Edge-exchangeable probabilistic network models generate edges as an i.i.d.~sequence from a discrete measure, providing a simple means for statistical inference of latent network properties. The measure is often constructed using the self-product of a realization from a Bayesian nonparametric (BNP) discrete prior; but unlike in standard BNP models, the self-product measure prior is not conjugate the likelihood, hindering the development of exact simulation and inference algorithms. Approximation via finite truncation of the discrete measure is a straightforward alternative, but incurs an unknown approximation error. In this paper, we develop methods for forward simulation and posterior inference in random self-product-measure models based on truncation, and provide theoretical guarantees on the quality of the results as a function of the truncation level. The techniques we present are general and extend to the broader class of discrete Bayesian nonparametric models.
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We propose a doubly robust approach to characterizing treatment effect heterogeneity in observational studies. We utilize posterior distributions for both the propensity score and outcome regression models to provide valid inference on the conditional average treatment effect even when high-dimensional or nonparametric models are used. We show that our approach leads to conservative inference in finite samples or under model misspecification, and provides a consistent variance estimator when both models are correctly specified. In simulations, we illustrate the utility of these results in difficult settings such as high-dimensional covariate spaces or highly flexible models for the propensity score and outcome regression. Lastly, we analyze environmental exposure data from NHANES to identify how the effects of these exposures vary by subject-level characteristics.
For rare events described in terms of Markov processes, truly unbiased estimation of the rare event probability generally requires the avoidance of numerical approximations of the Markov process. Recent work in the exact and $\varepsilon$-strong simulation of diffusions, which can be used to almost surely constrain sample paths to a given tolerance, suggests one way to do this. We specify how such algorithms can be combined with the classical multilevel splitting method for rare event simulation. This provides unbiased estimations of the probability in question. We discuss the practical feasibility of the algorithm with reference to existing $\varepsilon$-strong methods and provide proof-of-concept numerical examples.
This paper presents a topological learning-theoretic perspective on causal inference by introducing a series of topologies defined on general spaces of structural causal models (SCMs). As an illustration of the framework we prove a topological causal hierarchy theorem, showing that substantive assumption-free causal inference is possible only in a meager set of SCMs. Thanks to a known correspondence between open sets in the weak topology and statistically verifiable hypotheses, our results show that inductive assumptions sufficient to license valid causal inferences are statistically unverifiable in principle. Similar to no-free-lunch theorems for statistical inference, the present results clarify the inevitability of substantial assumptions for causal inference. An additional benefit of our topological approach is that it easily accommodates SCMs with infinitely many variables. We finally suggest that the framework may be helpful for the positive project of exploring and assessing alternative causal-inductive assumptions.
Random effect models are popular statistical models for detecting and correcting spurious sample correlations due to hidden confounders in genome-wide gene expression data. In applications where some confounding factors are known, estimating simultaneously the contribution of known and latent variance components in random effect models is a challenge that has so far relied on numerical gradient-based optimizers to maximize the likelihood function. This is unsatisfactory because the resulting solution is poorly characterized and the efficiency of the method may be suboptimal. Here we prove analytically that maximum-likelihood latent variables can always be chosen orthogonal to the known confounding factors, in other words, that maximum-likelihood latent variables explain sample covariances not already explained by known factors. Based on this result we propose a restricted maximum-likelihood method which estimates the latent variables by maximizing the likelihood on the restricted subspace orthogonal to the known confounding factors, and show that this reduces to probabilistic PCA on that subspace. The method then estimates the variance-covariance parameters by maximizing the remaining terms in the likelihood function given the latent variables, using a newly derived analytic solution for this problem. Compared to gradient-based optimizers, our method attains greater or equal likelihood values, can be computed using standard matrix operations, results in latent factors that don't overlap with any known factors, and has a runtime reduced by several orders of magnitude. Hence the restricted maximum-likelihood method facilitates the application of random effect modelling strategies for learning latent variance components to much larger gene expression datasets than possible with current methods.
It is difficult to use subsampling with variational inference in hierarchical models since the number of local latent variables scales with the dataset. Thus, inference in hierarchical models remains a challenge at large scale. It is helpful to use a variational family with structure matching the posterior, but optimization is still slow due to the huge number of local distributions. Instead, this paper suggests an amortized approach where shared parameters simultaneously represent all local distributions. This approach is similarly accurate as using a given joint distribution (e.g., a full-rank Gaussian) but is feasible on datasets that are several orders of magnitude larger. It is also dramatically faster than using a structured variational distribution.
The presence of measurement error is a widespread issue which, when ignored, can render the results of an analysis unreliable. Numerous corrections for the effects of measurement error have been proposed and studied, often under the assumption of a normally distributed, additive measurement error model. One such correction is the simulation extrapolation method, which provides a flexible way of correcting for the effects of error in a wide variety of models, when the errors are approximately normally distributed. However, in many situations observed data are non-symmetric, heavy-tailed, or otherwise highly non-normal. In these settings, correction techniques relying on the assumption of normality are undesirable. We propose an extension to the simulation extrapolation method which is nonparametric in the sense that no specific distributional assumptions are required on the error terms. The technique is implemented when either validation data or replicate measurements are available, and it shares the general structure of the standard simulation extrapolation procedure, making it immediately accessible for those familiar with this technique.
In this work we consider the problem of releasing a differentially private statistical summary that resides on a Riemannian manifold. We present an extension of the Laplace or K-norm mechanism that utilizes intrinsic distances and volumes on the manifold. We also consider in detail the specific case where the summary is the Fr\'echet mean of data residing on a manifold. We demonstrate that our mechanism is rate optimal and depends only on the dimension of the manifold, not on the dimension of any ambient space, while also showing how ignoring the manifold structure can decrease the utility of the sanitized summary. We illustrate our framework in two examples of particular interest in statistics: the space of symmetric positive definite matrices, which is used for covariance matrices, and the sphere, which can be used as a space for modeling discrete distributions.
The exponential-family random graph models (ERGMs) have emerged as an important framework for modeling social and other networks. ERGMs for valued networks are less well-studied than their unvalued counterparts, and pose particular computational challenges. Networks with edge values on the non-negative integers (count-valued networks) are an important such case, with applications ranging from migration and trade flow data to data on frequency of interactions and encounters. Here, we propose an efficient maximum pseudo-likelihood estimation (MPLE) scheme for count-valued ERGMs, and compare its performance with existing Contrastive Divergence (CD) and Monte Carlo Maximum Likelihood Estimation (MCMLE) approaches via a simulation study based on migration flow networks in two U.S states. Our results suggest that edge value variance is a key factor in method performance, with high-variance edges posing a particular challenge for CD. MCMLE can work well but requires careful seeding in the high-variance case, and the MPLE itself performs well when edge variance is high.
This work focuses on combining nonparametric topic models with Auto-Encoding Variational Bayes (AEVB). Specifically, we first propose iTM-VAE, where the topics are treated as trainable parameters and the document-specific topic proportions are obtained by a stick-breaking construction. The inference of iTM-VAE is modeled by neural networks such that it can be computed in a simple feed-forward manner. We also describe how to introduce a hyper-prior into iTM-VAE so as to model the uncertainty of the prior parameter. Actually, the hyper-prior technique is quite general and we show that it can be applied to other AEVB based models to alleviate the {\it collapse-to-prior} problem elegantly. Moreover, we also propose HiTM-VAE, where the document-specific topic distributions are generated in a hierarchical manner. HiTM-VAE is even more flexible and can generate topic distributions with better variability. Experimental results on 20News and Reuters RCV1-V2 datasets show that the proposed models outperform the state-of-the-art baselines significantly. The advantages of the hyper-prior technique and the hierarchical model construction are also confirmed by experiments.
Discrete random structures are important tools in Bayesian nonparametrics and the resulting models have proven effective in density estimation, clustering, topic modeling and prediction, among others. In this paper, we consider nested processes and study the dependence structures they induce. Dependence ranges between homogeneity, corresponding to full exchangeability, and maximum heterogeneity, corresponding to (unconditional) independence across samples. The popular nested Dirichlet process is shown to degenerate to the fully exchangeable case when there are ties across samples at the observed or latent level. To overcome this drawback, inherent to nesting general discrete random measures, we introduce a novel class of latent nested processes. These are obtained by adding common and group-specific completely random measures and, then, normalising to yield dependent random probability measures. We provide results on the partition distributions induced by latent nested processes, and develop an Markov Chain Monte Carlo sampler for Bayesian inferences. A test for distributional homogeneity across groups is obtained as a by product. The results and their inferential implications are showcased on synthetic and real data.
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