# Conditional PDF of joint density

Assume we have some conditional pdf given by: $$p(x,y\mid z,c)$$

then:

$$p(x,y\mid z,c)=p(x\mid y,z,c)p(y\mid z,c)$$

What is th reasoning behind this formula? Could you give me step by step explanation? What is exactly joint probability density of $x,y$ given some $z,c$ (what properties do we use to derive above formula)?

Second question (similiar in some way): $$p(x\mid z,c)=\int_\theta p(x,\theta\mid z,c) \, d\theta$$

What properties exactly do we use here? Obviously I have intuition behind these formulas (marginal and conditional probability rules), but I want exact properties which were used in derivation above.

## 1 Answer

Ok so I have answer for first question:

Assume that x,y=A and z,c=B, then $$p(x,y|z,c)=p(A|B)=\frac{p(A,B)}{p(B)}=\frac{p(x,y,z,c)}{p(z,c)}=\frac{p(x|y,z,c)p(y|z,c)p(z|c)p(c)}{p(z|c)p(c)}=p(x|y,z,c)p(y|z,c)$$ (by joint probability rule and chain rule of probability)

And I solved second question in similiar manner: $$p(x|z,c)=p(x|A)=\frac{p(x,A)}{p(A)}=\frac{p(x,z,c)}{p(z,c)}=\frac{\int{p(x,\theta,z,c)d\theta}}{p(z,c)}=\frac{\int{p(D,F)d\theta}}{p(z,c)}=\frac{\int{p(x,\theta|z,c)p(z,c)d\theta}}{p(z,c)}=\frac{p(z,c)\int{p(x,\theta|z,c)d\theta}}{p(z,c)}=\int{p(x,\theta|z,c)d\theta}$$

where $D=x,\theta$