Context: we have a large number of individuals characterized by two binary traits; call these $T$ with values $\{0,1\}$, and $T'$ with values $\{0',1'\}$. So there are four types of individuals: $00'$, $01'$, $10'$, $11'$, which appear in the population with unknown relative frequencies $f_{00'}$, $f_{01'}$, $f_{10'}$, $f_{11'}$ summing up to one.

Suppose that our degree of belief about these frequencies (assumed continuous) is expressed by a Dirichlet distribution with parameters $(Aa_{00'}, Aa_{01'}, Aa_{10'}, Aa_{11'})$, the $a$s summing up to one: $$\mathrm{p}[f_{00'}, f_{01'}, f_{10'}, f_{11'} \mid A, (a_{00'}, a_{01'}, a_{10'}, a_{11'})] \propto \prod_{i=0}^1\prod_{j'=0'}^{1'} f_{ij'}^{A a_{ij'}-1}\;\delta\bigl({\textstyle\sum_{ij'}}f_{ij'}-1\bigr).$$

We can also consider the marginal frequencies of individuals having trait $T'$ only, for example: $f_{0'} \equiv f_{00'} + f_{10'}$ and $f_{1'} \equiv f_{01'} + f_{11'}$. Owing to the "aggregation" property of the Dirichlet distribution (Kotz & al 2000, also Basu & al 1982), these marginal frequencies also have a Dirichlet distribution with parameters $\bigl(A(a_{00'} + a_{10'}), A(a_{01'} + a_{11'})\bigr)$ (a Beta distribution).

Question: Consider now the conditional frequencies of trait $T$ given $T'$, for example $$f_{1\mid 0'} \equiv \frac{f_{10'}}{f_{00'}+f_{10'}}.$$ What distribution expresses our degree of belief about such a conditional frequency, given the context above?

While I sit down and calculate (or sample), I'd be grateful for any literature or calculation hints on this. Thank you!

References:

– Basu, de Bragança Pereira: On the Bayesian analysis of categorical data: the problem of nonresponse (1982) https://doi.org/10.1016/0378-3758(82)90004-0, §§ 3–4

– Kotz, Balakrishnan, Johnson: Continuous Multivariate Distributions. Vol. 1 (2nd ed. Wiley 2000), §49.1.