# Difference between p(t,alpha,beta|y) and p(t|alpha,beta,y) Hierarchical model example from Gelman

The question surely has broader relevance, however it has come to me while studying 5.3 Bayesian analysis of conjugate hierarchical models of Bayesian Data Analysis (3rd edition) by Gelman.

t will stand for theta, a for alpha and b for beta.
So, we have a collection of j experiments with a binomial distribution yj~Bin(nj,tj). As these experiments are slightly different but related, we establish that the distribution of their parameter tj ~ Beta(alpha, beta). The Beta function is mainly chosen because it is the conjugate of the Binomial.

It follows,
Joint distribution:

P(t,a,b|y) ~ P(a,b)*P(t|a,b)*P(y|t,a,b)


Remember that P(t|a,b) is a Beta distribution and that P(y|t,a,b) is a Binomial distribution, which is too long to write without Latex.

The joint density:
P(t|a,b,y) is defined as solving the multiplication of the beta and binomial distributions, which is something we need no further information to do.

So I fail to see the difference between p(t|a,b,y) and p(t,a,b|y). The book continues dividing these distributions to obtain p(a,b|y)

Any help?