
Well, I guess I have to work a bit more to answer those questions…. to be continued…
The first idea (used in the previous post, 
is somehow the new parameter of the distribution. Then, a conjugate prior
for the parameter 
(where 
is the dimension of 
are i.i. with distribution 
, assume that the a priori distribution of 
is 
, , then the a posteriori distribution of 
is still Beta, with parameters
, assume that the a prioridistribution of 
, assume that the a priori distribution of 
, assume that the a priori distribution of 
, then the a posteriori distribution of 
, assume that the a prioridistribution of 
assume that the a prioridistribution of 
, then the a posteriori distribution of 
assume that 
, then
, assume that the a prioridistribution of 
, then the a posteriori distribution of 
, but 
, where 
(for some bijective transformation). The distribution (density) of 
denote Fisher information of parameter 
), it seems natural to consider
The idea is quite simple. We need a prior distribution so that
