Question

In Metropolis algorithm, the transition matrix is

Tji = min 1, p(x;) p(x))
3.a Given a distribution
= p = a2 аз
with a₁ < a₂ < a3. Use the Metropolis algorithm to write down the transition matrix. Show that this obeys the detailed balance principle for (i, j) = (1, 2), (2, 3), (1, 3)

Answer 1

The Metropolis algorithm uses a transition matrix to determine the probabilities of transitioning between states. We can write down the transition matrix for a given distribution by calculating the acceptance probabilities for each transition. By showing that the transition matrix satisfies the detailed balance principle, we can demonstrate equilibrium in the system.

Step-by-step explanation:

In the Metropolis algorithm, the transition matrix determines the probability of transitioning from one state to another.

In this case, we have a distribution p = a^2 * a^3, where a₁ < a₂ < a₃.

To write down the transition matrix, we need to calculate the acceptance probabilities for each transition. The acceptance probability for the (i,j) transition is given by:

p(i→j) = min(1, p(xj) / p(xi)

Using this formula, we can calculate the transition matrix for each pair of (i,j). For example, p(1→2) = min(1, (a₂^2 * a₃) / (a₁^2 * a₃))

By calculating the acceptance probabilities for (1,2), (2,3), and (1,3), we can show that the transition matrix satisfies the detailed balance principle, which ensures that the system reaches an equilibrium state.