I'm a doctor trying to understand Bayesian stats because I'm tired of poking needles in patients when I know that tests don't have excellent specificities and sensitivities.
I need help with a mixture model:
y_a ~ Poisson(θ_a)
y_b ~ Poisson(θ_b)
where θ_a = θ and θ_b = γ and γ is the ratio θ_a/θ_b
θ ~ gamma(a_θ,b_θ)
γ ~ gamma(a_γ,b_γ)
What is the full conditional of θ given y_a, y_b and γ? What is the full conditional of γ given y_a, y_b and θ?
I'm told that p(θ|y_a, y_b, γ) ≡ p(θ|y_a) but I don't understand how that is possible under this parameterization. Or is that wrong?
Thanks for help.