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Suppose I have $K$ classes with distribution $\theta$ over $\{1,...,K\}$ and an underlying domain $D$ on which each class defines a categorical distribution $\phi_i$.

Given a draw $i\sim\theta$ and $x\sim\phi_i$, where only $x$ is observed, I want to update both $\theta$ and the $\phi_i$'s. The posterior for $\theta$ is easy:

$\hat{\theta}(i) = P(i\mid x) \propto P(x\mid i)\cdot P(i) = \phi_i(x)\cdot \theta(i)$

Is it possible to compute a posterior for $\phi_i$ as well? From the definition of the posterior, it seems that it should be:

$\hat{\phi}_i(\tilde{x}) = P(\tilde{x}\mid x, i) \propto P(x\mid \tilde{x}, i)\cdot P(\tilde{x} \mid i) = \phi_i(x) \cdot \phi_i(\tilde{x})$.

but something about that just seems wrong. Shouldn't $\theta$ appear in the numerator somewhere? Am I interpreting the likelihood term wrong?

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2 Answers 2

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You seem to be talking about posterior predictive distribution, i.e. the a posteriori distribution of the data. You don't see $\theta$, because it is marginalized over possible parameter values. The distribution of the data given some particular parameter value is the likelihood function $P(x|i) = \phi_i(x)$.

Regarding your comments, I guess the other thing that you may mean is Bayesian updating. Given some data we update a prior

$$ p(\theta | x) \propto p(x | \theta)\,p(\theta) $$

next, knowing this you can use the posterior as a prior to be updated with new data $\tilde{x}$,

$$ p(\theta|x,\tilde{x}) \propto p(\tilde{x}|\theta,x) \, p(\theta|x) $$

By the chain rule, this can done in one step. So you just plug-in the posterior estimates given $x$ as a prior for the likelihood for $\tilde{x}$, this may be what you are asking about.

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  • $\begingroup$ Yes, I see how the posterior predictive is related, but would still like to condition on $i$ so that I can use it to update the $\phi_i$ parameter. It seems what I want to do is update the likelihood function. Perhaps that's not possible or I need more hyperparameters? $\endgroup$
    – Bill
    Feb 11, 2020 at 15:59
  • $\begingroup$ @Bill likelihood is conditioned on $i$, so I'm not sure what you mean? $\endgroup$
    – Tim
    Feb 11, 2020 at 16:24
  • $\begingroup$ right, but $P(\tilde{x}\mid i)$ is just $\phi_i(\tilde{x})$. I want to update the $\phi_i$'s with something like $\hat{\phi}_i(\tilde{x}) = P(\tilde{x}\mid x,i)$, which I guess is the posterior predictive conditioned on $i$? $\endgroup$
    – Bill
    Feb 11, 2020 at 16:38
  • $\begingroup$ @Bill I can't see how would this differ from likelihood? $\endgroup$
    – Tim
    Feb 11, 2020 at 17:10
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The model considered there is a mixture $$X\sim\sum_{i=1}^K \theta_i \sum_{j=1}^J \phi_{ij}\mathbb{I}_j(x)$$ The posterior distribution on the parameters of the model is thus $$\pi(\theta,\phi|x_1,\ldots,x_n)\propto\pi(\theta,\phi)\prod_{k=1}^n \sum_{i=1}^K \theta_i \sum_{j=1}^J \phi_{ij}\mathbb{I}_j(x_k)$$ with no manageable closed form expression for the marginal posteriors $\pi(\theta|x_1,\ldots,x_n)$ and $\pi(\phi|x_1,\ldots,x_n)$ or even the conditional posteriors $\pi(\theta|x_1,\ldots,x_n,\phi)$ and $\pi(\phi|x_1,\ldots,x_n,\theta)$ (except when $n$ is very small).

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