Template Models

• Languages that specify how variables inherit dependency model from template
• Dynamic Bayesian Networks
• Object-relational models
  • Directed
    • Plate Models
  • Undirected

Temporal Models

• Time granularity

• Markov Assumption $$ P(X^{0:T}) = P(X^{(0)}) \prod_{t=0}^{T-1} P(X^{(t+1)}|X^{(0:t)}) $$ If $(X^{(t+1)} \bot X^{(0:t-1)}|X^{(t)})$ $$ P(X^{0:T}) = P(X^{(0)}) \prod_{t=0}^{T-1} P(X^{(t+1)}|X^{(t)}) $$

• Semi-Markov Model

• Time Invariance
• Template Transition Model
  • Initial State Distribution
• 2-Time-Slice Bayesian Network
• Ground Network

Dynamic Bayesian Networks (DBNS)

• DBNS are a compact representation for encoding structured distributions over arbitrarily long temporal trajectories
• They make assumptions that may require appropriate model (re)design:
  • Markov assumption
  • Time invariance

Hidden Markov Models (HMM)

Useful in recognition

• HMMs can be viewed as a subclass of DBNs
• HMMs seem unstructured at the level of random variables
• HMM structure typically manifests in sparsity and repeated elements within the transition matrix
• HMMs are used in a wide variety of applications for modeling sequences

Plate Models

• Nested Plate
• Overlapping Plate
• Explicit Parameter Sharing
• Collective Inference
• Plate Dependency Model
• Ground Network

Summary

• Template for an infinite set of BNs, each induced by a different set of domain objects.
• Parameters and structure are reused within a BN and across different BNs
• Models encode correlations across multiple objects, allowing collective inference
• Multiple "languages", each with different tradeoffs in expressive power

Structured CPDs

• Determininistic CPDs
• Tree-structured CPDs
• Logistic CPDs & Generalizations
• Noisy OR/AND
• Linear Gaussians & Generalizations

Context-Specific Independence

If $P \models (X \perp_c Y|Z,c) $, we have: $$ P(X,Y|Z,c) = P(X|Z,c)P(Y|Z,c) $$ $$ P(X|Y,Z,c) = P(X|Z,c) $$ $$ P(Y|X,Z,c) = P(Y|Z,c) $$

When independence happens? X is determined or X and Y are seriously independent (They have no correlation any more).

Tree-structured CPDs

• Multiplexer CPD
  $P(Y|A,Z_1,...,Z_k)$, where $A$ is a selector to determine which $Z_i$ will affect

Summary

• Compact CPD representation that captures context-specific dependencies
• Relevant in multiple applications:
  • Hardware configuration variables
  • Medical settings
  • Dependence on agent's action
  • Perceptual ambiguity