# Can likelihood be changed when the prior changes?

I have a data which follows gamma distribution and want to know the uncertainty of the parameters of this data.

•Data∼Gamma(α,β)

•Parameters

α∼Gamma(kα,θα)

β∼Gamma(kβ,θβ)

I used Winbugs (code below).

model{
for (i in 1:N){
Y[i] ~ dgamma(k, theta)
}

k ~ dgamma(0.1, 0.1)
theta ~ dgamma(0.1, 0.1)
}


1.To plot the likelihood, first, I used a uniform prior then divided the posterior by the prior which makes the posterior same as the likelihood. (Figure1)

2.Next I changed the prior several times and looked what happens, but the problem is when I plotted the likelihood by dividing the posterior by the prior, likelihood changes which shouldn’t, whenever I change the prior. (Figure 2, 3)

Question

Is it possible that the likelihood changes when the prior changes? If the prior is too narrow, is there a possibility that the posterior might be wrong? Can somebody help me what the problem is?

• Why do you think the likelihood would remain constant when you change the prior? As you pointed out, it is proportional to the quotient of the posterior by the prior, so if the prior changes it seems reasonable that the likelihood would as well.
– Matt
Mar 26, 2014 at 18:43
• I'm not good at statistics but in my understanding the likelihood is dependent on data not the prior. Am I wrong?
– moon
Mar 26, 2014 at 20:25
• I think this is a Bayesian/frequentist divide. If you use a classical statistical test, then the likelihood does not depend on anything except the data. If you use a prior, then it does and is equal to the other case with a uniform prior (as your graph shows).
– Matt
Mar 26, 2014 at 20:37