Poisson Equation
A central equation in Markov chain theory, of the form (I−P)g = f − π(f). Its solution corresponds to the optimal control variate and captures the cumulative correlations of the Markov chain to eliminate redundancy between adjacent samples. Closed-form solutions are generally unavailable for high-dimensional target distributions.
Timeline (last 90 days)
A truly effective control variate requires solving the Poisson equation of the sampler, which has no closed-form solution in the vast majority of cases.
The Poisson equation takes the form (I - P)g = f - π(f), where P is the transition kernel, π(f) is the expectation of f under the stationary distribution, and the solution g is the desired control variate.
A closed-form solution to the Poisson equation exists only when the transition kernel P has an analytically describable eigenfunction expansion or a known spectral decomposition.
The paper proves that the Markov kernel and the solution to its Poisson equation are correspondingly transformed under any bijection.
All Facts (4)
A truly effective control variate requires solving the Poisson equation of the sampler, which has no closed-form solution in the vast majority of cases.
50%UnverifiedThe Poisson equation takes the form (I - P)g = f - π(f), where P is the transition kernel, π(f) is the expectation of f under the stationary distribution, and the solution g is the desired control variate.
50%UnverifiedA closed-form solution to the Poisson equation exists only when the transition kernel P has an analytically describable eigenfunction expansion or a known spectral decomposition.
50%UnverifiedThe paper proves that the Markov kernel and the solution to its Poisson equation are correspondingly transformed under any bijection.
50%