bayes theorem

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.