This research considers a scalable inference for spatial data modeled through Gaussian intrinsic conditional autoregressive (ICAR) structures. The classical estimation method, restricted maximum likelihood (REML), requires repeated inversion and factorization of large, sparse precision matrices, which makes this computation costly. To sort this problem out, we propose a variational restricted maximum likelihood (VREML) framework that approximates the intractable marginal likelihood using a Gaussian variational distribution. By constructing an evidence lower bound (ELBO) on the restricted likelihood, we derive a computationally efficient coordinate-ascent algorithm for jointly estimating the spatial random effects and variance components. In this article, we theoretically establish the monotone convergence of ELBO and mathematically exhibit that the variational family is exact under Gaussian ICAR settings, which is an indication of nullifying approximation error at the posterior level. We empirically establish the supremacy of our VREML over MLE and INLA.

Existing machine learning frameworks for compliance monitoring -- Markov Logic Networks, Probabilistic Soft Logic, supervised models -- share a fundamental paradigm: they treat observed data as ground truth and attempt to approximate rules from it. This assumption breaks down in rule-governed domains such as taxation or regulatory compliance, where authoritative rules are known a priori and the true challenge is to infer the latent state of rule activation, compliance, and parametric drift from partial and noisy observations. We propose Rule-State Inference (RSI), a Bayesian framework that inverts this paradigm by encoding regulatory rules as structured priors and casting compliance monitoring as posterior inference over a latent rule-state space S = {(a_i, c_i, delta_i)}, where a_i captures rule activation, c_i models the compliance rate, and delta_i quantifies parametric drift. We prove three theoretical guarantees: (T1) RSI absorbs regulatory changes in O(1) time via a prior ratio correction, independently of dataset size; (T2) the posterior is Bernstein-von Mises consistent, converging to the true rule state as observations accumulate; (T3) mean-field variational inference monotonically maximizes the Evidence Lower BOund (ELBO). We instantiate RSI on the Togolese fiscal system and introduce RSI-Togo-Fiscal-Synthetic v1.0, a benchmark of 2,000 synthetic enterprises grounded in real OTR regulatory rules (2022-2025). Without any labeled training data, RSI achieves F1=0.519 and AUC=0.599, while absorbing regulatory changes in under 1ms versus 683-1082ms for full model retraining -- at least a 600x speedup.

When used as a surrogate objective for maximum likelihood estimation in latent variable models, the evidence lower bound (ELBO) produces state-of-the-art results. Inspired by this, we consider the extension of the ELBO to a family of lower bounds defined by a particle filter's estimator of the marginal likelihood, the filtering variational objectives (FIVOs). FIVOs take the same arguments as the ELBO, but can exploit a model's sequential structure to form tighter bounds. We present results that relate the tightness of FIVO's bound to the variance of the particle filter's estimator by considering the generic case of bounds defined as log-transformed likelihood estimators. Experimentally, we show that training with FIVO results in substantial improvements over training the same model architecture with the ELBO on sequential data.

Variational Auto-Encoders (VAE) have become very popular techniques to perform inference and learning in latent variable models as they allow us to leverage the rich representational power of neural networks to obtain flexible approximations of the posterior of latent variables as well as tight evidence lower bounds (ELBO). Combined with stochastic variational inference, this provides a methodology scaling to large datasets. However, for this methodology to be practically efficient, it is necessary to obtain low-variance unbiased estimators of the ELBO and its gradients with respect to the parameters of interest. While the use of Markov chain Monte Carlo (MCMC) techniques such as Hamiltonian Monte Carlo (HMC) has been previously suggested to achieve this [23, 26], the proposed methods require specifying reverse kernels which have a large impact on performance. Additionally, the resulting unbiased estimator of the ELBO for most MCMC kernels is typically not amenable to the reparameterization trick. We show here how to optimally select reverse kernels in this setting and, by building upon Hamiltonian Importance Sampling (HIS) [17], we obtain a scheme that provides low-variance unbiased estimators of the ELBO and its gradients using the reparameterization trick. This allows us to develop a Hamiltonian Variational Auto-Encoder (HVAE). This method can be re-interpreted as a target-informed normalizing flow [20] which, within our context, only requires a few evaluations of the gradient of the sampled likelihood and trivial Jacobian calculations at each iteration.

To do so, we first form a family of unnormalized densities (or a "path") parameterized by 2 [0, 1] between the two distributions of interest

Diffusion models have gained traction as powerful algorithms for synthesizing high-quality images. Central to these algorithms is the diffusion process, a set of equations which maps data to noise in a way that can significantly affect performance. In this paper, we explore whether the diffusion process can be learned from data.

Then, we intend to calculate the constraint part. The algorithm for Licence method for single-target interventiion scenario is shown in Algorithm 1. The details of experimental baselines are demonstrated as follows. AIT [11] is an active learning method that utilize f-score to select intervention queries. REAL fidelity means the model always choose the highest fidelity to conduct experiments.