There has been growing interest in the AI community for precise uncertainty quantification. Conditional density models f(y|x), where x represents potentially high-dimensional features, are an integral part of uncertainty quantification in prediction and Bayesian inference. However, it is challenging to assess conditional density estimates and gain insight into modes of failure. While existing diagnostic tools can determine whether an approximated conditional density is compatible overall with a data sample, they lack a principled framework for identifying, locating, and interpreting the nature of statistically significant discrepancies over the entire feature space. In this paper, we present rigorous and easy-to-interpret diagnostics such as (i) the "Local Coverage Test" (LCT), which distinguishes an arbitrarily misspecified model from the true conditional density of the sample, and (ii) "Amortized Local P-P plots" (ALP) which can quickly provide interpretable graphical summaries of distributional differences at any location x in the feature space. Our validation procedures scale to high dimensions and can potentially adapt to any type of data at hand. We demonstrate the effectiveness of LCT and ALP through a simulated experiment and applications to prediction and parameter inference for image data.
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Physically-inspired latent force models offer an interpretable alternative to purely data driven tools for inference in dynamical systems. They carry the structure of differential equations and the flexibility of Gaussian processes, yielding interpretable parameters and dynamics-imposed latent functions. However, the existing inference techniques associated with these models rely on the exact computation of posterior kernel terms which are seldom available in analytical form. Most applications relevant to practitioners, such as Hill equations or diffusion equations, are hence intractable. In this paper, we overcome these computational problems by proposing a variational solution to a general class of non-linear and parabolic partial differential equation latent force models. Further, we show that a neural operator approach can scale our model to thousands of instances, enabling fast, distributed computation. We demonstrate the efficacy and flexibility of our framework by achieving competitive performance on several tasks where the kernels are of varying degrees of tractability.
Modern high-dimensional point process data, especially those from neuroscience experiments, often involve observations from multiple conditions and/or experiments. Networks of interactions corresponding to these conditions are expected to share many edges, but also exhibit unique, condition-specific ones. However, the degree of similarity among the networks from different conditions is generally unknown. Existing approaches for multivariate point processes do not take these structures into account and do not provide inference for jointly estimated networks. To address these needs, we propose a joint estimation procedure for networks of high-dimensional point processes that incorporates easy-to-compute weights in order to data-adaptively encourage similarity between the estimated networks. We also propose a powerful hierarchical multiple testing procedure for edges of all estimated networks, which takes into account the data-driven similarity structure of the multi-experiment networks. Compared to conventional multiple testing procedures, our proposed procedure greatly reduces the number of tests and results in improved power, while tightly controlling the family-wise error rate. Unlike existing procedures, our method is also free of assumptions on dependency between tests, offers flexibility on p-values calculated along the hierarchy, and is robust to misspecification of the hierarchical structure. We verify our theoretical results via simulation studies and demonstrate the application of the proposed procedure using neuronal spike train data.
This paper defines a novel Bayesian inverse problem to infer an infinite-dimensional uncertain operator appearing in a differential equation, whose action on an observable state variable affects its dynamics. Inference is made tractable by parametrizing the operator using its eigendecomposition. The plausibility of operator inference in the sparse data regime is explored in terms of an uncertain, generalized diffusion operator appearing in an evolution equation for a contaminant's transport through a heterogeneous porous medium. Sparse data are augmented with prior information through the imposition of deterministic constraints on the eigendecomposition and the use of qualitative information about the system in the definition of the prior distribution. Limited observations of the state variable's evolution are used as data for inference, and the dependence on the solution of the inverse problem is studied as a function of the frequency of observations, as well as on whether or not the data is collected as a spatial or time series.
We consider the problem of estimating the parameters a Gaussian Mixture Model with K components of known weights, all with an identity covariance matrix. We make two contributions. First, at the population level, we present a sharper analysis of the local convergence of EM and gradient EM, compared to previous works. Assuming a separation of $\Omega(\sqrt{\log K})$, we prove convergence of both methods to the global optima from an initialization region larger than those of previous works. Specifically, the initial guess of each component can be as far as (almost) half its distance to the nearest Gaussian. This is essentially the largest possible contraction region. Our second contribution are improved sample size requirements for accurate estimation by EM and gradient EM. In previous works, the required number of samples had a quadratic dependence on the maximal separation between the K components, and the resulting error estimate increased linearly with this maximal separation. In this manuscript we show that both quantities depend only logarithmically on the maximal separation.
Derivative based optimization methods are efficient at solving optimal control problems near local optima. However, their ability to converge halts when derivative information vanishes. The inference approach to optimal control does not have strict requirements on the objective landscape. However, sampling, the primary tool for solving such problems, tends to be much slower in computation time. We propose a new method that combines second order methods with inference. We utilise the Kullback Leibler (KL) control framework to formulate an inference problem that computes the optimal controls from an adaptive distribution approximating the solution of the second order method. Our method allows for combining simple convex and non convex cost functions. This simplifies the process of cost function design and leverages the strengths of both inference and second order optimization. We compare our method to Model Predictive Path Integral (MPPI) and iterative Linear Quadratic Regulator (iLQG), outperforming both in sample efficiency and quality on manipulation and obstacle avoidance tasks.
We propose a framework for Bayesian Likelihood-Free Inference (LFI) based on Generalized Bayesian Inference. To define the generalized posterior, we use Scoring Rules (SRs), which evaluate probabilistic models given an observation. As in LFI we can sample from the model (but not evaluate the likelihood), we employ SRs with easy empirical estimators. Our framework includes novel approaches and popular LFI techniques (such as Bayesian Synthetic Likelihood), which benefit from the generalized Bayesian interpretation. Our method enjoys posterior consistency in a well-specified setting when a strictly-proper SR is used (i.e., one whose expectation is uniquely minimized when the model corresponds to the data generating process). Further, we prove a finite sample generalization bound and outlier robustness for the Kernel and Energy Score posteriors, and propose a strategy suitable for the LFI setup for tuning the learning rate in the generalized posterior. We run simulations studies with pseudo-marginal Markov Chain Monte Carlo (MCMC) and compare with related approaches, which we show do not enjoy robustness and consistency.
Beam search is the default decoding strategy for many sequence generation tasks in NLP. The set of approximate K-best items returned by the algorithm is a useful summary of the distribution for many applications; however, the candidates typically exhibit high overlap and may give a highly biased estimate for expectations under our model. These problems can be addressed by instead using stochastic decoding strategies. In this work, we propose a new method for turning beam search into a stochastic process: Conditional Poisson stochastic beam search. Rather than taking the maximizing set at each iteration, we sample K candidates without replacement according to the conditional Poisson sampling design. We view this as a more natural alternative to Kool et. al. 2019's stochastic beam search (SBS). Furthermore, we show how samples generated under the CPSBS design can be used to build consistent estimators and sample diverse sets from sequence models. In our experiments, we observe CPSBS produces lower variance and more efficient estimators than SBS, even showing improvements in high entropy settings.
We propose a new method of estimation in topic models, that is not a variation on the existing simplex finding algorithms, and that estimates the number of topics K from the observed data. We derive new finite sample minimax lower bounds for the estimation of A, as well as new upper bounds for our proposed estimator. We describe the scenarios where our estimator is minimax adaptive. Our finite sample analysis is valid for any number of documents (n), individual document length (N_i), dictionary size (p) and number of topics (K), and both p and K are allowed to increase with n, a situation not handled well by previous analyses. We complement our theoretical results with a detailed simulation study. We illustrate that the new algorithm is faster and more accurate than the current ones, although we start out with a computational and theoretical disadvantage of not knowing the correct number of topics K, while we provide the competing methods with the correct value in our simulations.
Dynamic topic models (DTMs) model the evolution of prevalent themes in literature, online media, and other forms of text over time. DTMs assume that word co-occurrence statistics change continuously and therefore impose continuous stochastic process priors on their model parameters. These dynamical priors make inference much harder than in regular topic models, and also limit scalability. In this paper, we present several new results around DTMs. First, we extend the class of tractable priors from Wiener processes to the generic class of Gaussian processes (GPs). This allows us to explore topics that develop smoothly over time, that have a long-term memory or are temporally concentrated (for event detection). Second, we show how to perform scalable approximate inference in these models based on ideas around stochastic variational inference and sparse Gaussian processes. This way we can train a rich family of DTMs to massive data. Our experiments on several large-scale datasets show that our generalized model allows us to find interesting patterns that were not accessible by previous approaches.
Owing to the recent advances in "Big Data" modeling and prediction tasks, variational Bayesian estimation has gained popularity due to their ability to provide exact solutions to approximate posteriors. One key technique for approximate inference is stochastic variational inference (SVI). SVI poses variational inference as a stochastic optimization problem and solves it iteratively using noisy gradient estimates. It aims to handle massive data for predictive and classification tasks by applying complex Bayesian models that have observed as well as latent variables. This paper aims to decentralize it allowing parallel computation, secure learning and robustness benefits. We use Alternating Direction Method of Multipliers in a top-down setting to develop a distributed SVI algorithm such that independent learners running inference algorithms only require sharing the estimated model parameters instead of their private datasets. Our work extends the distributed SVI-ADMM algorithm that we first propose, to an ADMM-based networked SVI algorithm in which not only are the learners working distributively but they share information according to rules of a graph by which they form a network. This kind of work lies under the umbrella of `deep learning over networks' and we verify our algorithm for a topic-modeling problem for corpus of Wikipedia articles. We illustrate the results on latent Dirichlet allocation (LDA) topic model in large document classification, compare performance with the centralized algorithm, and use numerical experiments to corroborate the analytical results.
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