 Solution

A random-walk update proposes ${\theta }^{\prime }=\theta +\eta$, with an increment distribution independent of the current state. If its density is symmetric, $q({\theta }^{\prime }\mid \theta )=q(\theta \mid {\theta }^{\prime })$, reducing the Metropolis–Hastings acceptance probability to $\min \{1,\pi ({\theta }^{\prime })/\pi (\theta )\}$. For nonsymmetric increments the proposal-density ratio must remain.

Such updates need little global knowledge of the target and adapt naturally to its local scale. Their limitation is a tuning tradeoff: very small increments accept often but move slowly, while very large increments are often rejected. They can also have difficulty moving between distant modes. Covariance-scaled increments help with anisotropic targets, but the chain still explores through local steps.