To show that $p(x, y \mid z) = p(x \mid z) p(y \mid z)$ when $x$ and $y$ are conditionally independent given $z$, we can proceed step by step as follows:

Step 1: Definition of Conditional Independence

Conditional independence between $x$ and $y$ given $z$ means that the occurrence of $x$ does not affect the probability of $y$ happening, given that $z$ has occurred. Mathematically, this is expressed as: $p(x \mid y, z) = p(x \mid z)$ Similarly, you can write: $p(y \mid x, z) = p(y \mid z)$

Step 2: Joint Probability and Conditional Probability

The joint probability $p(x, y \mid z)$ can be written as: $p(x, y \mid z) = p(x \mid y, z) \cdot p(y \mid z)$ This is a direct application of the chain rule of probability.

Step 3: Apply Conditional Independence

Using the conditional independence $p(x \mid y, z) = p(x \mid z)$, substitute this into the equation: $p(x, y \mid z) = p(x \mid z) \cdot p(y \mid z)$ This result shows that the joint conditional probability $p(x, y \mid z)$ factors into the product of the individual conditional probabilities $p(x \mid z)$ and $p(y \mid z)$.

Conclusion

Thus, we have proven that: $p(x, y \mid z) = p(x \mid z) \cdot p(y \mid z)$ when $x$ and $y$ are conditionally independent given $z$.

Would you like further details or have any questions?

Here are 5 relative questions to consider:

1. What is the significance of conditional independence in probability theory?
2. How would the result change if $x$ and $y$ were not conditionally independent given $z$?
3. Can you give an example where conditional independence is used in real-world scenarios?
4. How does the chain rule of probability relate to Bayes' Theorem?
5. What other properties follow from the assumption of conditional independence?

Tip: Always ensure to clearly state the assumptions like conditional independence when solving probability problems, as they simplify the derivation process significantly.