Question

In the Metropolis-Hastings algorithm, if the proposal distribution is symmetric, what specific value does the acceptance probability ratio simplify to when moving from the current state to a new state?

Accepted Answer

In the Metropolis-Hastings algorithm, the acceptance probability alpha determines whether to move from a current state x to a proposed state x prime. This probability is defined as the minimum of one and the ratio of the target distribution at x prime multiplied by the proposal distribution of x given x prime, divided by the target distribution at x multiplied by the proposal distribution of x prime given x. A proposal distribution is considered symmetric if the probability of proposing state x prime from state x is identical to the probability of proposing state x from state x prime. When this condition holds, the proposal distribution terms in the numerator and the denominator are equal and cancel each other out entirely. Consequently, the acceptance probability simplifies to the minimum of one and the ratio of the target distribution at x prime divided by the target distribution at x. This specific form is known as the Metropolis algorithm, where the decision to accept or reject the move depends solely on the relative density of the target distribution at the new state versus the current state.

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In the Metropolis-Hastings algorithm, if the proposal distribution is symmetric, what specific value does the acceptance probability ratio simplify to when moving from the current state to a new state?