# Bayesian networks

## History

2011 ACM Turing Award

• named after Thomas Bayes

uses Bayes' conditioning

• informal variant in 1913

John Henry Wigmore

## Motivation

• model of the problem domain probability distribution
• probability theory provides a consistent calculus
• intuitively interpretable
• handles missing values
• flexible applicability

## Conditional Independence

Two random variables X and Y are conditionally independent given a third random variable Z if and only if they are independent in their conditional probability distribution given Z.

P(X|Y,Z) = P(X|Z)
P(XY,Z)=P(XZ)

## Bayesian network

Each node is conditionally independent of its nondescendants given its parents

## Chain rule

P(X_1,X_2, ... ,X_n) = P(X_1)P(X_2|X_1)P(X_3|X_1,X_2) ... P(X_n|...)
P(X1,X2,...,Xn)=P(X1)P(X2X1)P(X3X1,X2)...P(Xn...)
P(X_1,X_2,...,X_n) = \prod(X_i| parents(X_i))
P(X1,X2,...,Xn)=(Xiparents(Xi))

# Naive bayes

## Inference

• deriving logical conclusions from premises known or assumed to be true
• BNs have all necessary information
• can compute any subset of variables from any other
• generally NP-Hard
• exact inference
• Monte Carlo methods

P(R=T|G=T) = ?
P(R=TG=T)=?

# Demo

## Learning BN

• estimate parameters
• learn structure

## Estimating parameters

• Maximum likelihood for complete data
• EM for incomplete data

## Expectation maximization

• initialize parameters ignoring missing data
• repeat until model converges

E  - calculate missing values using learned model

M - relearn model with new (computed) data

## Structure learning

• state space search
• score based
• initially no connections or expert made
• penalty for each connection
• must avoid cycles
• correlation between attributes, MAP
• using ML resluts in maximum network

## BNs and timeseries

• dynamic BNs
• each point in time, timeslice, is BN
• conditional dependencies between and within timeslices

## Applications of BNs

• victims identification
• oil exloration
• wireless 3G and 4G codecs
• spam filtering
• cancer risk modeling
• biomonitoring
• decision support systems

# Bonaparte

Using Bonaparte, all victims of the plane crashes in Tripoli (2010) and the Ukraine (MH17, 2014) were identified. In 2012 Bonaparte was used to solve a notorious 13 years old cold case (the Vaatstra Case). Recently Bonaparte was used to identify a serial rapist in Utrecht (2014).

Bayesian Network for Disaster Victim
Identification

# Petrophysical decision system

Estimates the type of soil and the probability that it contains oil, gas or other valuable minerals, based on drilling measurements. The system is based on a Bayesian network where the probability computation is done using a Monte Carlo sampling method.

Used by Shell.

By Martin Barus

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