Forslagsfordeling

Forslagsfordeling refers to the probability distribution from which candidate states are drawn in a Markov Chain Monte Carlo (MCMC) algorithm. The design of a good proposal distribution is crucial for the efficiency of many sampling methods, especially the Metropolis–Hastings algorithm.

In the Metropolis–Hastings framework, starting from a current state \(x\), a new state \(y\) is proposed by sampling from a proposal density \(q(y|x)\). The proposal is accepted with probability \(\alpha(x,y) = \min\bigl(1, \frac{\pi(y)q(x|y)}{\pi(x)q(y|x)}\bigr)\), where \(\pi\) is the target distribution. If the proposal is symmetric, the ratio \(q(x|y)/q(y|x)\) equals one and the acceptance probability simplifies to the ratio of target densities. As a result, many implementations use symmetric proposals such as a Gaussian centered at the current state.

The choice of proposal impacts both convergence speed and computational cost. A proposal that is too narrow results in high acceptance but slow exploration, whereas a too wide proposal yields low acceptance and wasted computational effort. Adaptive MCMC methods adjust the proposal covariance over time to match the target distribution’s scale. In high-dimensional problems, specialized proposals like the Hamiltonian or slice samplers can exploit gradient or slice information to improve performance.

In practice, the proposal distribution is tuned empirically or through theoretical criteria such as optimal acceptance rates (often around 23% for high-dimensional Metropolis–Hastings). Good proposal design reduces autocorrelation in the Markov chain, leading to more accurate estimates of posterior expectations.