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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.

Prove step by step the equation p(x, y | z) = p(x | z)p(y | z)

This step-by-step guide walks through the proof of conditional independence: p(x, y | z) = p(x | z)p(y | z). It includes explanations of the chain rule for probabilities and simplifying assumptions, designed for undergraduate students or advanced high school learners.

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