Will I ever be a bayesian statistician ? (part 1)
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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
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In order to be sure that we understood, consider now a heads and tails problem, i.e.
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prior=dbeta(u,a,b) posterior=dbeta(u,a+y,n-y+b)The estimator proposed is then the expected value of that conditional distribution,
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On the graphs below, we consider the following heads/tails sample
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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) }
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 ?
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