Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 224 / 1 / d

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- Solution d

Solution

The Markov property gives

I (X_{1}; X_{4} ∣ X_{3}) = I (X_{2}; X_{4} ∣ X_{3}) = 0

. The chain rule therefore yields

I (X_{1}; X_{3}) - I (X_{1}; X_{4}) = I (X_{1}; X_{3} ∣ X_{4})

and

I (X_{2}; X_{3}) - I (X_{2}; X_{4}) = I (X_{2}; X_{3} ∣ X_{4}) .

Conditional on

X_{4}

, the factorization of the chain still gives

X_{1} \to X_{2} \to X_{3}

. The conditional data-processing inequality thus gives

I (X_{1}; X_{3} ∣ X_{4}) \leq I (X_{2}; X_{3} ∣ X_{4}) .

Rearranging proves

I (X_{1}; X_{3}) + I (X_{2}; X_{4}) \leq I (X_{1}; X_{4}) + I (X_{2}; X_{3}) .

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