Multilevel and multicanonical sampling

Choose boundary data B and average the remaining degrees of freedom conditionally on B. The tower property guarantees that the outer expectation of this conditional average equals the original expectation. The variance decomposition separates fluctuation between boundary conditions from noise remaining inside each conditional ensemble. If the inner conditional expectation could be computed exactly, the second contribution would disappear. In practice it is estimated with finite inner samples, so its uncertainty and computational cost remain in the algorithm.

Drawing many inner samples at one fixed boundary does not create many independent outer configurations. The persistent boundary variance gives a floor that only additional outer sampling can reduce. Estimator correlations can also appear if subdomain updates share random numbers or reuse states. Thus a claimed variance improvement must specify both levels’ sampling schedule and compare cost at equal total precision. Conditional averaging preserves correctness only when its conditional measures and the subsequent outer averaging are constructed consistently.

For a local lattice action, holding an adequate separating set of boundary variables fixed can make disjoint interiors conditionally independent. A suitable observable built from factors in those interiors can then use products of separately estimated conditional averages. The boundary must include every variable needed by interactions crossing the separation. Longer-range couplings or a nonlocal effective determinant can invalidate a naive partition, even when the original microscopic theory had local interactions.

In pure lattice gauge theory this idea underlies powerful multilevel estimates of suitable Wilson-loop and related observables. Such demonstrations depend on the action and factorization structure and should not be advertised as an unchanged solution for arbitrary dynamical-fermion observables. At the simple probability level, if X and Y are conditionally independent given B, their conditional product average factorizes. Distinguishing that conditional statement from unconditional independence is essential: averaging over a shared fluctuating boundary can correlate X and Y strongly.

For discrete energy levels with density of states g, assign each configuration a positive multicanonical weight W depending on its energy. The resulting energy marginal is proportional to g times W, so a weight inverse to g makes it flat over the specified range. This differs from assigning equal weight to every configuration, which usually favors energy levels with many states. To recover a canonical expectation, reweight each sampled configuration by its Boltzmann factor divided by W, retaining a normalized ratio.

The density of states is generally unknown and must be estimated or tuned. A clean production design fixes the chosen weights after tuning; an adaptive algorithm needs its own convergence justification rather than an automatic stationary-chain claim during adaptation. Flatness in energy does not ensure rapid passage through barriers in other coordinates, and restricted energy ranges can exclude target support. Positive auxiliary bias also does not erase a fluctuating physical sign. The binomial sample-mean lab below illustrates only the reduction of an independent inner-average variance, without outer boundaries or field configurations.