Appendix A provides derivations supporting Section 3 in the main paper. In this section we provide detailed derivations of the ST -DGMRF joint distribution, for both first-order transition models (Section A.1) and higher-order transition models (Section A.2). A.1 Joint distribution The LDS (see Section 2.2 and 3.1 in the main paper) defines a joint distribution over system states First, note that Eq. (1) can be written as a set of linear equations x We make use of this property in the DGMRF formulation and in the conjugate gradient method. Eq. 11 is converted into a discrete-time dynamical system by approximating ρ We consider two ST -DGMRF variants that capture different amounts of prior knowledge. DGMRF transition matrices can be parameterized accordingly. The air quality dataset is based on hourly PM2.5 measurements obtained from [ The raw PM2.5 measurements are log-transformed and standardized to zero mean and unit Ca. 50% of the nodes are masked out (purple nodes within We use a simple MLP with one hidden layer of width 16 with ReLU activations and no output non-linearity. The DGMRF parameters are not shared across time, allowing for dynamically changing spatial covariance patterns.

V ariational inference (VI) is a method to approximate the computationally intractable posterior distributions that arise in Bayesian statistics.

We carefully design the potential functions to enable relational reasoning among different objects according to knowledge graph embedding methods.

Semi-implicit variational inference (SIVI) enhances the expressiveness of variational families through hierarchical semi-implicit distributions, but the intractability of their densities makes standard ELBO-based optimization biased. Recent score-matching approaches to SIVI (SIVI-SM) address this issue via a minimax formulation, at the expense of an additional lower-level optimization problem. In this paper, we propose kernel semi-implicit variational inference (KSIVI), a principled and tractable alternative that eliminates the lower-level optimization by leveraging kernel methods. We show that when optimizing over a reproducing kernel Hilbert space, the lower-level problem admits an explicit solution, reducing the objective to the kernel Stein discrepancy (KSD). Exploiting the hierarchical structure of semi-implicit distributions, the resulting KSD objective can be efficiently optimized using stochastic gradient methods. We establish optimization guarantees via variance bounds on Monte Carlo gradient estimators and derive statistical generalization bounds of order $\tilde{\mathcal{O}}(1/\sqrt{n})$. We further introduce a multi-layer hierarchical extension that improves expressiveness while preserving tractability. Empirical results on synthetic and real-world Bayesian inference tasks demonstrate the effectiveness of KSIVI.

We use the posterior from Equation (11), the likelihood from Equation (12), and Jensen's inequality These objectives lead to the model, policy, and critic losses. In this section, we describe the architecture of our sequential latent variable model. The parameters of the convolution layers are shared among both distributions. The latent variables have 32 and 256 dimensions, respectively, i.e. Before the agent starts learning on the task, the model is first pretrained using a small amount of random data.

In this work, we propose Uncorrected Hamiltonian Annealing (UHA), a differentiable alternative to Hamiltonian AIS.

Variational inference plays a vital role in learning graphical models, especially on large-scale datasets. Much of its success depends on a proper choice of auxiliary distribution class for posterior approximation. However, how to pursue an auxiliary distribution class that achieves both good approximation ability and computation efficiency remains a core challenge. In this paper, we proposed coupled variational Bayes which exploits the primal-dual view of the ELBO with the variational distribution class generated by an optimization procedure, which is termed optimization embedding.