In science and engineering, we often work with models designed for accurate prediction of variables of interest. Recognizing that these models are approximations of reality, it becomes desirable to apply multiple models to the same data and integrate their outcomes. In this paper, we operate within the Bayesian paradigm, relying on Gaussian processes as our models. These models generate predictive probability density functions (pdfs), and the objective is to integrate them systematically, employing both linear and log-linear pooling. We introduce novel approaches for log-linear pooling, determining input-dependent weights for the predictive pdfs of the Gaussian processes. The aggregation of the pdfs is realized through Monte Carlo sampling, drawing samples of weights from their posterior. The performance of these methods, as well as those based on linear pooling, is demonstrated using a synthetic dataset.

Benchmark datasets used for image classification tend to have very low levels of label noise. When Bayesian neural networks are trained on these datasets, they often underfit, misrepresenting the aleatoric uncertainty of the data. A common solution is to cool the posterior, which improves fit to the training data but is challenging to interpret from a Bayesian perspective. We explore whether posterior tempering can be replaced by a confidence-inducing prior distribution. First, we introduce a "DirClip" prior that is practical to sample and nearly matches the performance of a cold posterior. Second, we introduce a "confidence prior" that directly approximates a cold likelihood in the limit of decreasing temperature but cannot be easily sampled. Lastly, we provide several general insights into confidence-inducing priors, such as when they might diverge and how fine-tuning can mitigate numerical instability.

Language models (LMs) may appear insensitive to word order changes in natural language understanding (NLU) tasks. In this paper, we propose that linguistic redundancy can explain this phenomenon, whereby word order and other linguistic cues such as case markers provide overlapping and thus redundant information. Our hypothesis is that models exhibit insensitivity to word order when the order provides redundant information, and the degree of insensitivity varies across tasks. We quantify how informative word order is using mutual information (MI) between unscrambled and scrambled sentences. Our results show the effect that the less informative word order is, the more consistent the model's predictions are between unscrambled and scrambled sentences. We also find that the effect varies across tasks: for some tasks, like SST-2, LMs' prediction is almost always consistent with the original one even if the Pointwise-MI (PMI) changes, while for others, like RTE, the consistency is near random when the PMI gets lower, i.e., word order is really important.

Bayesian approaches to reconstructing the evolutionary history of languages rely on the tree model, which assumes that these languages descended from a common ancestor and underwent modifications over time. However, this assumption can be violated to different extents due to contact and other factors. Understanding the degree to which this assumption is violated is crucial for validating the accuracy of phylolinguistic inference. In this paper, we propose a simple sanity check: projecting a reconstructed tree onto a space generated by principal component analysis. By using both synthetic and real data, we demonstrate that our method effectively visualizes anomalies, particularly in the form of jogging.

Cyclical MCMC is a novel MCMC framework recently proposed by Zhang et al. (2019) to address the challenge posed by high-dimensional multimodal posterior distributions like those arising in deep learning. The algorithm works by generating a nonhomogeneous Markov chain that tracks - cyclically in time - tempered versions of the target distribution. We show in this work that cyclical MCMC converges to the desired probability distribution in settings where the Markov kernels used are fast mixing, and sufficiently long cycles are employed. However in the far more common settings of slow mixing kernels, the algorithm may fail to produce samples from the desired distribution. In particular, in a simple mixture example with unequal variance where powering is known to produce slow mixing kernels, we show by simulation that cyclical MCMC fails to converge to the desired limit. Finally, we show that cyclical MCMC typically estimates well the local shape of the target distribution around each mode, even when we do not have convergence to the target.

In recent years, denoising problems have become intertwined with the development of deep generative models. In particular, diffusion models are trained like denoisers, and the distribution they model coincide with denoising priors in the Bayesian picture. However, denoising through diffusion-based posterior sampling requires the noise level and covariance to be known, preventing blind denoising. We overcome this limitation by introducing Gibbs Diffusion (GDiff), a general methodology addressing posterior sampling of both the signal and the noise parameters. Assuming arbitrary parametric Gaussian noise, we develop a Gibbs algorithm that alternates sampling steps from a conditional diffusion model trained to map the signal prior to the family of noise distributions, and a Monte Carlo sampler to infer the noise parameters. Our theoretical analysis highlights potential pitfalls, guides diagnostic usage, and quantifies errors in the Gibbs stationary distribution caused by the diffusion model. We showcase our method for 1) blind denoising of natural images involving colored noises with unknown amplitude and spectral index, and 2) a cosmology problem, namely the analysis of cosmic microwave background data, where Bayesian inference of "noise" parameters means constraining models of the evolution of the Universe.

Human knowledge is largely implicit and relational -- do we have a friend in common? can I walk from here to there? In this work, we leverage the combinatorial structure of graphs to quantify human priors over such relational data. Our experiments focus on two domains that have been continuously relevant over evolutionary timescales: social interaction and spatial navigation. We find that some features of the inferred priors are remarkably consistent, such as the tendency for sparsity as a function of graph size. Other features are domain-specific, such as the propensity for triadic closure in social interactions. More broadly, our work demonstrates how nonclassical statistical analysis of indirect behavioral experiments can be used to efficiently model latent biases in the data.

Bayesian optimal experimental design (OED) seeks to conduct the most informative experiment under budget constraints to update the prior knowledge of a system to its posterior from the experimental data in a Bayesian framework. Such problems are computationally challenging because of (1) expensive and repeated evaluation of some optimality criterion that typically involves a double integration with respect to both the system parameters and the experimental data, (2) suffering from the curse-of-dimensionality when the system parameters and design variables are high-dimensional, (3) the optimization is combinatorial and highly non-convex if the design variables are binary, often leading to non-robust designs. To make the solution of the Bayesian OED problem efficient, scalable, and robust for practical applications, we propose a novel joint optimization approach. This approach performs simultaneous (1) training of a scalable conditional normalizing flow (CNF) to efficiently maximize the expected information gain (EIG) of a jointly learned experimental design (2) optimization of a probabilistic formulation of the binary experimental design with a Bernoulli distribution. We demonstrate the performance of our proposed method for a practical MRI data acquisition problem, one of the most challenging Bayesian OED problems that has high-dimensional (320 $\times$ 320) parameters at high image resolution, high-dimensional (640 $\times$ 386) observations, and binary mask designs to select the most informative observations.

A common network inference problem, arising from real-world data constraints, is how to infer a dynamic network from its time-aggregated adjacency matrix and time-varying marginals (i.e., row and column sums). Prior approaches to this problem have repurposed the classic iterative proportional fitting (IPF) procedure, also known as Sinkhorn's algorithm, with promising empirical results. However, the statistical foundation for using IPF has not been well understood: under what settings does IPF provide principled estimation of a dynamic network from its marginals, and how well does it estimate the network? In this work, we establish such a setting, by identifying a generative network model whose maximum likelihood estimates are recovered by IPF. Our model both reveals implicit assumptions on the use of IPF in such settings and enables new analyses, such as structure-dependent error bounds on IPF's parameter estimates. When IPF fails to converge on sparse network data, we introduce a principled algorithm that guarantees IPF converges under minimal changes to the network structure. Finally, we conduct experiments with synthetic and real-world data, which demonstrate the practical value of our theoretical and algorithmic contributions.

Conditional diffusion models have shown remarkable performance in various generative tasks, but training them requires large-scale datasets that often contain noise in conditional inputs, a.k.a. noisy labels. This noise leads to condition mismatch and quality degradation of generated data. This paper proposes Transition-aware weighted Denoising Score Matching (TDSM) for training conditional diffusion models with noisy labels, which is the first study in the line of diffusion models. The TDSM objective contains a weighted sum of score networks, incorporating instance-wise and time-dependent label transition probabilities. We introduce a transition-aware weight estimator, which leverages a time-dependent noisy-label classifier distinctively customized to the diffusion process. Through experiments across various datasets and noisy label settings, TDSM improves the quality of generated samples aligned with given conditions. Furthermore, our method improves generation performance even on prevalent benchmark datasets, which implies the potential noisy labels and their risk of generative model learning. Finally, we show the improved performance of TDSM on top of conventional noisy label corrections, which empirically proving its contribution as a part of label-noise robust generative models. Our code is available at: https://github.com/byeonghu-na/tdsm.