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Nonparametric Bayes dynamic modeling of relational data

Daniele Durante, David B. Dunson

arXiv:1311.4669v1stat.ML

TL;DR

Dynamic relational data raise questions about inference on evolving relationship structure and prediction, while prior dynamic approaches were limited in scope or time setting. This paper uses a nonparametric Bayesian latent-space model with continuously evolving coordinates. It applies the model to financial-market co-movements during the recent crisis and reports insights into the market network.

Abstract

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Symmetric binary matrices representing relations among entities are commonly collected in many areas. Our focus is on dynamically evolving binary relational matrices, with interest being in inference on the relationship structure and prediction. We propose a nonparametric Bayesian dynamic model, which reduces dimensionality in characterizing the binary matrix through a lower-dimensional latent space representation, with the latent coordinates evolving in continuous time via Gaussian processes. By using a logistic mapping function from the probability matrix space to the latent relational space, we obtain a flexible and computational tractable formulation. Employing P\`olya-Gamma data augmentation, an efficient Gibbs sampler is developed for posterior computation, with the dimension of the latent space automatically inferred. We provide some theoretical results on flexibility of the model, and illustrate performance via simulation experiments. We also consider an application to co-movements in world financial markets.

151. Introduction

The introduction motivates dynamic modeling of one-mode binary relations, where existing approaches face limitations in flexibility, computation, or continuous-time treatment. The paper embeds entities in a low-dimensional latent space with continuously evolving coordinates and develops posterior computation that automatically infers the space’s dimension.

151. Introduction

  • The paper treats financial-market co-movements as dynamic relational data rather than focusing on time-varying covariance or correlation matrices.Two indices are linked when they move in the same direction and unlinked when they move in opposite directions.Source: 151. Introduction · pp. 1–2
  • Earlier dynamic relational models were limited to a few proposals, often restricted to specific applications and discrete-time evolution.For binary similarity data, existing approaches could also yield out-of-range predictions or many tied ranks in non-metric MDS.Source: 151. Introduction · p. 2
  • Existing alternatives include ERGMs with computationally demanding estimation and possible degeneracy, and dynamic block models with Gaussian-assumption or linearization concerns.The introduction also notes that Bayesian extensions of choice-data methods lack scalability for selecting latent-space dimension by cross-validation.Source: 151. Introduction · p. 3
  • The model represents one-mode binary relations by embedding entities in a low-dimensional Euclidean space whose coordinates evolve continuously through Gaussian processes.A logistic mapping links edge probabilities to latent-coordinate dot products.Source: 151. Introduction · p. 3
  • A Gibbs sampler is presented as converging to the exact posterior while automatically inferring the latent-space dimension.The introduction contrasts this with related dynamic latent-space approaches that rely on multiple approximation layers.Source: 151. Introduction · p. 3

962. Dynamic Latent Space Model

The model represents time-varying symmetric binary relations through a baseline and low-dimensional latent coordinates, with a logistic link mapping similarities to probabilities. Gaussian-process priors and shrinkage support continuous-time modeling and learning the latent dimension.

972.1 Notation and Motivation

  • Latent coordinates reduce the need to model a separate time-varying similarity process for every pair, borrowing information across units to ease computation as V grows.The target is a prior over time-varying probability matrices that supports dimensionality reduction, unequal observation spacing, missing values, and prediction with uncertainty.Source: 1092.2 Latent space dynamic model… · p. 4, 972.1 Notation and Motivation · p. 4

1092.2 Latent space dynamic model formulation

  • The model forms each pair’s similarity from a baseline and the inner product of its units’ latent coordinates, then maps similarity to probability with a logistic link.Units with coordinates pointing in the same directions are more likely to be similar; coordinates in opposite directions correspond to lower similarity probability.Source: 1092.2 Latent space dynamic model… · p. 4
  • For H ≥ V, every symmetric probability matrix can be represented by the latent-coordinate specification, supporting the model’s stated flexibility.The paper describes the specification as nonparametric given sufficiently flexible priors.Source: 1092.2 Latent space dynamic model… · p. 5

1602.3 Prior Specification

  • Gaussian-process latent coordinates extend the factor-shrinkage approach to time-varying factors, while shrinkage favors later coordinate trajectories near zero.The prior is designed to support continuous-time analysis, unequal spacing, missing values, and learning the latent-space dimension.Source: 1602.3 Prior Specification · p. 6
  • On compact T, the induced prior assigns positive probability to every neighborhood of continuous symmetric probability-matrix trajectories.The paper states this support result for Ππ by combining the support result for ΠS with the continuous increasing probability link.Source: 1602.3 Prior Specification · pp. 6–7

1993. Posterior computation

Pólya–Gamma augmentation turns the logistic likelihood into a Gaussian-mixture representation, yielding conjugate conditional updates. The Gibbs sampler updates augmentation variables, the baseline and latent coordinates, and shrinkage parameters.

1993. Posterior computation

  • Pólya–Gamma augmentation represents binomial likelihoods parameterized by log-odds as Gaussian mixtures, enabling full conditional conjugacy for Gaussian-prior coefficients.The augmented variables also have Pólya–Gamma conditional distributions, supporting a Gibbs sampler without analytic approximations.Source: 1993. Posterior computation · p. 7
  • The sampler updates Pólya–Gamma variables, then the baseline and each unit’s time-varying latent coordinates using conditional Gaussian posteriors.The coordinate update is framed as a logistic regression with a linear predictor.Source: 1993. Posterior computation · p. 8
  • After coordinate updates, the sampler draws global shrinkage hyperparameters and imputes missing binary similarities from their conditional distribution.The listed steps also provide a strategy for posterior prediction.Source: 1993. Posterior computation · p. 9

5. Given X(t) and µ(t) sample each missing value from its conditional distribution

Future probability matrices are predicted by extending the observed data with a missing matrix and using posterior samples from the resulting Markov chain.

5. Given X(t) and µ(t) sample each missing value from its conditional distribution

  • For a future time tT+1, the procedure targets the predictive posterior distribution of π(tT+1).The setting is tT+1 > tT, after observing the similarity matrices at times in T0.Source: §5 · p. 9
  • The procedure appends a new matrix of missing values at tT+1 to the observed matrices before carrying out posterior computation.The appended matrix represents the unobserved future outcomes.Source: §5 · p. 9
  • Predictive inference uses samples from the Markov chain obtained after adding that missing matrix to the dataset.The target is the predictive posterior for the future probability matrix.Source: §5 · p. 9

2524. Simulation Study

In simulation, the model reconstructs probability trajectories and predicts missing outcomes; its ROC curve has area 0.87. Compared with independent time-series estimates, network modeling yields less bias and variance.

2614.1 Estimating Performance

  • The posterior means capture the true probability matrices, including at t = 40 when the observed matrix is missing and prediction is assessed.The comparison is reported for selected times, with Figure 2 showing true matrices, posterior means, and absolute differences.Source: 2614.1 Estimating Performance · p. 10, Figure 2 · p. 10
  • 0.87 area under the ROC curve accompanies the posterior-mean probability estimates in the simulation.Figure 3 compares true probabilities with posterior means and plots the ROC curve using estimated probabilities and observed data.Source: 2614.1 Estimating Performance · p. 10, Figure 3 · p. 11
  • Compared with estimates using only each dyad’s time series, the network model avoids the reported oversmoothing and larger HPD intervals.The authors report accurate estimates and posterior distributions concentrating rapidly around the true processes when network structure is used.Source: 2614.1 Estimating Performance · pp. 10–11, Figure 4 · p. 11
  • The model accurately selects latent dimension in this simulation, and similar results hold across the reported GP length-scale sensitivity settings.The later factor trajectories have limited influence; reported sensitivity settings are κµ = κx = 0.03, 0.1, and 0.5.Source: 2614.1 Estimating Performance · p. 11

2995. Application to co-movements among National Stock Market Indices

The application models quarterly co-movement probabilities among national stock indices and interprets their evolution through financial crises. It finds changing cross-market connections, including weaker Greek ties during the sovereign-debt crisis.

2995. Application to co-movements among National Stock Market Indices

3285.1 Model Interpretation

  • The baseline process rises persistently during the US housing-bubble growth and burst and the initial turmoil before the 2008 global financial crisis.The authors connect this pattern to increasing market interconnection between 2004 and 2007.Source: 3285.1 Model Interpretation · p. 13
  • The estimated co-movement trajectories achieve a ROC area of 0.79, and the authors report that the model adaptively learns the data structure.Figure 6 displays observed data, estimated trajectories, and posterior predictive trajectories with 0.95 highest posterior density intervals.Source: 3285.1 Model Interpretation · p. 13, Figure 6 · p. 13
  • Co-movement probabilities among the USA and Greece drop sharply around the Greek debt crisis, unlike the similar evolving patterns of Germany and France.The authors interpret the reduction in connections with a country in crisis as an empirical result.Source: 3285.1 Model Interpretation · pp. 13–14
  • During the 2008 financial crisis, estimated co-movement probabilities indicate stronger connections across markets; during the Greek debt crisis, Greece has low connections with most markets but strong ties to Spain and Italy.The authors describe the crisis-period increases as highlighting a financial contagion effect.Source: Figure 7 · p. 14, 3285.1 Model Interpretation · p. 14
  • The overall network links countries with similar financial economies; China has lower interconnections, while European markets and Asian Tigers have stronger networks.The three networks average estimated probabilities over the full period, the global financial crisis, and the Greek debt crisis.Source: 3285.1 Model Interpretation · p. 14

3726. Discussion

The discussion presents a continuous-time latent-coordinate model for dynamic binary similarity matrices, with shrinkage for automatic dimension learning and Pólya–Gamma augmentation for Gibbs computation. It also notes applications, theoretical and empirical contributions, and scope limits.

3726. Discussion

  • The model represents binary similarity through dot products of latent coordinates that evolve continuously in time under Gaussian process priors.The model borrows information across time and the network structure while reducing dimensionality.Source: 3726. Discussion · pp. 14–15
  • A shrinkage hyperprior automatically learns latent-space dimension and supports a parsimonious specification that avoids the risk of over-parameterization..Source: 3726. Discussion · p. 15
  • Pólya–Gamma data augmentation yields a Gibbs sampler with full conditional conjugate posterior distributions for posterior computation.The authors describe this computation as promising for scaling to moderately large numbers of units.Source: 3726. Discussion · p. 15
  • Scaling to large numbers of time points could be improved using stochastic differential equation models to approximate the Gaussian process prior.The authors also identify discrete-valued dynamic matrices as a direction for future models.Source: 3726. Discussion · p. 15
  • The paper provides theoretical flexibility results, evaluates performance in simulations, and applies the model to time-varying financial co-movement data.The application offers insights on financial-market networks during the recent crisis.Source: 3726. Discussion · p. 15
  • Dynamic social networks and time-varying binary evaluations are identified as natural application areas..Source: 3726. Discussion · p. 15

Papers, International Monetary Fund 46, 167–195. 400

The supplied section is a reference list spanning relational and network models, financial time series, and Bayesian latent-variable methods.

References

4907. Appendix

The proof establishes positive support for approximating a target function by reducing the argument to finitely many neighborhoods and controlling approximation through continuity and prior support.

4907. Appendix

  • Compactness of T supplies a finite subcover of ε₀-balls, reducing the approximation argument to finitely many neighborhoods.The proof first covers T with ε₀-balls and then considers each ball independently.Source: 4907. Appendix · p. 18
  • Continuity of the target and of the latent-score matrix provides neighborhood radii that keep their approximation errors below ε/3.The matrix X(t)X(t)′ is almost surely continuous because its entries are finite sums of products of almost surely continuous Gaussian-process coordinates; the baseline is also almost surely continuous.Source: 4907. Appendix · pp. 18–19
  • Wishart support for the latent rank-one terms, together with large support for the baseline prior, yields positive probability for the required approximations.The proof combines positivity for the three terms to complete the argument.Source: 4907. Appendix · p. 19
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