Time Varying Dynamic Bayesian Network for Nonstationary Events Modeling and Online Inference

Wang et al
Presented by: Guilherme dos Santos

Introduction

Introduction

  • Modeling the evolution of temporal sequences is of great interest in many areas.

  • Dynamic Bayesian Networks (DBN).

    • Unrolling a BN in time. (Bold assumption)

    • Inadequate if the relationships change through time.

  • Alternative: Incorporating the temporal variation of network structure and parameter.

    Problem

    Data scarcity

Proposed model

  • A novel time varying dynamic Bayesian network (TVDBN) model for online inference of the underlying distribution of nonstationary sequences.
  • Both the structure and parameter of network become random variables that can change through time.
  • Particle filter to dynamically infer the hidden states of network.
  • Multinomial and Gaussian distributions.

Time-Varying Dynamic Bayesian Network

DBN

Time Varying Network Representation

  • Network structure and parameter can vary. (\(G[t]\),\(\mathbf{\Theta}[t]\))

Example

Network Transition Distribution

Structure

  • Sequence of network structures \(\{G[t]\}\) as a Markov chain.
  • State-space: all valid directed acyclic graphs that connect data nodes across adjacent time epochs or within the same time epoch.

Example

Parameters (Multinomial)

Parameters (Multinomial)

Parameters (Gaussian)

Sequential Monte Carlo Inference on TVDBN

Sequential Monte Carlo

\[\small{\mathbf{s}_t=[\mathbf{X}[t], G[t], \mathbf{\Theta}[t]]}\]

\[\small{\begin{aligned} & p\left(\mathbf{s}_t \mid \mathbf{s}_{t-1}\right)=p(G[t] \mid G[t-1]) p(\boldsymbol{\Theta}[t] \mid \boldsymbol{\Theta}[t-1], G[t]) \\ & \times p(\mathbf{X}[t] \mid \mathbf{X}[t-1], \boldsymbol{\Theta}[t], G[t]) \end{aligned}}\]

\[\small{p\left(\mathbf{s}_t \mid \mathbf{o}_{1: t}\right) \approx \sum_{i=1}^{N_s} w_t^i \delta\left(\mathbf{s}_t-\mathbf{s}_t^i\right)}\]

\[\small{\mathbf{s}_t^i \sim q\left(\mathbf{s}_t \mid \mathbf{s}_{t-1}^i, \mathbf{o}_t\right)}\]

\[w_t^i \propto w_{t-1}^i \frac{p\left(\mathbf{o}_t \mid \mathbf{s}_t^i\right) p\left(\mathbf{s}_t^i \mid \mathbf{s}_{t-1}^i\right)}{q\left(\mathbf{s}_t^i \mid \mathbf{s}_{t-1}^i, \mathbf{o}_t\right)}\]

Experimental Results

Simulation

Simulation Results

Simulation Results

Simulation Results

Simulation Results

Simulation Results

Simulation Results

Random Initialization

Estimated network distribution at time 150 with random structure initialization at time 101. (a) Average posterior probability of true network structure G[150] = 15. (b) KL divergence for p(X[150] | .). Median, minimal, maximal trials are shown.

Simulation Results

  • “Convergence is defined to occur when the posterior probability of true structure is higher than 0.95 and the KL divergence from true distribution is less than 0.01.”

The time needed to converge to true network status from different initial network structures.

Simulation Results

Active Camera Tracking

Results

Results

Results

Multiple Targets

Video

Types of interaction

Posterior probabilities

Estimated structure

Confusion matrices

Precision

\[precision = \frac{tp}{tp+ fp}\]

Conclusion and Discussion

  • Contributions
    • The proposed TVDBN model works online. It can adaptively track the current state of network with the latest data observation.
    • The algorithm provides a general framework that can be applied to nonstationary sequences of any distribution.
    • The effectiveness of the proposed model was validated with extensive experiments.
  • Brief Discussion on scalability.