Free energy evaluation using marginalized annealed importance sampling

The evaluation of the free energy of a stochastic model is considered a significant issue in various fields of physics and machine learning. However, the exact free energy evaluation is computationally infeasible because the free energy expression includes an intractable partition function. Annealed importance sampling (AIS) is a type of importance sampling based on the Markov chain Monte Carlo method that is similar to a simulated annealing and can effectively approximate the free energy. This study proposes an AIS-based approach, which is referred to as marginalized AIS (mAIS). The statistical efficiency of mAIS is investigated in detail based on theoretical and numerical perspectives. Based on the investigation, it is proved that mAIS is more effective than AIS under a certain condition.

Figure 1

State transition, from the previous to next states, according to Eq. (39).

Figure 2

Square-grid graph can be regarded as a bipartite graph.

Figure 3

State transitions on a bipartite-type MRF based on the one-sample approximation of Eqs. (44) and (45): The transitions of (a) mAIS and (b) AIS. The transitions between the layers (illustrated by the arrows) are the blocked Gibbs sampling based on $P_k\left(\mathit{h}\right|\mathit{v})$ and $P_k\left(\mathit{v}\right|\mathit{h})$.

Figure 4

APEs versus $1/T$ for $K=10$, 30, and 60. These plots represent the average values over 1000 experiments.

Figure 5

Distributions of $lnr$ for several $\alpha$ values when $1/T=0.5$: (a) $K=10$, (b) $K=30$, and (c) $K=60$.

Figure 6

Distributions of $lnr$ for several $\alpha$ values when $1/T=1$: (a) $K=10$, (b) $K=30$, and (c) $K=60$.

Figure 7

Distributions of $lnr$ for several $\alpha$ values when $1/T=2$: (a) $K=10$, (b) $K=30$, and (c) $K=60$.

Figure 8

Illustration of Roussel's sampling method [28]. The transition from ${\mathit{h}}_{\mathrm{old}}$ to ${\mathit{h}}_{\mathrm{new}}$ is performed using the MH method on $\mathit{h}$ space. The transitions between the two layers are the same as the blocked Gibbs sampling. This sampling procedure can be used as ${\tau }_k\left({\mathit{v}}^{\prime }\right|\mathit{v})$.

Figure 9

APEs versus $K$ on the RBM with the Gaussian interactions: (a) $1/T=2$ and (b) $1/T=20$. These plots represent the average values over 1000 experiments.

Figure 10

APEs versus $K$ on the RBM with the Hopfield-type interactions: (a) $1/T=2$ and (b) $1/T=20$. These plots represent the average values over 1000 experiments.