Assume we want to use Gibbs sampling to get an estimate of the parameter , and that we have the following expression for the conditional posterior of :
If I am not mistaken, this means that the conditional posterior is proportional to the PDF of a Gamma distribution, which should then imply that:
where c is the normalising constant for the distribution. In that case, the tricky part comes now. Does the above expression mean that:
or is this where I get lost? A mixture of gut feeling and extensive searches here on Cross Validated and Google in general are telling me that I am missing something, that I have to use the normalising constant c somehow to adjust the observations generated by simulating from . If my gut and browsing history are in the right, exactly what is the adjustment? I've tried to figure it out but I think I might have just stared myself blind at this point.
Also, to avoid as much confusion as possible on my part, in this case, the normalising constant c would be
or have I misunderstood this? (Obs: here, simply denotes the Gamma function, not the Gamma distribution.)