= Solution
A <random-walk update> proposes $\theta'=\theta+\eta$, with an increment distribution independent of the current state. If its density is symmetric, $q(\theta'\mid\theta)=q(\theta\mid\theta')$, reducing the <Metropolis–Hastings acceptance probability> to $\min\{1,\pi(\theta')/\pi(\theta)\}$. 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.
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