Gradient-based proposals for the exchange algorithm without special likelihood assumptions

Develop gradient-based proposal mechanisms within the exchange algorithm for sampling doubly-intractable posteriors that do not rely on special forms of the likelihood p_theta(x), while preserving the correct stationary distribution and remaining practically implementable without access to the normalizing constant or its gradient.

Background

The exchange algorithm targets doubly-intractable posteriors of the form pi(theta|x) proportional to pi(theta) f_theta(x) / Z(theta), where Z(theta) is intractable. Gradient-based proposals (e.g., MALA/HMC) can dramatically improve mixing, but are difficult here because gradients involve the unknown log Z(theta).

A special case admits a workaround when p_theta(x) has an exponential family form, allowing an unbiased estimate of the gradient using synthetic data. However, beyond such special forms, there is no general gradient-based proposal construction known that maintains detailed balance for the exchange algorithm. The authors explicitly state that extending gradient-based proposals to the general case remains open.