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Yee Whye Teh

Yee Whye Teh appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.

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Machine LearningComputationArtificial IntelligenceMethodologyApplicationsComputation and LanguageComputer Visionmath.PRmath.STStatistics TheoryInformation RetrievalProgramming Languages

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Published work

59 published item(s)

preprint2026arXiv

From Backward Spreading to Forward Replay: Revisiting Target Construction in LLM Parameter Editing

LLM parameter editing methods commonly rely on computing an ideal target hidden-state at a target layer (referred as anchor point) and distributing the target vector to multiple preceding layers (commonly known as backward spreading) for cooperative editing. Although widely used for a long time, its underlying basis have not been systematically investigated. In this paper, we first conduct a systematic study of its foundations, which helps clarify its capability boundaries, practical considerations, and potential failure modes. Then, we propose a simple and elegant alternative that replaces backward spreading with forward-propagation. Instead of optimizing the target at the last editing layer, we optimize the anchor point at the first editing layer, and then propagate it forward to obtain accurate and mutually compatible target hidden-states for all subsequent editing layers. This approach achieves the same computational complexity as existing methods while producing more accurate layer-wise targets. Our method is simple, without interfering with either the computation of the initial target hidden state or any other components of the subsequent editing pipeline, and thus constituting a benefit for a wide range of LLM parameter editing methods.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

Meta-Learning Objectives for Preference Optimization

Evaluating preference optimization (PO) algorithms on LLM alignment is a challenging task that presents prohibitive costs, noise, and several variables like model size and hyper-parameters. In this work, we show that it is possible to gain insights on the efficacy of PO algorithm on simpler benchmarks. We design a diagnostic suite of MuJoCo tasks and datasets, which we use to systematically evaluate PO algorithms, establishing a more controlled and cheaper benchmark. We then propose a novel family of PO algorithms based on mirror descent, which we call Mirror Preference Optimization (MPO). Through evolutionary strategies, we search this class to discover algorithms specialized to specific properties of preference datasets, such as mixed-quality or noisy data. We demonstrate that our discovered PO algorithms outperform all known algorithms in the targeted MuJoCo settings. Finally, based on the insights gained from our MuJoCo experiments, we design a PO algorithm that significantly outperform existing baselines in an LLM alignment task.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

Metropolis-Adjusted Diffusion Models

Sampling from score-based diffusion models incurs bias due to both time discretisation and the approximation of the score function. A common strategy for reducing this bias is to apply corrector steps based on the unadjusted Langevin algorithm (ULA) at each noise level within a predictor-corrector framework. However, ULA is itself a biased sampler, as it discretises a continuous diffusion process. In this work, we consider adjusted Langevin correctors that employ Metropolis--Hastings (MH) or Barker's accept-reject steps to correct for this bias. Since the target density ratio typically required by MH-based algorithms is unavailable, we propose methods that instead utilise the score function to compute the correct acceptance probability. We introduce the first exact method for adjusting Langevin corrections in diffusion models, based on a two-coin Bernoulli factory algorithm. We also propose an efficient approximation based on Simpson's rule that achieves accuracy of order $5/2$ in the step size at near-zero marginal cost. We demonstrate that these procedures improve sample quality on both synthetic and image datasets, yielding consistent gains in Fréchet Inception Distance (FID) on the latter.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

Selective Safety Steering via Value-Filtered Decoding

While large language models (LLMs) are trained to align with human values, their generations may still violate safety constraints. A growing line of work addresses this problem by modifying the model's sampling policy at decoding time using a safety reward. However, existing decoding-time steering methods often intervene unnecessarily, modifying generations that would have been safe under the base model. Such unnecessary interventions are undesirable, as they can distort key properties of the base model such as helpfulness, fluency, style, and coherence. We propose a new test-time steering method designed to reduce such unnecessary interventions while improving the safety of unsafe responses. Our approach filters tokens using a value-based safety criterion and provides an explicit bound on the probability of false interventions. A single threshold hyperparameter controls this bound, allowing practitioners to trade off higher rates of unnecessary intervention for better output safety. Across multiple datasets and experiments, we show that our value-filtered decoding method outperforms existing baselines, achieving better trade-offs between safety, helpfulness, and similarity to the base model.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

Verifier-Backed Hard Problem Generation for Mathematical Reasoning

Large Language Models (LLMs) demonstrate strong capabilities for solving scientific and mathematical problems, yet they struggle to produce valid, challenging, and novel problems - an essential component for advancing LLM training and enabling autonomous scientific research. Existing problem generation approaches either depend on expensive human expert involvement or adopt naive self-play paradigms, which frequently yield invalid problems due to reward hacking. This work introduces VHG, a verifier-enhanced hard problem generation framework built upon three-party self-play. By integrating an independent verifier into the conventional setter-solver duality, our design constrains the setter's reward to be jointly determined by problem validity (evaluated by the verifier) and difficulty (assessed by the solver). We instantiate two verifier variants: a Hard symbolic verifier and a Soft LLM-based verifier, with evaluations conducted on indefinite integral tasks and general mathematical reasoning tasks. Experimental results show that VHG substantially outperforms all baseline methods by a clear margin.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Amortized Rejection Sampling in Universal Probabilistic Programming

Naive approaches to amortized inference in probabilistic programs with unbounded loops can produce estimators with infinite variance. This is particularly true of importance sampling inference in programs that explicitly include rejection sampling as part of the user-programmed generative procedure. In this paper we develop a new and efficient amortized importance sampling estimator. We prove finite variance of our estimator and empirically demonstrate our method's correctness and efficiency compared to existing alternatives on generative programs containing rejection sampling loops and discuss how to implement our method in a generic probabilistic programming framework.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Bayesian Nonparametrics for Sparse Dynamic Networks

In this paper we propose a Bayesian nonparametric approach to modelling sparse time-varying networks. A positive parameter is associated to each node of a network, which models the sociability of that node. Sociabilities are assumed to evolve over time, and are modelled via a dynamic point process model. The model is able to capture long term evolution of the sociabilities. Moreover, it yields sparse graphs, where the number of edges grows subquadratically with the number of nodes. The evolution of the sociabilities is described by a tractable time-varying generalised gamma process. We provide some theoretical insights into the model and apply it to three datasets: a simulated network, a network of hyperlinks between communities on Reddit, and a network of co-occurences of words in Reuters news articles after the September 11th attacks.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Generative Models as Distributions of Functions

Generative models are typically trained on grid-like data such as images. As a result, the size of these models usually scales directly with the underlying grid resolution. In this paper, we abandon discretized grids and instead parameterize individual data points by continuous functions. We then build generative models by learning distributions over such functions. By treating data points as functions, we can abstract away from the specific type of data we train on and construct models that are agnostic to discretization. To train our model, we use an adversarial approach with a discriminator that acts on continuous signals. Through experiments on a wide variety of data modalities including images, 3D shapes and climate data, we demonstrate that our model can learn rich distributions of functions independently of data type and resolution.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Meta-Learning Sparse Compression Networks

Recent work in Deep Learning has re-imagined the representation of data as functions mapping from a coordinate space to an underlying continuous signal. When such functions are approximated by neural networks this introduces a compelling alternative to the more common multi-dimensional array representation. Recent work on such Implicit Neural Representations (INRs) has shown that - following careful architecture search - INRs can outperform established compression methods such as JPEG (e.g. Dupont et al., 2021). In this paper, we propose crucial steps towards making such ideas scalable: Firstly, we employ state-of-the-art network sparsification techniques to drastically improve compression. Secondly, introduce the first method allowing for sparsification to be employed in the inner-loop of commonly used Meta-Learning algorithms, drastically improving both compression and the computational cost of learning INRs. The generality of this formalism allows us to present results on diverse data modalities such as images, manifolds, signed distance functions, 3D shapes and scenes, several of which establish new state-of-the-art results.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

On Incorporating Inductive Biases into VAEs

We explain why directly changing the prior can be a surprisingly ineffective mechanism for incorporating inductive biases into VAEs, and introduce a simple and effective alternative approach: Intermediary Latent Space VAEs(InteL-VAEs). InteL-VAEs use an intermediary set of latent variables to control the stochasticity of the encoding process, before mapping these in turn to the latent representation using a parametric function that encapsulates our desired inductive bias(es). This allows us to impose properties like sparsity or clustering on learned representations, and incorporate human knowledge into the generative model. Whereas changing the prior only indirectly encourages behavior through regularizing the encoder, InteL-VAEs are able to directly enforce desired characteristics. Moreover, they bypass the computation and encoder design issues caused by non-Gaussian priors, while allowing for additional flexibility through training of the parametric mapping function. We show that these advantages, in turn, lead to both better generative models and better representations being learned.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Riemannian Diffusion Schrödinger Bridge

Score-based generative models exhibit state of the art performance on density estimation and generative modeling tasks. These models typically assume that the data geometry is flat, yet recent extensions have been developed to synthesize data living on Riemannian manifolds. Existing methods to accelerate sampling of diffusion models are typically not applicable in the Riemannian setting and Riemannian score-based methods have not yet been adapted to the important task of interpolation of datasets. To overcome these issues, we introduce \emph{Riemannian Diffusion Schrödinger Bridge}. Our proposed method generalizes Diffusion Schrödinger Bridge introduced in \cite{debortoli2021neurips} to the non-Euclidean setting and extends Riemannian score-based models beyond the first time reversal. We validate our proposed method on synthetic data and real Earth and climate data.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

Noise Contrastive Meta-Learning for Conditional Density Estimation using Kernel Mean Embeddings

Current meta-learning approaches focus on learning functional representations of relationships between variables, i.e. on estimating conditional expectations in regression. In many applications, however, we are faced with conditional distributions which cannot be meaningfully summarized using expectation only (due to e.g. multimodality). Hence, we consider the problem of conditional density estimation in the meta-learning setting. We introduce a novel technique for meta-learning which combines neural representation and noise-contrastive estimation with the established literature of conditional mean embeddings into reproducing kernel Hilbert spaces. The method is validated on synthetic and real-world problems, demonstrating the utility of sharing learned representations across multiple conditional density estimation tasks.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

Understanding the 2016 US Presidential Election using ecological inference and distribution regression with census microdata

We combine fine-grained spatially referenced census data with the vote outcomes from the 2016 US presidential election. Using this dataset, we perform ecological inference using distribution regression (Flaxman et al, KDD 2015) with a multinomial-logit regression so as to model the vote outcome Trump, Clinton, Other / Didn't vote as a function of demographic and socioeconomic features. Ecological inference allows us to estimate "exit poll" style results like what was Trump's support among white women, but for entirely novel categories. We also perform exploratory data analysis to understand which census variables are predictive of voting for Trump, voting for Clinton, or not voting for either. All of our methods are implemented in Python and R, and are available online for replication.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

A Unified Stochastic Gradient Approach to Designing Bayesian-Optimal Experiments

We introduce a fully stochastic gradient based approach to Bayesian optimal experimental design (BOED). Our approach utilizes variational lower bounds on the expected information gain (EIG) of an experiment that can be simultaneously optimized with respect to both the variational and design parameters. This allows the design process to be carried out through a single unified stochastic gradient ascent procedure, in contrast to existing approaches that typically construct a pointwise EIG estimator, before passing this estimator to a separate optimizer. We provide a number of different variational objectives including the novel adaptive contrastive estimation (ACE) bound. Finally, we show that our gradient-based approaches are able to provide effective design optimization in substantially higher dimensional settings than existing approaches.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

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.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Divide, Conquer, and Combine: a New Inference Strategy for Probabilistic Programs with Stochastic Support

Universal probabilistic programming systems (PPSs) provide a powerful framework for specifying rich probabilistic models. They further attempt to automate the process of drawing inferences from these models, but doing this successfully is severely hampered by the wide range of non--standard models they can express. As a result, although one can specify complex models in a universal PPS, the provided inference engines often fall far short of what is required. In particular, we show that they produce surprisingly unsatisfactory performance for models where the support varies between executions, often doing no better than importance sampling from the prior. To address this, we introduce a new inference framework: Divide, Conquer, and Combine, which remains efficient for such models, and show how it can be implemented as an automated and generic PPS inference engine. We empirically demonstrate substantial performance improvements over existing approaches on three examples.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Exploiting Hierarchy for Learning and Transfer in KL-regularized RL

As reinforcement learning agents are tasked with solving more challenging and diverse tasks, the ability to incorporate prior knowledge into the learning system and to exploit reusable structure in solution space is likely to become increasingly important. The KL-regularized expected reward objective constitutes one possible tool to this end. It introduces an additional component, a default or prior behavior, which can be learned alongside the policy and as such partially transforms the reinforcement learning problem into one of behavior modelling. In this work we consider the implications of this framework in cases where both the policy and default behavior are augmented with latent variables. We discuss how the resulting hierarchical structures can be used to implement different inductive biases and how their modularity can benefit transfer. Empirically we find that they can lead to faster learning and transfer on a range of continuous control tasks.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Functional Regularisation for Continual Learning with Gaussian Processes

We introduce a framework for Continual Learning (CL) based on Bayesian inference over the function space rather than the parameters of a deep neural network. This method, referred to as functional regularisation for Continual Learning, avoids forgetting a previous task by constructing and memorising an approximate posterior belief over the underlying task-specific function. To achieve this we rely on a Gaussian process obtained by treating the weights of the last layer of a neural network as random and Gaussian distributed. Then, the training algorithm sequentially encounters tasks and constructs posterior beliefs over the task-specific functions by using inducing point sparse Gaussian process methods. At each step a new task is first learnt and then a summary is constructed consisting of (i) inducing inputs -- a fixed-size subset of the task inputs selected such that it optimally represents the task -- and (ii) a posterior distribution over the function values at these inputs. This summary then regularises learning of future tasks, through Kullback-Leibler regularisation terms. Our method thus unites approaches focused on (pseudo-)rehearsal with those derived from a sequential Bayesian inference perspective in a principled way, leading to strong results on accepted benchmarks.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

MetaFun: Meta-Learning with Iterative Functional Updates

We develop a functional encoder-decoder approach to supervised meta-learning, where labeled data is encoded into an infinite-dimensional functional representation rather than a finite-dimensional one. Furthermore, rather than directly producing the representation, we learn a neural update rule resembling functional gradient descent which iteratively improves the representation. The final representation is used to condition the decoder to make predictions on unlabeled data. Our approach is the first to demonstrates the success of encoder-decoder style meta-learning methods like conditional neural processes on large-scale few-shot classification benchmarks such as miniImageNet and tieredImageNet, where it achieves state-of-the-art performance.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Non-exchangeable feature allocation models with sublinear growth of the feature sizes

Feature allocation models are popular models used in different applications such as unsupervised learning or network modeling. In particular, the Indian buffet process is a flexible and simple one-parameter feature allocation model where the number of features grows unboundedly with the number of objects. The Indian buffet process, like most feature allocation models, satisfies a symmetry property of exchangeability: the distribution is invariant under permutation of the objects. While this property is desirable in some cases, it has some strong implications. Importantly, the number of objects sharing a particular feature grows linearly with the number of objects. In this article, we describe a class of non-exchangeable feature allocation models where the number of objects sharing a given feature grows sublinearly, where the rate can be controlled by a tuning parameter. We derive the asymptotic properties of the model, and show that such model provides a better fit and better predictive performances on various datasets.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Probabilistic symmetries and invariant neural networks

Treating neural network inputs and outputs as random variables, we characterize the structure of neural networks that can be used to model data that are invariant or equivariant under the action of a compact group. Much recent research has been devoted to encoding invariance under symmetry transformations into neural network architectures, in an effort to improve the performance of deep neural networks in data-scarce, non-i.i.d., or unsupervised settings. By considering group invariance from the perspective of probabilistic symmetry, we establish a link between functional and probabilistic symmetry, and obtain generative functional representations of probability distributions that are invariant or equivariant under the action of a compact group. Our representations completely characterize the structure of neural networks that can be used to model such distributions and yield a general program for constructing invariant stochastic or deterministic neural networks. We demonstrate that examples from the recent literature are special cases, and develop the details of the general program for exchangeable sequences and arrays.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Uncertainty Estimation Using a Single Deep Deterministic Neural Network

We propose a method for training a deterministic deep model that can find and reject out of distribution data points at test time with a single forward pass. Our approach, deterministic uncertainty quantification (DUQ), builds upon ideas of RBF networks. We scale training in these with a novel loss function and centroid updating scheme and match the accuracy of softmax models. By enforcing detectability of changes in the input using a gradient penalty, we are able to reliably detect out of distribution data. Our uncertainty quantification scales well to large datasets, and using a single model, we improve upon or match Deep Ensembles in out of distribution detection on notable difficult dataset pairs such as FashionMNIST vs. MNIST, and CIFAR-10 vs. SVHN.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Variational Bayesian Optimal Experimental Design

Bayesian optimal experimental design (BOED) is a principled framework for making efficient use of limited experimental resources. Unfortunately, its applicability is hampered by the difficulty of obtaining accurate estimates of the expected information gain (EIG) of an experiment. To address this, we introduce several classes of fast EIG estimators by building on ideas from amortized variational inference. We show theoretically and empirically that these estimators can provide significant gains in speed and accuracy over previous approaches. We further demonstrate the practicality of our approach on a number of end-to-end experiments.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

A characterization of product-form exchangeable feature probability functions

We characterize the class of exchangeable feature allocations assigning probability $V_{n,k}\prod_{l=1}^{k}W_{m_{l}}U_{n-m_{l}}$ to a feature allocation of $n$ individuals, displaying $k$ features with counts $(m_{1},\ldots,m_{k})$ for these features. Each element of this class is parametrized by a countable matrix $V$ and two sequences $U$ and $W$ of non-negative weights. Moreover, a consistency condition is imposed to guarantee that the distribution for feature allocations of $n-1$ individuals is recovered from that of $n$ individuals, when the last individual is integrated out. In Theorem 1.1, we prove that the only members of this class satisfying the consistency condition are mixtures of the Indian Buffet Process over its mass parameter $γ$ and mixtures of the Beta--Bernoulli model over its dimensionality parameter $N$. Hence, we provide a characterization of these two models as the only, up to randomization of the parameters, consistent exchangeable feature allocations having the required product form.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

A nonparametric HMM for genetic imputation and coalescent inference

Genetic sequence data are well described by hidden Markov models (HMMs) in which latent states correspond to clusters of similar mutation patterns. Theory from statistical genetics suggests that these HMMs are nonhomogeneous (their transition probabilities vary along the chromosome) and have large support for self transitions. We develop a new nonparametric model of genetic sequence data, based on the hierarchical Dirichlet process, which supports these self transitions and nonhomogeneity. Our model provides a parameterization of the genetic process that is more parsimonious than other more general nonparametric models which have previously been applied to population genetics. We provide truncation-free MCMC inference for our model using a new auxiliary sampling scheme for Bayesian nonparametric HMMs. In a series of experiments on male X chromosome data from the Thousand Genomes Project and also on data simulated from a population bottleneck we show the benefits of our model over the popular finite model fastPHASE, which can itself be seen as a parametric truncation of our model. We find that the number of HMM states found by our model is correlated with the time to the most recent common ancestor in population bottlenecks. This work demonstrates the flexibility of Bayesian nonparametrics applied to large and complex genetic data.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

DR-ABC: Approximate Bayesian Computation with Kernel-Based Distribution Regression

Performing exact posterior inference in complex generative models is often difficult or impossible due to an expensive to evaluate or intractable likelihood function. Approximate Bayesian computation (ABC) is an inference framework that constructs an approximation to the true likelihood based on the similarity between the observed and simulated data as measured by a predefined set of summary statistics. Although the choice of appropriate problem-specific summary statistics crucially influences the quality of the likelihood approximation and hence also the quality of the posterior sample in ABC, there are only few principled general-purpose approaches to the selection or construction of such summary statistics. In this paper, we develop a novel framework for this task using kernel-based distribution regression. We model the functional relationship between data distributions and the optimal choice (with respect to a loss function) of summary statistics using kernel-based distribution regression. We show that our approach can be implemented in a computationally and statistically efficient way using the random Fourier features framework for large-scale kernel learning. In addition to that, our framework shows superior performance when compared to related methods on toy and real-world problems.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Gaussian Processes for Survival Analysis

We introduce a semi-parametric Bayesian model for survival analysis. The model is centred on a parametric baseline hazard, and uses a Gaussian process to model variations away from it nonparametrically, as well as dependence on covariates. As opposed to many other methods in survival analysis, our framework does not impose unnecessary constraints in the hazard rate or in the survival function. Furthermore, our model handles left, right and interval censoring mechanisms common in survival analysis. We propose a MCMC algorithm to perform inference and an approximation scheme based on random Fourier features to make computations faster. We report experimental results on synthetic and real data, showing that our model performs better than competing models such as Cox proportional hazards, ANOVA-DDP and random survival forests.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Image Retrieval with a Bayesian Model of Relevance Feedback

A content-based image retrieval system based on multinomial relevance feedback is proposed. The system relies on an interactive search paradigm where at each round a user is presented with k images and selects the one closest to their ideal target. Two approaches, one based on the Dirichlet distribution and one based the Beta distribution, are used to model the problem motivating an algorithm that trades exploration and exploitation in presenting the images in each round. Experimental results show that the new approach compares favourably with previous work.

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preprint2016arXiv

Mondrian Forests for Large-Scale Regression when Uncertainty Matters

Many real-world regression problems demand a measure of the uncertainty associated with each prediction. Standard decision forests deliver efficient state-of-the-art predictive performance, but high-quality uncertainty estimates are lacking. Gaussian processes (GPs) deliver uncertainty estimates, but scaling GPs to large-scale data sets comes at the cost of approximating the uncertainty estimates. We extend Mondrian forests, first proposed by Lakshminarayanan et al. (2014) for classification problems, to the large-scale non-parametric regression setting. Using a novel hierarchical Gaussian prior that dovetails with the Mondrian forest framework, we obtain principled uncertainty estimates, while still retaining the computational advantages of decision forests. Through a combination of illustrative examples, real-world large-scale datasets, and Bayesian optimization benchmarks, we demonstrate that Mondrian forests outperform approximate GPs on large-scale regression tasks and deliver better-calibrated uncertainty assessments than decision-forest-based methods.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Poisson Random Fields for Dynamic Feature Models

We present the Wright-Fisher Indian buffet process (WF-IBP), a probabilistic model for time-dependent data assumed to have been generated by an unknown number of latent features. This model is suitable as a prior in Bayesian nonparametric feature allocation models in which the features underlying the observed data exhibit a dependency structure over time. More specifically, we establish a new framework for generating dependent Indian buffet processes, where the Poisson random field model from population genetics is used as a way of constructing dependent beta processes. Inference in the model is complex, and we describe a sophisticated Markov Chain Monte Carlo algorithm for exact posterior simulation. We apply our construction to develop a nonparametric focused topic model for collections of time-stamped text documents and test it on the full corpus of NIPS papers published from 1987 to 2015.

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preprint2016arXiv

Posterior Consistency for a Non-parametric Survival Model under a Gaussian Process Prior

In this paper, we prove almost surely consistency of a Survival Analysis model, which puts a Gaussian process, mapped to the unit interval, as a prior on the so-called hazard function. We assume our data is given by survival lifetimes $T$ belonging to $\mathbb{R}^{+}$, and covariates on $[0,1]^d$, where $d$ is an arbitrary dimension. We define an appropriate metric for survival functions and prove posterior consistency with respect to this metric. Our proof is based on an extension of the theorem of Schwartz (1965), which gives general conditions for proving almost surely consistency in the setting of non i.i.d random variables. Due to the nature of our data, several results for Gaussian processes on $\mathbb{R}^+$ are proved which may be of independent interest.

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preprint2016arXiv

Relativistic Monte Carlo

Hamiltonian Monte Carlo (HMC) is a popular Markov chain Monte Carlo (MCMC) algorithm that generates proposals for a Metropolis-Hastings algorithm by simulating the dynamics of a Hamiltonian system. However, HMC is sensitive to large time discretizations and performs poorly if there is a mismatch between the spatial geometry of the target distribution and the scales of the momentum distribution. In particular the mass matrix of HMC is hard to tune well. In order to alleviate these problems we propose relativistic Hamiltonian Monte Carlo, a version of HMC based on relativistic dynamics that introduce a maximum velocity on particles. We also derive stochastic gradient versions of the algorithm and show that the resulting algorithms bear interesting relationships to gradient clipping, RMSprop, Adagrad and Adam, popular optimisation methods in deep learning. Based on this, we develop relativistic stochastic gradient descent by taking the zero-temperature limit of relativistic stochastic gradient Hamiltonian Monte Carlo. In experiments we show that the relativistic algorithms perform better than classical Newtonian variants and Adam.

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preprint2016arXiv

The Mondrian Kernel

We introduce the Mondrian kernel, a fast random feature approximation to the Laplace kernel. It is suitable for both batch and online learning, and admits a fast kernel-width-selection procedure as the random features can be re-used efficiently for all kernel widths. The features are constructed by sampling trees via a Mondrian process [Roy and Teh, 2009], and we highlight the connection to Mondrian forests [Lakshminarayanan et al., 2014], where trees are also sampled via a Mondrian process, but fit independently. This link provides a new insight into the relationship between kernel methods and random forests.

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preprint2015arXiv

Consistency and fluctuations for stochastic gradient Langevin dynamics

Applying standard Markov chain Monte Carlo (MCMC) algorithms to large data sets is computationally expensive. Both the calculation of the acceptance probability and the creation of informed proposals usually require an iteration through the whole data set. The recently proposed stochastic gradient Langevin dynamics (SGLD) method circumvents this problem by generating proposals which are only based on a subset of the data, by skipping the accept-reject step and by using decreasing step-sizes sequence $(δ_m)_{m \geq 0}$. %Under appropriate Lyapunov conditions, We provide in this article a rigorous mathematical framework for analysing this algorithm. We prove that, under verifiable assumptions, the algorithm is consistent, satisfies a central limit theorem (CLT) and its asymptotic bias-variance decomposition can be characterized by an explicit functional of the step-sizes sequence $(δ_m)_{m \geq 0}$. We leverage this analysis to give practical recommendations for the notoriously difficult tuning of this algorithm: it is asymptotically optimal to use a step-size sequence of the type $δ_m \asymp m^{-1/3}$, leading to an algorithm whose mean squared error (MSE) decreases at rate $\mathcal{O}(m^{-1/3})$

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preprint2015arXiv

Expectation Particle Belief Propagation

We propose an original particle-based implementation of the Loopy Belief Propagation (LPB) algorithm for pairwise Markov Random Fields (MRF) on a continuous state space. The algorithm constructs adaptively efficient proposal distributions approximating the local beliefs at each note of the MRF. This is achieved by considering proposal distributions in the exponential family whose parameters are updated iterately in an Expectation Propagation (EP) framework. The proposed particle scheme provides consistent estimation of the LBP marginals as the number of particles increases. We demonstrate that it provides more accurate results than the Particle Belief Propagation (PBP) algorithm of Ihler and McAllester (2009) at a fraction of the computational cost and is additionally more robust empirically. The computational complexity of our algorithm at each iteration is quadratic in the number of particles. We also propose an accelerated implementation with sub-quadratic computational complexity which still provides consistent estimates of the loopy BP marginal distributions and performs almost as well as the original procedure.

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preprint2015arXiv

Mondrian Forests: Efficient Online Random Forests

Ensembles of randomized decision trees, usually referred to as random forests, are widely used for classification and regression tasks in machine learning and statistics. Random forests achieve competitive predictive performance and are computationally efficient to train and test, making them excellent candidates for real-world prediction tasks. The most popular random forest variants (such as Breiman's random forest and extremely randomized trees) operate on batches of training data. Online methods are now in greater demand. Existing online random forests, however, require more training data than their batch counterpart to achieve comparable predictive performance. In this work, we use Mondrian processes (Roy and Teh, 2009) to construct ensembles of random decision trees we call Mondrian forests. Mondrian forests can be grown in an incremental/online fashion and remarkably, the distribution of online Mondrian forests is the same as that of batch Mondrian forests. Mondrian forests achieve competitive predictive performance comparable with existing online random forests and periodically re-trained batch random forests, while being more than an order of magnitude faster, thus representing a better computation vs accuracy tradeoff.

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preprint2015arXiv

Particle Gibbs for Bayesian Additive Regression Trees

Additive regression trees are flexible non-parametric models and popular off-the-shelf tools for real-world non-linear regression. In application domains, such as bioinformatics, where there is also demand for probabilistic predictions with measures of uncertainty, the Bayesian additive regression trees (BART) model, introduced by Chipman et al. (2010), is increasingly popular. As data sets have grown in size, however, the standard Metropolis-Hastings algorithms used to perform inference in BART are proving inadequate. In particular, these Markov chains make local changes to the trees and suffer from slow mixing when the data are high-dimensional or the best fitting trees are more than a few layers deep. We present a novel sampler for BART based on the Particle Gibbs (PG) algorithm (Andrieu et al., 2010) and a top-down particle filtering algorithm for Bayesian decision trees (Lakshminarayanan et al., 2013). Rather than making local changes to individual trees, the PG sampler proposes a complete tree to fit the residual. Experiments show that the PG sampler outperforms existing samplers in many settings.

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preprint2015arXiv

Rediscovery of Good-Turing estimators via Bayesian nonparametrics

The problem of estimating discovery probabilities originated in the context of statistical ecology, and in recent years it has become popular due to its frequent appearance in challenging applications arising in genetics, bioinformatics, linguistics, designs of experiments, machine learning, etc. A full range of statistical approaches, parametric and nonparametric as well as frequentist and Bayesian, has been proposed for estimating discovery probabilities. In this paper we investigate the relationships between the celebrated Good-Turing approach, which is a frequentist nonparametric approach developed in the 1940s, and a Bayesian nonparametric approach recently introduced in the literature. Specifically, under the assumption of a two parameter Poisson-Dirichlet prior, we show that Bayesian nonparametric estimators of discovery probabilities are asymptotically equivalent, for a large sample size, to suitably smoothed Good-Turing estimators. As a by-product of this result, we introduce and investigate a methodology for deriving exact and asymptotic credible intervals to be associated with the Bayesian nonparametric estimators of discovery probabilities. The proposed methodology is illustrated through a comprehensive simulation study and the analysis of Expressed Sequence Tags data generated by sequencing a benchmark complementary DNA library.

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preprint2015arXiv

Scaled subordinators and generalizations of the Indian buffet process

We study random families of subsets of $\mathbb{N}$ that are similar to exchangeable random partitions, but do not require constituent sets to be disjoint: Each element of ${\mathbb{N}}$ may be contained in multiple subsets. One class of such objects, known as Indian buffet processes, has become a popular tool in machine learning. Based on an equivalence between Indian buffet and scale-invariant Poisson processes, we identify a random scaling variable whose role is similar to that played in exchangeable partition models by the total mass of a random measure. Analogous to the construction of exchangeable partitions from normalized subordinators, random families of sets can be constructed from randomly scaled subordinators. Coupling to a heavy-tailed scaling variable induces a power law on the number of sets containing the first $n$ elements. Several examples, with properties desirable in applications, are derived explicitly. A relationship to exchangeable partitions is made precise as a correspondence between scaled subordinators and Poisson-Kingman measures, generalizing a result of Arratia, Barbour and Tavare on scale-invariant processes.

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preprint2015arXiv

The Mondrian Process for Machine Learning

This report is concerned with the Mondrian process and its applications in machine learning. The Mondrian process is a guillotine-partition-valued stochastic process that possesses an elegant self-consistency property. The first part of the report uses simple concepts from applied probability to define the Mondrian process and explore its properties. The Mondrian process has been used as the main building block of a clever online random forest classification algorithm that turns out to be equivalent to its batch counterpart. We outline a slight adaptation of this algorithm to regression, as the remainder of the report uses regression as a case study of how Mondrian processes can be utilized in machine learning. In particular, the Mondrian process will be used to construct a fast approximation to the computationally expensive kernel ridge regression problem with a Laplace kernel. The complexity of random guillotine partitions generated by a Mondrian process and hence the complexity of the resulting regression models is controlled by a lifetime hyperparameter. It turns out that these models can be efficiently trained and evaluated for all lifetimes in a given range at once, without needing to retrain them from scratch for each lifetime value. This leads to an efficient procedure for determining the right model complexity for a dataset at hand. The limitation of having a single lifetime hyperparameter will motivate the final Mondrian grid model, in which each input dimension is endowed with its own lifetime parameter. In this model we preserve the property that its hyperparameters can be tweaked without needing to retrain the modified model from scratch.

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preprint2014arXiv

Adaptive Reconfiguration Moves for Dirichlet Mixtures

Bayesian mixture models are widely applied for unsupervised learning and exploratory data analysis. Markov chain Monte Carlo based on Gibbs sampling and split-merge moves are widely used for inference in these models. However, both methods are restricted to limited types of transitions and suffer from torpid mixing and low accept rates even for problems of modest size. We propose a method that considers a broader range of transitions that are close to equilibrium by exploiting multiple chains in parallel and using the past states adaptively to inform the proposal distribution. The method significantly improves on Gibbs and split-merge sampling as quantified using convergence diagnostics and acceptance rates. Adaptive MCMC methods which use past states to inform the proposal distribution has given rise to many ingenious sampling schemes for continuous problems and the present work can be seen as an important first step in bringing these benefits to partition-based problems

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preprint2014arXiv

Asynchronous Anytime Sequential Monte Carlo

We introduce a new sequential Monte Carlo algorithm we call the particle cascade. The particle cascade is an asynchronous, anytime alternative to traditional particle filtering algorithms. It uses no barrier synchronizations which leads to improved particle throughput and memory efficiency. It is an anytime algorithm in the sense that it can be run forever to emit an unbounded number of particles while keeping within a fixed memory budget. We prove that the particle cascade is an unbiased marginal likelihood estimator which means that it can be straightforwardly plugged into existing pseudomarginal methods.

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preprint2014arXiv

Bayesian Nonparametric Crowdsourcing

Crowdsourcing has been proven to be an effective and efficient tool to annotate large datasets. User annotations are often noisy, so methods to combine the annotations to produce reliable estimates of the ground truth are necessary. We claim that considering the existence of clusters of users in this combination step can improve the performance. This is especially important in early stages of crowdsourcing implementations, where the number of annotations is low. At this stage there is not enough information to accurately estimate the bias introduced by each annotator separately, so we have to resort to models that consider the statistical links among them. In addition, finding these clusters is interesting in itself as knowing the behavior of the pool of annotators allows implementing efficient active learning strategies. Based on this, we propose in this paper two new fully unsupervised models based on a Chinese Restaurant Process (CRP) prior and a hierarchical structure that allows inferring these groups jointly with the ground truth and the properties of the users. Efficient inference algorithms based on Gibbs sampling with auxiliary variables are proposed. Finally, we perform experiments, both on synthetic and real databases, to show the advantages of our models over state-of-the-art algorithms.

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preprint2014arXiv

Bayesian nonparametric Plackett-Luce models for the analysis of preferences for college degree programmes

In this paper we propose a Bayesian nonparametric model for clustering partial ranking data. We start by developing a Bayesian nonparametric extension of the popular Plackett-Luce choice model that can handle an infinite number of choice items. Our framework is based on the theory of random atomic measures, with the prior specified by a completely random measure. We characterise the posterior distribution given data, and derive a simple and effective Gibbs sampler for posterior simulation. We then develop a Dirichlet process mixture extension of our model and apply it to investigate the clustering of preferences for college degree programmes amongst Irish secondary school graduates. The existence of clusters of applicants who have similar preferences for degree programmes is established and we determine that subject matter and geographical location of the third level institution characterise these clusters.

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preprint2013arXiv

Belief Optimization for Binary Networks: A Stable Alternative to Loopy Belief Propagation

We present a novel inference algorithm for arbitrary, binary, undirected graphs. Unlike loopy belief propagation, which iterates fixed point equations, we directly descend on the Bethe free energy. The algorithm consists of two phases, first we update the pairwise probabilities, given the marginal probabilities at each unit,using an analytic expression. Next, we update the marginal probabilities, given the pairwise probabilities by following the negative gradient of the Bethe free energy. Both steps are guaranteed to decrease the Bethe free energy, and since it is lower bounded, the algorithm is guaranteed to converge to a local minimum. We also show that the Bethe free energy is equal to the TAP free energy up to second order in the weights. In experiments we confirm that when belief propagation converges it usually finds identical solutions as our belief optimization method. However, in cases where belief propagation fails to converge, belief optimization continues to converge to reasonable beliefs. The stable nature of belief optimization makes it ideally suited for learning graphical models from data.

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preprint2013arXiv

Discovering Multiple Constraints that are Frequently Approximately Satisfied

Some high-dimensional data.sets can be modelled by assuming that there are many different linear constraints, each of which is Frequently Approximately Satisfied (FAS) by the data. The probability of a data vector under the model is then proportional to the product of the probabilities of its constraint violations. We describe three methods of learning products of constraints using a heavy-tailed probability distribution for the violations.

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preprint2013arXiv

Fast MCMC sampling for Markov jump processes and extensions

Markov jump processes (or continuous-time Markov chains) are a simple and important class of continuous-time dynamical systems. In this paper, we tackle the problem of simulating from the posterior distribution over paths in these models, given partial and noisy observations. Our approach is an auxiliary variable Gibbs sampler, and is based on the idea of uniformization. This sets up a Markov chain over paths by alternately sampling a finite set of virtual jump times given the current path and then sampling a new path given the set of extant and virtual jump times using a standard hidden Markov model forward filtering-backward sampling algorithm. Our method is exact and does not involve approximations like time-discretization. We demonstrate how our sampler extends naturally to MJP-based models like Markov-modulated Poisson processes and continuous-time Bayesian networks and show significant computational benefits over state-of-the-art MCMC samplers for these models.

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preprint2013arXiv

Inferring ground truth from multi-annotator ordinal data: a probabilistic approach

A popular approach for large scale data annotation tasks is crowdsourcing, wherein each data point is labeled by multiple noisy annotators. We consider the problem of inferring ground truth from noisy ordinal labels obtained from multiple annotators of varying and unknown expertise levels. Annotation models for ordinal data have been proposed mostly as extensions of their binary/categorical counterparts and have received little attention in the crowdsourcing literature. We propose a new model for crowdsourced ordinal data that accounts for instance difficulty as well as annotator expertise, and derive a variational Bayesian inference algorithm for parameter estimation. We analyze the ordinal extensions of several state-of-the-art annotator models for binary/categorical labels and evaluate the performance of all the models on two real world datasets containing ordinal query-URL relevance scores, collected through Amazon's Mechanical Turk. Our results indicate that the proposed model performs better or as well as existing state-of-the-art methods and is more resistant to `spammy' annotators (i.e., annotators who assign labels randomly without actually looking at the instance) than popular baselines such as mean, median, and majority vote which do not account for annotator expertise.

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preprint2013arXiv

MCMC for Normalized Random Measure Mixture Models

This paper concerns the use of Markov chain Monte Carlo methods for posterior sampling in Bayesian nonparametric mixture models with normalized random measure priors. Making use of some recent posterior characterizations for the class of normalized random measures, we propose novel Markov chain Monte Carlo methods of both marginal type and conditional type. The proposed marginal samplers are generalizations of Neal's well-regarded Algorithm 8 for Dirichlet process mixture models, whereas the conditional sampler is a variation of those recently introduced in the literature. For both the marginal and conditional methods, we consider as a running example a mixture model with an underlying normalized generalized Gamma process prior, and describe comparative simulation results demonstrating the efficacies of the proposed methods.

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preprint2013arXiv

Top-down particle filtering for Bayesian decision trees

Decision tree learning is a popular approach for classification and regression in machine learning and statistics, and Bayesian formulations---which introduce a prior distribution over decision trees, and formulate learning as posterior inference given data---have been shown to produce competitive performance. Unlike classic decision tree learning algorithms like ID3, C4.5 and CART, which work in a top-down manner, existing Bayesian algorithms produce an approximation to the posterior distribution by evolving a complete tree (or collection thereof) iteratively via local Monte Carlo modifications to the structure of the tree, e.g., using Markov chain Monte Carlo (MCMC). We present a sequential Monte Carlo (SMC) algorithm that instead works in a top-down manner, mimicking the behavior and speed of classic algorithms. We demonstrate empirically that our approach delivers accuracy comparable to the most popular MCMC method, but operates more than an order of magnitude faster, and thus represents a better computation-accuracy tradeoff.

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preprint2012arXiv

A Fast and Simple Algorithm for Training Neural Probabilistic Language Models

In spite of their superior performance, neural probabilistic language models (NPLMs) remain far less widely used than n-gram models due to their notoriously long training times, which are measured in weeks even for moderately-sized datasets. Training NPLMs is computationally expensive because they are explicitly normalized, which leads to having to consider all words in the vocabulary when computing the log-likelihood gradients. We propose a fast and simple algorithm for training NPLMs based on noise-contrastive estimation, a newly introduced procedure for estimating unnormalized continuous distributions. We investigate the behaviour of the algorithm on the Penn Treebank corpus and show that it reduces the training times by more than an order of magnitude without affecting the quality of the resulting models. The algorithm is also more efficient and much more stable than importance sampling because it requires far fewer noise samples to perform well. We demonstrate the scalability of the proposed approach by training several neural language models on a 47M-word corpus with a 80K-word vocabulary, obtaining state-of-the-art results on the Microsoft Research Sentence Completion Challenge dataset.

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preprint2012arXiv

Bayesian nonparametric models for ranked data

We develop a Bayesian nonparametric extension of the popular Plackett-Luce choice model that can handle an infinite number of choice items. Our framework is based on the theory of random atomic measures, with the prior specified by a gamma process. We derive a posterior characterization and a simple and effective Gibbs sampler for posterior simulation. We develop a time-varying extension of our model, and apply it to the New York Times lists of weekly bestselling books.

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preprint2012arXiv

Bayesian Rose Trees

Hierarchical structure is ubiquitous in data across many domains. There are many hierarchical clustering methods, frequently used by domain experts, which strive to discover this structure. However, most of these methods limit discoverable hierarchies to those with binary branching structure. This limitation, while computationally convenient, is often undesirable. In this paper we explore a Bayesian hierarchical clustering algorithm that can produce trees with arbitrary branching structure at each node, known as rose trees. We interpret these trees as mixtures over partitions of a data set, and use a computationally efficient, greedy agglomerative algorithm to find the rose trees which have high marginal likelihood given the data. Lastly, we perform experiments which demonstrate that rose trees are better models of data than the typical binary trees returned by other hierarchical clustering algorithms.

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preprint2012arXiv

Fast MCMC sampling for Markov jump processes and continuous time Bayesian networks

Markov jump processes and continuous time Bayesian networks are important classes of continuous time dynamical systems. In this paper, we tackle the problem of inferring unobserved paths in these models by introducing a fast auxiliary variable Gibbs sampler. Our approach is based on the idea of uniformization, and sets up a Markov chain over paths by sampling a finite set of virtual jump times and then running a standard hidden Markov model forward filtering-backward sampling algorithm over states at the set of extant and virtual jump times. We demonstrate significant computational benefits over a state-of-the-art Gibbs sampler on a number of continuous time Bayesian networks.

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preprint2012arXiv

Hybrid Variational/Gibbs Collapsed Inference in Topic Models

Variational Bayesian inference and (collapsed) Gibbs sampling are the two important classes of inference algorithms for Bayesian networks. Both have their advantages and disadvantages: collapsed Gibbs sampling is unbiased but is also inefficient for large count values and requires averaging over many samples to reduce variance. On the other hand, variational Bayesian inference is efficient and accurate for large count values but suffers from bias for small counts. We propose a hybrid algorithm that combines the best of both worlds: it samples very small counts and applies variational updates to large counts. This hybridization is shown to significantly improve testset perplexity relative to variational inference at no computational cost.

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preprint2012arXiv

On Smoothing and Inference for Topic Models

Latent Dirichlet analysis, or topic modeling, is a flexible latent variable framework for modeling high-dimensional sparse count data. Various learning algorithms have been developed in recent years, including collapsed Gibbs sampling, variational inference, and maximum a posteriori estimation, and this variety motivates the need for careful empirical comparisons. In this paper, we highlight the close connections between these approaches. We find that the main differences are attributable to the amount of smoothing applied to the counts. When the hyperparameters are optimized, the differences in performance among the algorithms diminish significantly. The ability of these algorithms to achieve solutions of comparable accuracy gives us the freedom to select computationally efficient approaches. Using the insights gained from this comparative study, we show how accurate topic models can be learned in several seconds on text corpora with thousands of documents.

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preprint2012arXiv

Structured Region Graphs: Morphing EP into GBP

GBP and EP are two successful algorithms for approximate probabilistic inference, which are based on different approximation strategies. An open problem in both algorithms has been how to choose an appropriate approximation structure. We introduce 'structured region graphs', a formalism which marries these two strategies, reveals a deep connection between them, and suggests how to choose good approximation structures. In this formalism, each region has an internal structure which defines an exponential family, whose sufficient statistics must be matched by the parent region. Reduction operators on these structures allow conversion between EP and GBP free energies. Thus it is revealed that all EP approximations on discrete variables are special cases of GBP, and conversely that some wellknown GBP approximations, such as overlapping squares, are special cases of EP. Furthermore, region graphs derived from EP have a number of good structural properties, including maxent-normality and overall counting number of one. The result is a convenient framework for producing high-quality approximations with a user-adjustable level of complexity

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preprint2011arXiv

Learning Item Trees for Probabilistic Modelling of Implicit Feedback

User preferences for items can be inferred from either explicit feedback, such as item ratings, or implicit feedback, such as rental histories. Research in collaborative filtering has concentrated on explicit feedback, resulting in the development of accurate and scalable models. However, since explicit feedback is often difficult to collect it is important to develop effective models that take advantage of the more widely available implicit feedback. We introduce a probabilistic approach to collaborative filtering with implicit feedback based on modelling the user's item selection process. In the interests of scalability, we restrict our attention to tree-structured distributions over items and develop a principled and efficient algorithm for learning item trees from data. We also identify a problem with a widely used protocol for evaluating implicit feedback models and propose a way of addressing it using a small quantity of explicit feedback data.

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preprint2010arXiv

Mixed Cumulative Distribution Networks

Directed acyclic graphs (DAGs) are a popular framework to express multivariate probability distributions. Acyclic directed mixed graphs (ADMGs) are generalizations of DAGs that can succinctly capture much richer sets of conditional independencies, and are especially useful in modeling the effects of latent variables implicitly. Unfortunately there are currently no good parameterizations of general ADMGs. In this paper, we apply recent work on cumulative distribution networks and copulas to propose one one general construction for ADMG models. We consider a simple parameter estimation approach, and report some encouraging experimental results.

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