How can I use conditional probability to get: $p(y, x\mid \theta, \phi) = p(y\mid \theta)p(x\mid y, \phi)$, for random variables $y,x,\theta, \phi$

Question

If we have random variables $y, x, \theta, \phi$, how can I use Bayes' Rule to get:

$$ p(y, x\mid \theta, \phi) = p(y\mid \theta)p(x\mid y, \phi) $$?

I tried to factor it out but was stuck and can't seem to get the right combination going. Is there a special trick here? Thanks.

Answer 1

It's likely not correct:

$$p(y, x\mid \theta, \phi) = \frac{p(y, x, \theta, \phi)}{p(\theta, \phi)}$$

$$p(y\mid \theta) = \frac{p(y, \theta)}{p(\theta)}$$

$$p(x\mid y, \phi) = \frac{p(x, y, \phi)}{p(y, \phi)}$$

$$\to p(y\mid \theta)p(x\mid y, \phi) = \frac{p(x, y, \phi)p(y, \theta)}{p(y, \phi)p(\theta)}$$

If we have some additional conditions like something is independent of some other things we could have:

$$\to p(y\mid \theta)p(x\mid y, \phi) = \frac{p(x, y, \phi)p(y) p(\theta)}{p(y) p(\phi)p(\theta)}$$

$$\to p(y\mid \theta)p(x\mid y, \phi) = \frac{p(x, y, \phi) p(\theta)}{ p(\phi)p(\theta)}$$

$$\to p(y\mid \theta)p(x\mid y, \phi) = \frac{p(y, x, \theta, \phi)}{p(\theta, \phi)}$$