Lets imagine I am comparing two groups of animals (treatment/control). There is previous data from cell cultures indicating the treatment should have a positive effect. This gives me "prior component 1". There are also two previous studies very similar to my own. One of them had an effect of 5 +/- 1 (prior component 2), the other of 1 +/- 2 (prior component 3). I feel the cell culture data is highly convincing, and that the prior component 3 is not such a reliable study. So I choose some weights of 3,1, and .5 for each and multiply.
1) To calculate the "overall prior" do I simply add these together as shown in the lower right panel?
2) Am I supposed to normalize these components before adding them?
I then calculate a likelihood function for my current data as shown in the upper panel.
3) How do I combine this information with the prior information shown in the first figure? For the lower panel I simply multiplied overall prior*likelihood.
4) I then want to make a decision based on this outcome. If I believe the effect is between -1 and 1 then I will stop studying the drug. If the effect is < -1 then I would perform new study A, if the effect is > 1 I will perform new study B.
5) Obviously there are a number of ways of choosing a decision (% density between -1 and 1, etc) Is there a best choice?
6) I feel I am doing something incorrectly, but maybe not. Is there a name for what I am trying to accomplish?
If it helps I am trying to use the framework proposed by Richard Royall:
1) The likelihood function tells me "how to interpret this body of observations as evidence"
2) The likelihood function + priors tells be "what I should believe"
3) The likelihood function + priors +cost/benefit determines ""what I should do".
Royall R (1997) Statistical evidence: a likelihood paradigm (Chapman & Hall/CRC)
While the priors used here are subjective/nebulous they are built out of simple building blocks (uniform and normal distributions) that mathematically unsophisticated researchers can understand quickly. I think they convey my thought processes as a researcher well. Others may of course know of different background information. They should be able to build their own "compound prior" which may lead to a different decision than mine, but we should always agree on the likelihood function.
This approach (if implemented correctly, which I am not sure I am doing here), appears to me to model the actual thought processes of researchers and thus be suitable for scientific inference. The steps map to the common sections found in scientific papers. The priors are the introduction, the likelihood is the results, and the posterior probability is the discussion.
#Generate Priors x<-seq(-10,10,by=.1) y1<-dunif(seq(0,10,by=.1), min=-10, max=10) y1<-c(rep(0,length(x)-length(y1)),y1) y2<-dnorm(x, mean=5, sd=1) y3<-dnorm(x, mean=1, sd=2) #Weights for Priors wt1<-3 wt2<-1 wt3<-.5 #Final Priors y1<-y1*wt1 y2<-y2*wt2 y3<-y3*wt3 #Sum to get overall Prior y<-y1+y2+y3 #Likelihood function for "current data" lik<-10*dnorm(x, mean=1, sd=1) #Updated Posterior Probability? prob<-lik*y par(mfrow=c(2,2)) plot(x,y1, ylim=c(0,1), type="l", lwd=4, ylab="Density", xlab="Effect", main="Prior Component 1") plot(x,y2, ylim=c(0,1), type="l", lwd=4, ylab="Density", xlab="Effect", main="Prior Component 2") plot(x,y3, ylim=c(0,1), type="l", lwd=4, ylab="Density", xlab="Effect", main="Prior Component 3") plot(x,y, ylim=c(0,1), type="l", lwd=4, ylab="Density", xlab="Effect", main="Overall Prior") dev.new() par(mfrow=c(2,1)) plot(x,lik, type="l", lwd=4, col="Red", ylab="Likelihood", xlab="Effect", main="Likelihood") plot(x,prob, type="l", lwd=4, col="Blue", ylab="Probability", xlab="Effect", main="Posterior Probability?") abline(v=c(-1,1), lty=2, lwd=3)