I have a discrete variable $x_a$ for which the likelihood function is Bernoulli. Within the Hybrid Bayesian Network, $Parents(x_a)$ includes both continuous and discrete variables.

Is there a conjugate prior for the two logistic equation parameters $\theta_0$ and $\theta_1$ that I can use for each instantiation of $DiscreteParents(x_a)$?

What have I tried?

I have thought about using a separate linear Gaussian for a) each instantiation of $DiscreteParents(x_a)$ and b) each of the two parameters (i.e. $\theta_0=shape$ and $\theta_1=scale$) in the logistic equation:

$\theta_0 \sim \mathcal{N}(\mu_0, \sigma_0^2)$ and $\theta_1 \sim \mathcal{N}(\mu_1, \sigma_1^2)$, where $\mu_n$ is a linear Gaussian of $ContinuousParents(x_a)$ and $\sigma_n^2$ is known.

What I can't figure out is how to update the prior analytically in the presence of data.


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