The equation you provided is:

$p(x, y | z) = p(x | z) p(y | z)$

This equation represents the conditional independence of $x$ and $y$ given $z$. In other words, it suggests that if we know $z$, then knowing $x$ does not give us any additional information about $y$ and vice versa. Let's go step by step to prove this.

Step 1: Conditional Joint Distribution

The left-hand side of the equation represents the joint probability of $x$ and $y$ conditioned on $z$. By the definition of conditional probability, the joint probability can be expressed as:

$p(x, y | z) = \frac{p(x, y, z)}{p(z)}$

where $p(x, y, z)$ is the joint distribution of $x$, $y$, and $z$, and $p(z)$ is the marginal distribution of $z$.

Step 2: Conditional Independence Assumption

Now, let's assume that $x$ and $y$ are conditionally independent given $z$. This means:

$p(x, y | z) = p(x | z) p(y | z)$

This is precisely the equation we are trying to prove.

Step 3: Apply the Chain Rule for Probabilities

By the chain rule for conditional probabilities, we know that the joint distribution of two variables given a third variable $z$ can be written as:

$p(x, y | z) = p(x | y, z) p(y | z)$

This says that the probability of $x$ and $y$ given $z$ can be factored into the conditional probability of $x$ given both $y$ and $z$, and the probability of $y$ given $z$.

Step 4: Conditional Independence Implies Simplification

Under the assumption that $x$ and $y$ are conditionally independent given $z$, knowing $y$ does not affect the probability of $x$ given $z$. This implies:

$p(x | y, z) = p(x | z)$

Thus, the joint probability simplifies as:

$p(x, y | z) = p(x | z) p(y | z)$

Conclusion

Therefore, assuming that $x$ and $y$ are conditionally independent given $z$, the expression:

$p(x, y | z) = p(x | z) p(y | z)$

holds true, and this completes the proof.

Would you like further details on conditional independence or any other clarification? Here are some related questions for further exploration:

1. What is the chain rule for conditional probabilities?
2. How does conditional independence simplify complex probability calculations?
3. Can you provide a practical example of conditional independence?
4. What happens when variables are not conditionally independent?
5. How does this concept relate to Bayesian networks?

Tip: When working with conditional probabilities, it's important to clearly identify the conditioning variable and any assumptions (like independence) to simplify the expressions.