arXiv subjects

Vinayak Rao

Publications and source records attributed to Vinayak Rao.

At least 19 recordsLinked to original sources

arXiv2026

Detecting Jumps on a Tree: a Hierarchical Normalized Stable Process Model for Evolution of Discrete Distributions

This work focuses on clustering populations whose hierarchical dependency structure can be described by a tree. A particular example that is the focus of this work is a phylogenetic tree, whose nodes represent different biological species. In this problem, clustering the populations at the leaves of the tree is equivalent to identifying branches in the tree where the populations at the parent and child node have significantly different distributions. We construct a nonparametric Bayesian model based on the hierarchical normalized stable process and the Poisson process to exploit this hierarchical structure, with a key contribution being the ability to share statistical information between subpopulations. We develop an efficient particle MCMC algorithm to address computational challenges involved with posterior inference. We illustrate the efficacy of our proposed approach on both synthetic and real-world problems.

stat.ME↗

arXiv2026

Statistical Properties of Nonparametric MLE under Laplace Noise

Local differential privacy (LDP) protects individuals in a dataset by perturbing each measurement before release. For real-valued data, a widely used mechanism is additive Laplace noise. We study the problem of estimating the distribution of the latent confidential data from the privatized observations via the nonparametric maximum likelihood estimator (NPMLE) under an i.i.d. sampling model. We first show that under the Laplace convolution model, the NPMLE admits a finite-dimensional reformulation in which the support is restricted to the observation set. This reduces the original infinite-dimensional optimization over all mixing distributions to an $n$-dimensional convex optimization over mixture weights, where $n$ is the sample size. We then study the statistical convergence of the NPMLE under the 1-Wasserstein distance, and explicitly connect its convergence rate with the privacy noise scale. Allowing the privacy noise level to change with the sample size, our analysis shows that the NPMLE remains consistent when the Laplace noise grows at a rate slower than $n^{3/16}$. Conversely, when the Laplace noise is of the order $\sqrt n$ or larger, no estimator can achieve uniformly consistent recovery of the latent distribution.

math.ST↗

arXiv2026

Offline Estimation of Controlled Markov Chains: Minimaxity and Sample Complexity

In this work, we study a natural nonparametric estimator of the transition probability matrices of a finite controlled Markov chain. We consider an offline setting with a fixed dataset, collected using a so-called logging policy. We develop sample complexity bounds for the estimator and establish conditions for minimaxity. Our statistical bounds depend on the logging policy through its mixing properties. We show that achieving a particular statistical risk bound involves a subtle and interesting trade-off between the strength of the mixing properties and the number of samples. We demonstrate the validity of our results under various examples, such as ergodic Markov chains, weakly ergodic inhomogeneous Markov chains, and controlled Markov chains with non-stationary Markov, episodic, and greedy controls. Lastly, we use these sample complexity bounds to establish concomitant ones for offline evaluation of stationary Markov control policies.

stat.ML↗

arXiv2026

Neural Diffusion Intensity Models for Point Process Data

Cox processes model overdispersed point process data via a latent stochastic intensity, but both nonparametric estimation of the intensity model and posterior inference over intensity paths are typically intractable, relying on expensive MCMC methods. We introduce Neural Diffusion Intensity Models, a variational framework for Cox processes driven by neural SDEs. Our key theoretical result, based on enlargement of filtrations, shows that conditioning on point process observations preserves the diffusion structure of the latent intensity with an explicit drift correction. This guarantees the variational family contains the true posterior, so that ELBO maximization coincides with maximum likelihood estimation under sufficient model capacity. We design an amortized encoder architecture that maps variable-length event sequences to posterior intensity paths by simulating the drift-corrected SDE, replacing repeated MCMC runs with a single forward pass. Experiments on synthetic and real-world data demonstrate accurate recovery of latent intensity dynamics and posterior paths, with orders-of-magnitude speedups over MCMC-based methods.

cs.LG↗

arXiv2026

Bayesian Repulsive Mixture Modeling with Matérn Point Processes

Mixture models are a standard tool in statistical analyses, widely used for density modeling and model-based clustering. In this work, we propose a Bayesian mixture model with repulsion between mixture components. Such repulsion helps address the problem of overlapping or poorly separated clusters, and assists with model interpretibility and robustness. Our modeling approach introduces repulsion via a generalized Matérn type-III repulsive point process model, and proceeds by applying a dependent sequential thinning scheme to a latent Poisson point process. A key feature of our model is that in contrast to most existing approaches to modeling repulsion, efficient posterior inference is possible via a Gibbs sampler, one that exploits the latent Poisson of our problem. This novel sampler also allows posterior inference over the number of clusters, and is of independent interest even in standard clustering applications without repulsion. We demonstrate the utility of the proposed method on a number of synthetic and real-world problems.

stat.ME↗

arXiv2026

Exact Gibbs sampling for stochastic differential equations with gradient drift and constant diffusion

Stochastic differential equations (SDEs) are an important class of time-series models, used to describe stochastic systems evolving in continuous time. Simulating paths from these processes, particularly after conditioning on noisy observations of the latent path, remains a challenge. Existing methods often introduce bias through time-discretization, require involved rejection sampling or debiasing schemes or are restricted to a narrow family of diffusions. In this work, we propose an exact Markov chain Monte Carlo (MCMC) sampling algorithm that is applicable to a broad subset of all SDEs with unit diffusion coefficient; after suitable transformation, this includes an even larger class of multivariate SDEs and most 1-d SDEs. We develop a Gibbs sampling framework that allows exact MCMC for such diffusions, without any discretization error. We demonstrate how our MCMC methodology requires only fairly straightforward simulation steps. Our framework can be extended to include parameter simulation, and allows tools from the Gaussian process literature to be easily applied. We evaluate our method on synthetic and real datasets, demonstrating superior performance to particle MCMC approaches.

stat.CO↗

arXiv2025

Adaptive Estimation of the Transition Density of Controlled Markov Chains

Estimating the transition dynamics of controlled Markov chains is crucial in fields such as time series analysis, reinforcement learning, and system exploration. Traditional non-parametric density estimation methods often assume independent samples and require oracle knowledge of smoothness parameters like the Hölder continuity coefficient. These assumptions are unrealistic in controlled Markovian settings, especially when the controls are non-Markovian, since such parameters need to hold uniformly over all control values. To address this gap, we propose an adaptive estimator for the transition densities of controlled Markov chains that does not rely on prior knowledge of smoothness parameters or assumptions about the control sequence distribution. Our method builds upon recent advances in adaptive density estimation by selecting an estimator that minimizes a loss function {and} fitting the observed data well, using a constrained minimax criterion over a dense class of estimators. We validate the performance of our estimator through oracle risk bounds, employing both randomized and deterministic versions of the Hellinger distance as loss functions. This approach provides a robust and flexible framework for estimating transition densities in controlled Markovian systems without imposing strong assumptions.

math.ST↗

arXiv2024

MCMC for Bayesian nonparametric mixture modeling under differential privacy

Estimating the probability density of a population while preserving the privacy of individuals in that population is an important and challenging problem that has received considerable attention in recent years. While the previous literature focused on frequentist approaches, in this paper, we propose a Bayesian nonparametric mixture model under differential privacy (DP) and present two Markov chain Monte Carlo (MCMC) algorithms for posterior inference. One is a marginal approach, resembling Neal's algorithm 5 with a pseudo-marginal Metropolis-Hastings move, and the other is a conditional approach. Although our focus is primarily on local DP, we show that our MCMC algorithms can be easily extended to deal with global differential privacy mechanisms. Moreover, for some carefully chosen mechanisms and mixture kernels, we show how auxiliary parameters can be analytically marginalized, allowing standard MCMC algorithms (i.e., non-privatized, such as Neal's Algorithm 2) to be efficiently employed. Our approach is general and applicable to any mixture model and privacy mechanism. In several simulations and a real case study, we discuss the performance of our algorithms and evaluate different privacy mechanisms proposed in the frequentist literature.

stat.CO↗

arXiv2022

Marginally Constrained Nonparametric Bayesian Inference through Gaussian Processes

Nonparametric Bayesian models are used routinely as flexible and powerful models of complex data. Many times, a statistician may have additional informative beliefs about data distribution of interest, e.g., its mean or subset components, that is not part of, or even compatible with, the nonparametric prior. An important challenge is then to incorporate this partial prior belief into nonparametric Bayesian models. In this paper, we are motivated by settings where practitioners have additional distributional information about a subset of the coordinates of the observations being modeled. Our approach links this problem to that of conditional density modeling. Our main idea is a novel constrained Bayesian model, based on a perturbation of a parametric distribution with a transformed Gaussian process prior on the perturbation function. We also develop a corresponding posterior sampling method based on data augmentation. We illustrate the efficacy of our proposed constrained nonparametric Bayesian model in a variety of real-world scenarios including modeling environmental and earthquake data.

stat.ME↗

arXiv2022

Privacy-Aware Rejection Sampling

Differential privacy (DP) offers strong theoretical privacy guarantees, but implementations of DP mechanisms may be vulnerable to side-channel attacks, such as timing attacks. When sampling methods such as MCMC or rejection sampling are used to implement a mechanism, the runtime can leak private information. We characterize the additional privacy cost due to the runtime of a rejection sampler in terms of both $(ε,δ)$-DP as well as $f$-DP. We also show that unless the acceptance probability is constant across databases, the runtime of a rejection sampler does not satisfy $ε$-DP for any $ε$. We show that there is a similar breakdown in privacy with adaptive rejection samplers. We propose three modifications to the rejection sampling algorithm, with varying assumptions, to protect against timing attacks by making the runtime independent of the data. The modification with the weakest assumptions is an approximate sampler, introducing a small increase in the privacy cost, whereas the other modifications give perfect samplers. We also use our techniques to develop an adaptive rejection sampler for log-Hölder densities, which also has data-independent runtime. We give several examples of DP mechanisms that fit the assumptions of our methods and can thus be implemented using our samplers.

cs.CR↗

arXiv2021

Set Twister for Single-hop Node Classification

Node classification is a central task in relational learning, with the current state-of-the-art hinging on two key principles: (i) predictions are permutation-invariant to the ordering of a node's neighbors, and (ii) predictions are a function of the node's $r$-hop neighborhood topology and attributes, $r \geq 2$. Both graph neural networks and collective inference methods (e.g., belief propagation) rely on information from up to $r$-hops away. In this work, we study if the use of more powerful permutation-invariant functions can sometimes avoid the need for classifiers to collect information beyond $1$-hop. Towards this, we introduce a new architecture, the Set Twister, which generalizes DeepSets (Zaheer et al., 2017), a simple and widely-used permutation-invariant representation. Set Twister theoretically increases expressiveness of DeepSets, allowing it to capture higher-order dependencies, while keeping its simplicity and low computational cost. Empirically, we see accuracy improvements of Set Twister over DeepSets as well as a variety of graph neural networks and collective inference schemes in several tasks, while showcasing its implementation simplicity and computational efficiency.

cs.LG↗

arXiv2021

Contextual Unsupervised Outlier Detection in Sequences

This work proposes an unsupervised learning framework for trajectory (sequence) outlier detection that combines ranking tests with user sequence models. The overall framework identifies sequence outliers at a desired false positive rate (FPR), in an otherwise parameter-free manner. We evaluate our methodology on a collection of real and simulated datasets based on user actions at the websites last.fm and msnbc.com, where we know ground truth, and demonstrate improved accuracy over existing approaches. We also apply our approach to a large real-world dataset of Pinterest and Facebook users, where we find that users tend to re-share Pinterest posts of Facebook friends significantly more than other types of users, pointing to a potential influence of Facebook friendship on sharing behavior on Pinterest.

cs.LG↗

arXiv2020

An Exact Auxiliary Variable Gibbs Sampler for a Class of Diffusions

Stochastic differential equations (SDEs) or diffusions are continuous-valued continuous-time stochastic processes widely used in the applied and mathematical sciences. Simulating paths from these processes is usually an intractable problem, and typically involves time-discretization approximations. We propose an exact Markov chain Monte Carlo sampling algorithm that involves no such time-discretization error. Our sampler is applicable to the problem of prior simulation from an SDE, posterior simulation conditioned on noisy observations, as well as parameter inference given noisy observations. Our work recasts an existing rejection sampling algorithm for a class of diffusions as a latent variable model, and then derives an auxiliary variable Gibbs sampling algorithm that targets the associated joint distribution. At a high level, the resulting algorithm involves two steps: simulating a random grid of times from an inhomogeneous Poisson process, and updating the SDE trajectory conditioned on this grid. Our work allows the vast literature of Monte Carlo sampling algorithms from the Gaussian process literature to be brought to bear to applications involving diffusions. We study our method on synthetic and real datasets, where we demonstrate superior performance over competing methods.

stat.CO↗

arXiv2020

Efficient parameter sampling for Markov jump processes

Markov jump processes (MJPs) are continuous-time stochastic processes widely used in a variety of applied disciplines. Inference for MJPs typically proceeds via Markov chain Monte Carlo, the state-of-the-art being a uniformization-based auxiliary variable Gibbs sampler. This was designed for situations where the MJP parameters are known, and Bayesian inference over unknown parameters is typically carried out by incorporating it into a larger Gibbs sampler. This strategy of sampling parameters given path, and path given parameters can result in poor Markov chain mixing. In this work, we propose a simple and elegant algorithm to address this problem. Our scheme brings Metropolis-Hastings approaches for discrete-time hidden Markov models to the continuous-time setting, resulting in a complete and clean recipe for parameter and path inference in MJPs. In our experiments, we demonstrate superior performance over Gibbs sampling, as well as another popular approach, particle MCMC. We also show our sampler inherits geometric mixing from an `ideal' sampler that operates without computational constraints.

stat.CO↗

arXiv2019

Flexible Mixture Modeling on Constrained Spaces

This paper addresses challenges in flexibly modeling multimodal data that lie on constrained spaces. Such data are commonly found in spatial applications, such as climatology and criminology, where measurements are restricted to a geographical area. Other settings include domains where unsuitable recordings are discarded, such as flow-cytometry measurements. A simple approach to modeling such data is through the use of mixture models, especially nonparametric mixture models. Mixture models, while flexible and theoretically well-understood, are unsuitable for settings involving complicated constraints, leading to difficulties in specifying the component distributions and in evaluating normalization constants. Bayesian inference over the parameters of these models results in posterior distributions that are doubly-intractable. We address this problem via an algorithm based on rejection sampling and data augmentation. We view samples from a truncated distribution as outcomes of a rejection sampling scheme, where proposals are made from a simple mixture model and are rejected if they violate the constraints. Our scheme proceeds by imputing the rejected samples given mixture parameters and then resampling parameters given all samples. We study two modeling approaches: mixtures of truncated Gaussians and truncated mixtures of Gaussians, along with their associated Markov chain Monte Carlo sampling algorithms. We also discuss variations of the models, as well as approximations to improve mixing, reduce computational cost, and lower variance. We present results on simulated data and apply our algorithms to two problems; one involving flow-cytometry data, and the other, crime recorded in the city of Chicago.

stat.CO↗

arXiv2019

Relational Pooling for Graph Representations

This work generalizes graph neural networks (GNNs) beyond those based on the Weisfeiler-Lehman (WL) algorithm, graph Laplacians, and diffusions. Our approach, denoted Relational Pooling (RP), draws from the theory of finite partial exchangeability to provide a framework with maximal representation power for graphs. RP can work with existing graph representation models and, somewhat counterintuitively, can make them even more powerful than the original WL isomorphism test. Additionally, RP allows architectures like Recurrent Neural Networks and Convolutional Neural Networks to be used in a theoretically sound approach for graph classification. We demonstrate improved performance of RP-based graph representations over state-of-the-art methods on a number of tasks.

cs.LG↗

arXiv2019

Community detection over a heterogeneous population of non-aligned networks

Clustering and community detection with multiple graphs have typically focused on aligned graphs, where there is a mapping between nodes across the graphs (e.g., multi-view, multi-layer, temporal graphs). However, there are numerous application areas with multiple graphs that are only partially aligned, or even unaligned. These graphs are often drawn from the same population, with communities of potentially different sizes that exhibit similar structure. In this paper, we develop a joint stochastic blockmodel (Joint SBM) to estimate shared communities across sets of heterogeneous non-aligned graphs. We derive an efficient spectral clustering approach to learn the parameters of the joint SBM. We evaluate the model on both synthetic and real-world datasets and show that the joint model is able to exploit cross-graph information to better estimate the communities compared to learning separate SBMs on each individual graph.

cs.SI↗

arXiv2019

Janossy Pooling: Learning Deep Permutation-Invariant Functions for Variable-Size Inputs

We consider a simple and overarching representation for permutation-invariant functions of sequences (or multiset functions). Our approach, which we call Janossy pooling, expresses a permutation-invariant function as the average of a permutation-sensitive function applied to all reorderings of the input sequence. This allows us to leverage the rich and mature literature on permutation-sensitive functions to construct novel and flexible permutation-invariant functions. If carried out naively, Janossy pooling can be computationally prohibitive. To allow computational tractability, we consider three kinds of approximations: canonical orderings of sequences, functions with $k$-order interactions, and stochastic optimization algorithms with random permutations. Our framework unifies a variety of existing work in the literature, and suggests possible modeling and algorithmic extensions. We explore a few in our experiments, which demonstrate improved performance over current state-of-the-art methods.

cs.LG↗