I have four alphas for four parameters. The prior probability of each is a Gamma distribution with shape parameter 1/6 and scale parameter 1. The model is that the sum of alpha should equal 1.5. Is it the case that this model will inherently trend to give one of the alphas almost all of the 1.5, and the rest to be very low?
Looking at it from an oversimplified probabilistic perspective, two scenarios one where all the sum of alpha comes almost entirely from a single parameter, and one where sum of alpha comes similarly from all parameters:
Almost all from single parameter:
P(?1 > 1.2) * P(?2 < 0.1)^3 = (0.031) * (0.72)^3 = 0.011
Similarly from all parameters:
P(0.4 >?1 > 0.35)^4 = (0.014)^4 = 0.000000038
As above but with wider constraints on each alpha:
P(0.45 >?1 > 0.3)^4 = (0.043)^4 = 0.0000034
If so will any model constructed with a prior involving summing gamma distributed (with this shape an scale) alphas to equal a certain value lead to results where one of the alphas is responsible for almost the entire sum of alpha?
Looking at it from an oversimplified probabilistic perspective, two scenarios one where all the sum of alpha comes almost entirely from a single parameter, and one where sum of alpha comes similarly from all parameters:
Almost all from single parameter:
P(?1 > 1.2) * P(?2 < 0.1)^3 = (0.031) * (0.72)^3 = 0.011
Similarly from all parameters:
P(0.4 >?1 > 0.35)^4 = (0.014)^4 = 0.000000038
As above but with wider constraints on each alpha:
P(0.45 >?1 > 0.3)^4 = (0.043)^4 = 0.0000034
If so will any model constructed with a prior involving summing gamma distributed (with this shape an scale) alphas to equal a certain value lead to results where one of the alphas is responsible for almost the entire sum of alpha?
