# Changing a conditional probability to a deterministic function

by KRL   Last Updated June 20, 2019 02:19 AM

Suppose that we have a conditional density function $$p(y|x;\theta^*)$$, where $$\theta^*$$ represents distribution parameters and are assumed to be deterministic. Is it possible that we write this conditional density as a deterministic function of $$x$$ and $$\theta$$ where $$\theta$$ is a random variable independent of $$x$$? Furthermore, is this representation unique?

For example, if $$y$$ has a Gaussian distribution with mean $$x$$ and s.d. $$\sigma^*$$, we can write

$$y = x + \sigma,$$

where $$\sigma$$ has a Gaussian distribution with mean zero and s.d. $$\sigma$$. My question might be related to the question discussed here.

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