Bayesian probabilistic interference
- 1. Bayesian probabilistic interference Prof. Neeraj Bhargava Kapil Chauhan Department of Computer Science School of Engineering & Systems Sciences MDS University, Ajmer
- 2. Outline What is a Bayesian network? Why Bayesian networks are useful? Why learn a Bayesian network? Semanitcs What is probabilities Types of probabilities More probabilities
- 3. What is a Bayesian network? A simple, graphical notation for conditional independence assertions and hence for compact specification of full joint distributions Syntax: a set of nodes, one per variable a directed, acyclic graph (link ≈ "directly influences") a conditional distribution for each node given its parents: P (Xi | Parents (Xi)) In the simplest case, conditional distribution represented as a conditional probability table (CPT) giving the distribution over Xi for each combination of parent values
- 4. Example Topology of network encodes conditional independence assertions: Weather is independent of the other variables Toothache and Catch are conditionally independent given Cavity
- 5. Why is a Bayesian network? Expressive language Finite mixture models, Factor analysis, Kalman filter,… Intuitive language Can utilize causal knowledge in constructing models Domain experts comfortable building a network General purpose “inference” algorithms P(Bad Battery | Has Gas, Won’t Start) Exact: Modular specification leads to large computational efficiencies Approximate: “Loopy” belief propagation Gas Start Battery
- 6. Why Learning? data-based -Answer Wizard, Office 95, 97, & 2000 -Troubleshooters, Windows 98 & 2000 -Causal discovery -Data visualization -Concise model of data -Prediction knowledge-based (expert systems)
- 7. Semantics The full joint distribution is defined as the product of the local conditional distributions: P (X1, … ,Xn) = πi = 1 P (Xi | Parents(Xi)) e.g., P(j m a b e) = P (j | a) P (m | a) P (a | b, e) P (b) P (e) n
- 8. Constructing Bayesian networks 1. Choose an ordering of variables X1, … ,Xn 2. For i = 1 to n add Xi to the network select parents from X1, … ,Xi-1 such that P (Xi | Parents(Xi)) = P (Xi | X1, ... Xi-1) This choice of parents guarantees: P (X1, … ,Xn) = πi =1 P (Xi | X1, … , Xi-1) (chain rule) = πi =1P (Xi | Parents(Xi)) (by construction) n n
- 9. Assignment Explain Bayesian probabilistic interference with example.