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.
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)