Fields such as phylogenetics often work with a sort of abstracted family tree, called a phylogenetic tree, frequently abbreviated here as tree. These trees have different members of a population as their tips, and their branching points describe the relations between the tips and how recently they had a common ancestor. If some of the tips are censored, the tree topology simplifies in a process we refer to as restriction. If one has multiple trees restricted from the same original, uncensored tree, one may wish to reconstruct the original supertree. Suppose instead one has multiple probability distributions of restricted trees, then one may be interested in reconstructing the supertree probability distribution.

Why can the Expectation-Maximization (EM) algorithm for mixture models converge? Why can different initial parameters cause various convergence difficulties? The Q-L synchronization theory explains that the observed data log-likelihood L and the complete data log-likelihood Q are positively correlated; we can achieve maximum L by maximizing Q. According to this theory, the Deterministic Annealing EM (DAEM) algorithm's authors make great efforts to eliminate locally maximal Q for avoiding L's local convergence. However, this paper proves that in some cases, Q may and should decrease for L to increase; slow or local convergence exists only because of small samples and unfair competition. This paper uses marriage competition to explain different convergence difficulties and proposes the Fair Competition Principle (FCP) with an initialization map for improving initializations. It uses the rate-verisimilitude function, extended from the rate-distortion function, to explain the convergence of the EM and improved EM algorithms. This convergence proof adopts variational and iterative methods that Shannon et al. used for analyzing rate-distortion functions. The initialization map can vastly save both algorithms' running times for binary Gaussian mixtures. The FCP and the initialization map are useful for complicated mixtures but not sufficient; we need further studies for specific methods.

Existing approaches for generating human-aware agent behaviors have considered different measures of interpretability in isolation. Further, these measures have been studied under differing assumptions, thus precluding the possibility of designing a single framework that captures these measures under the same assumptions. In this paper, we present a unifying Bayesian framework that models a human observer's evolving beliefs about an agent and thereby define the problem of Generalized Human-Aware Planning. We will show that the definitions of interpretability measures like explicability, legibility and predictability from the prior literature fall out as special cases of our general framework. Through this framework, we also bring a previously ignored fact to light that the human-robot interactions are in effect open-world problems, particularly as a result of modeling the human's beliefs over the agent. Since the human may not only hold beliefs unknown to the agent but may also form new hypotheses about the agent when presented with novel or unexpected behaviors.

Reliability of machine learning (ML) systems is crucial in safety-critical applications such as healthcare, and uncertainty estimation is a widely researched method to highlight the confidence of ML systems in deployment. Sequential and parallel ensemble techniques have shown improved performance of ML systems in multi-modal settings by leveraging the feature sets together. We propose an uncertainty-aware boosting technique for multi-modal ensembling in order to focus on the data points with higher associated uncertainty estimates, rather than the ones with higher loss values. We evaluate this method on healthcare tasks related to Dementia and Parkinson's disease which involve real-world multi-modal speech and text data, wherein our method shows an improved performance. Additional analysis suggests that introducing uncertainty-awareness into the boosted ensembles decreases the overall entropy of the system, making it more robust to heteroscedasticity in the data, as well as better calibrating each of the modalities along with high quality prediction intervals. We open-source our entire codebase at https://github.com/usarawgi911/Uncertainty-aware-boosting

Conditioning is crucial in applied science when inference involving time series is involved. Belief calculus is an effective way of handling such inference in the presence of epistemic uncertainty -- unfortunately, different approaches to conditioning in the belief function framework have been proposed in the past, leaving the matter somewhat unsettled. Inspired by the geometric approach to uncertainty, in this paper we propose an approach to the conditioning of belief functions based on geometrically projecting them onto the simplex associated with the conditioning event in the space of all belief functions. We show here that such a geometric approach to conditioning often produces simple results with straightforward interpretations in terms of degrees of belief. This raises the question of whether classical approaches, such as for instance Dempster's conditioning, can also be reduced to some form of distance minimisation in a suitable space. The study of families of combination rules generated by (geometric) conditioning rules appears to be the natural prosecution of the presented research.

When constructing a Bayesian Machine Learning model, we might be faced with multiple different prior distributions and thus are required to properly consider them in a sensible manner in our model. While this situation is reasonably well explored for classical Bayesian Statistics, it appears useful to develop a corresponding method for complex Machine Learning problems. Given their underlying Bayesian framework and their widespread popularity, Gaussian Processes are a good candidate to tackle this task. We therefore extend the idea of Mixture models for Gaussian Process regression in order to work with multiple prior beliefs at once - both a analytical regression formula and a Sparse Variational approach are considered. In addition, we consider the usage of our approach to additionally account for the problem of prior misspecification in functional regression problems.

Topic models have been successfully used for analyzing text documents. However, with existing topic models, many documents are required for training. In this paper, we propose a neural network-based few-shot learning method that can learn a topic model from just a few documents. The neural networks in our model take a small number of documents as inputs, and output topic model priors. The proposed method trains the neural networks such that the expected test likelihood is improved when topic model parameters are estimated by maximizing the posterior probability using the priors based on the EM algorithm. Since each step in the EM algorithm is differentiable, the proposed method can backpropagate the loss through the EM algorithm to train the neural networks. The expected test likelihood is maximized by a stochastic gradient descent method using a set of multiple text corpora with an episodic training framework. In our experiments, we demonstrate that the proposed method achieves better perplexity than existing methods using three real-world text document sets.

We introduce distributed NLI, a new NLU task with a goal to predict the distribution of human judgements for natural language inference. We show that models can capture human judgement distribution by applying additional distribution estimation methods, namely, Monte Carlo (MC) Dropout, Deep Ensemble, Re-Calibration, and Distribution Distillation. All four of these methods substantially outperform the softmax baseline. We show that MC Dropout is able to achieve decent performance without any distribution annotations while Re-Calibration can further give substantial improvements when extra distribution annotations are provided, suggesting the value of multiple annotations for the example in modeling the distribution of human judgements. Moreover, MC Dropout and Re-Calibration can achieve decent transfer performance on out-of-domain data. Despite these improvements, the best results are still far below estimated human upper-bound, indicating that the task of predicting the distribution of human judgements is still an open, challenging problem with large room for future improvements. We showcase the common errors for MC Dropout and Re-Calibration. Finally, we give guidelines on the usage of these methods with different levels of data availability and encourage future work on modeling the human opinion distribution for language reasoning.

$\alpha$-posteriors and their variational approximations distort standard posterior inference by downweighting the likelihood and introducing variational approximation errors. We show that such distortions, if tuned appropriately, reduce the Kullback-Leibler (KL) divergence from the true, but perhaps infeasible, posterior distribution when there is potential parametric model misspecification. To make this point, we derive a Bernstein-von Mises theorem showing convergence in total variation distance of $\alpha$-posteriors and their variational approximations to limiting Gaussian distributions. We use these distributions to evaluate the KL divergence between true and reported posteriors. We show this divergence is minimized by choosing $\alpha$ strictly smaller than one, assuming there is a vanishingly small probability of model misspecification. The optimized value becomes smaller as the the misspecification becomes more severe. The optimized KL divergence increases logarithmically in the degree of misspecification and not linearly as with the usual posterior.

Hawkes processes are a popular means of modeling the event times of self-exciting phenomena, such as earthquake strikes or tweets on a topical subject. Classically, these models are fit to historical event time data via likelihood maximization. However, in many scenarios, the exact times of historical events are not recorded for either privacy (e.g., patient admittance to hospitals) or technical limitations (e.g., most transport data records the volume of vehicles passing loop detectors but not the individual times). The interval-censored setting denotes when only the aggregate counts of events at specific time intervals are observed. Fitting the parameters of interval-censored Hawkes processes requires designing new training objectives that do not rely on the exact event times. In this paper, we propose a model to estimate the parameters of a Hawkes process in interval-censored settings. Our model builds upon the existing Hawkes Intensity Process (HIP) of in several important directions. First, we observe that while HIP is formulated in terms of expected intensities, it is more natural to work instead with expected counts; further, one can express the latter as the solution to an integral equation closely related to the defining equation of HIP. Second, we show how a non-homogeneous Poisson approximation to the Hawkes process admits a tractable likelihood in the interval-censored setting; this approximation recovers the original HIP objective as a special case, and allows for the use of a broader class of Bregman divergences as loss function. Third, we explicate how to compute a tighter approximation to the ground truth in the likelihood. Finally, we show how our model can incorporate information about varying interval lengths. Experiments on synthetic and real-world data confirm our HIPPer model outperforms HIP and several other baselines on the task of interval-censored inference.