Multilayer Neural Networks (MNNs) are commonly trained using gradient descent-based methods, such as BackPropagation (BP). Inference in probabilistic graphical models is often done using variational Bayes methods, such as Expectation Propagation (EP). We show how an EP based approach can also be used to train deterministic MNNs. Specifically, we approximate the posterior of the weights given the data using a "mean-field" factorized distribution, in an online setting. Using online EP and the central limit theorem we find an analytical approximation to the Bayes update of this posterior, as well as the resulting Bayes estimates of the weights and outputs. Despite a different origin, the resulting algorithm, Expectation BackPropagation (EBP), is very similar to BP in form and efficiency. However, it has several additional advantages: (1) Training is parameter-free, given initial conditions (prior) and the MNN architecture. This is useful for large-scale problems, where parameter tuning is a major challenge.

We propose a novel information-theoretic approach for Bayesian optimization called Predictive Entropy Search (PES). At each iteration, PES selects the next evaluation point that maximizes the expected information gained with respect to the global maximum. PES codifies this intractable acquisition function in terms of the expected reduction in the differential entropy of the predictive distribution. This reformulation allows PES to obtain approximations that are both more accurate and efficient than other alternatives such as Entropy Search (ES). Furthermore, PES can easily perform a fully Bayesian treatment of the model hyperparameters while ES cannot. We evaluate PES in both synthetic and realworld applications, including optimization problems in machine learning, finance, biotechnology, and robotics. We show that the increased accuracy of PES leads to significant gains in optimization performance.

Weaknesses: - Central parts of the paper are unclear eg. in line 80 \log P_M (X; \theta) should be the negative cross entropy. The only quantitative results are on adaptation from USPS to MNIST in line 268. However, prior work [1] achieves 96.5% accuracy in comparison to the 55% accuracy achieved by the proposed method. It would be desirable to evaluate the proposed approach on the more complex Facades/Maps/Cityscapes using the MSE metric to facilitate comparison with AlignFlow and [1]. It is unclear how the inductive bias from each of the datasets influence the shared space.

After discussion, all reviewers, and the meta-reviewer, agree that the paper should be accepted. As the authors show, the method in its current form may not scale well to higher dimensions. While a method without this limitation would obviously be preferable, the reviewers agree that this limitation can be addressed in future work, where the connection with GANs that the authors establish may be helpful.

Offline preference alignment for language models such as Direct Preference Optimization (DPO) is favored for its effectiveness and simplicity, eliminating the need for costly reinforcement learning. Various offline algorithms have been developed for different data settings, yet they lack a unified understanding. In this study, we introduce Pior-Informed Preference Alignment (PIPA), a unified, RL-free probabilistic framework that formulates language model preference alignment as a Maximum Likelihood Estimation (MLE) problem with prior constraints. This method effectively accommodates both paired and unpaired data, as well as answer and step-level annotations. We illustrate that DPO and KTO are special cases with different prior constraints within our framework. By integrating different types of prior information, we developed two variations of PIPA: PIPA-M and PIPA-N. Both algorithms demonstrate a $3\sim10\%$ performance enhancement on the GSM8K and MATH benchmarks across all configurations, achieving these gains without additional training or computational costs compared to existing algorithms.

Large Language Models (LLMs) are prone to generating plausible yet incorrect responses, known as hallucinations. Effectively detecting hallucinations is therefore crucial for the safe deployment of LLMs. Recent research has linked hallucinations to model uncertainty, suggesting that hallucinations can be detected by measuring dispersion over answer distributions obtained from a set of samples drawn from a model. While drawing from the distribution over tokens defined by the model is a natural way to obtain samples, in this work, we argue that it is sub-optimal for the purpose of detecting hallucinations. We show that detection can be improved significantly by taking into account model uncertainty in the Bayesian sense. To this end, we propose a very simple and efficient approach that perturbs an appropriate subset of model parameters, or equivalently hidden unit activations, during sampling. We demonstrate its effectiveness across a wide range of datasets and model architectures.

In this work, we present a novel approach to system identification for dynamical systems, based on a specific class of Deep Gaussian Processes (Deep GPs). These models are constructed by interconnecting linear dynamic GPs (equivalent to stochastic linear time-invariant dynamical systems) and static GPs (to model static nonlinearities). Our approach combines the strengths of data-driven methods, such as those based on neural network architectures, with the ability to output a probability distribution. This offers a more comprehensive framework for system identification that includes uncertainty quantification. Using both simulated and real-world data, we demonstrate the effectiveness of the proposed approach.

We propose a construction of coresets based on a predictive view of Bayesian posterior inference (Fong et al., 2024; Fortini and Petrone, 2012). The main attraction of the approach is the model-agnostic nature - the method is valid with any inference model and independent of the specific inference goals, making it highly adaptable for a wide range of applications. Such adaptability is particularly valuable in the context of large-scale datasets, now commonplace in fields like genomics and astronomy. While this explosion of data offers incredible opportunities for discoveries, it also brings significant computational challenges. Tasks that were once straightforward, such as evaluating likelihoods several times have become increasingly difficult, making traditional data processing methods impractical. These obstacles have frequently pushed practitioners toward simpler statistical models that might not capture the full complexity of the data, disregarding expressiveness and flexibility that rich hierarchical and nonparametric models can offer.

SUMMARY: This paper studies the effect of noise correlation in some models of multi-output regression. It argues that a method that does not benefit from the correlation, such as Ordinary Least Squares (OLS), may perform much worse than a method that does, such as Maximum Likelihood Estimation (MLE). For certain linear models (Pooled model and Seemingly Unrelated Regression), which are studied in the paper, the MLE estimator requires the joint optimization of the covariance and regression weights. This is a non-convex problem. Alternative Minimization (AltMin) algorithm is an approach to solve the problem by iteratively optimizing the covariance and the weights.

The paper proposes a mechanism for explaining Bayesian inference and network plasticity in the brain using an algorithm very similar to Stochastic Gradient Langevin Dynamics. Clarity: The paper is well written. Even though my background is machine learning and not neuroscience, I was able to follow most of the paper. Originality: The mechanism itself is well studied in the machine learning literature where it is called Stochastic Gradient Langevin Dynamics (SGLD) (see Ref[1] and analysis in Ref[2]). This is also well known in physics where it is usually called the Langevin equation with noisy force (see e.g.