Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 28L / b

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- Solution b

Solution

Choose an initial value

X_{0}

. One sweep of the Gibbs sampler consists of

Y_{m} \sim f_{Y ∣ X} (\cdot ∣ X_{m - 1}), X_{m} \sim f_{X ∣ Y} (\cdot ∣ Y_{m}) .

After a burn-in period, the pairs

(Y_{m}, X_{m})

are retained as approximate samples. Standard irreducibility and recurrence conditions are needed for convergence from the chosen starting point.

The transition density from

(y, x)

to

(y^{'}, x^{'})

is

k ((y, x), (y^{'}, x^{'})) = f_{Y ∣ X} (y^{'} ∣ x) f_{X ∣ Y} (x^{'} ∣ y^{'}) .

Suppose

(Y, X)

currently has density

f_{X Y}

. The density after one sweep is

\int_{R^{2}} f_{X Y} (y, x) f_{Y ∣ X} (y^{'} ∣ x) f_{X ∣ Y} (x^{'} ∣ y^{'}) d y d x = f_{X ∣ Y} (x^{'} ∣ y^{'}) \int_{R} f_{X} (x) f_{Y ∣ X} (y^{'} ∣ x) d x = f_{X ∣ Y} (x^{'} ∣ y^{'}) f_{Y} (y^{'}) = f_{X Y} (x^{'}, y^{'}) .

This proves the stationarity of the two-coordinate Gibbs sampler:

f_{X Y} is stationary for the Gibbs transition kernel .

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