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.

A real-world application or setting involves interaction between different modalities (e.g., video, speech, text). In order to process the multimodal information automatically and use it for an end application, Multimodal Representation Learning (MRL) has emerged as an active area of research in recent times. MRL involves learning reliable and robust representations of information from heterogeneous sources and fusing them. However, in practice, the data acquired from different sources are typically noisy. In some extreme cases, a noise of large magnitude can completely alter the semantics of the data leading to inconsistencies in the parallel multimodal data. In this paper, we propose a novel method for multimodal representation learning in a noisy environment via the generalized product of experts technique. In the proposed method, we train a separate network for each modality to assess the credibility of information coming from that modality, and subsequently, the contribution from each modality is dynamically varied while estimating the joint distribution. We evaluate our method on two challenging benchmarks from two diverse domains: multimodal 3D hand-pose estimation and multimodal surgical video segmentation. We attain state-of-the-art performance on both benchmarks. Our extensive quantitative and qualitative evaluations show the advantages of our method compared to previous approaches.

Gaussian processes (GPs) are nonparametric Bayesian models that have been applied to regression and classification problems. One of the approaches to alleviate their cubic training cost is the use of local GP experts trained on subsets of the data. In particular, product-of-expert models combine the predictive distributions of local experts through a tractable product operation. While these expert models allow for massively distributed computation, their predictions typically suffer from erratic behaviour of the mean or uncalibrated uncertainty quantification. By calibrating predictions via a tempered softmax weighting, we provide a solution to these problems for multiple product-of-expert models, including the generalised product of experts and the robust Bayesian committee machine. Furthermore, we leverage the optimal transport literature and propose a new product-of-expert model that combines predictions of local experts by computing their Wasserstein barycenter, which can be applied to both regression and classification.

In this work, we propose a generalized product of experts (gPoE) framework for combining the predictions of multiple probabilistic models. We identify four desirable properties that are important for scalability, expressiveness and robustness, when learning and inferring with a combination of multiple models. Through analysis and experiments, we show that gPoE of Gaussian processes (GP) have these qualities, while no other existing combination schemes satisfy all of them at the same time. The resulting GP-gPoE is highly scalable as individual GP experts can be independently learned in parallel; very expressive as the way experts are combined depends on the input rather than fixed; the combined prediction is still a valid probabilistic model with natural interpretation; and finally robust to unreliable predictions from individual experts.