Cognitive Computer Vision Kingsley Sage [email protected] and Hilary Buxton [email protected] Prepared under ECVision Specific Action 8-3 .

Cognitive Computer Vision

Kingsley [email protected]

Hilary [email protected]

Prepared under ECVision Specific Action 8-3http://www.ecvision.org

Lecture 5

Reminder of probability theory Bayes rule Bayesian networks

So why is Bayes rule relevant to Cognitive CV?

Provides a well-founded methodology for reasoning with uncertainty

These methods are the basis for our model of perception guided by expectation

We can develop well-founded methods of learning rather than just being stuck with hand-coded models

Bayes rule: dealing with uncertainty

Rev. THOMAS BAYES1702-1761

Sources of uncertainty e.g.:– ignorance– complexity– physical randomness– vagueness

Use probability theory to reason about uncertainty

Be careful to understand what you mean by probability and use it consistently

– frequency analysis– belief

Probability theory - reminder

p(x): single continuous value in the range [0,1]. Think of either as “x is true in 0.7 of cases” (frequentist) of “I believe x = true with probability 0.7”

P(X): often (but not always) used to denote a distribution over a set of values, e.g. if X is discrete {x=true, x=false} then P(X) encompasses knowledge of both values. p(x=true) is then a single value.

Probability theory - reminder

Joint probability

Conditional probability

)().|(),(

)( as written also ),(

YpYXPYXP

YXPYXP

Y"given X" i.e. )|( YXp

Probability theory - reminder

Conditional independence

)().(),(

)()|( then iff

YPXPYXP

XpYXPYX

Marginalising

Y

YPYXPXP

YPYXPYXP

)().|()(

)().|(),(

Bayes rule – the basics

Y

X

)().|()().|(

),(),(

)().|(),(

)().|(),(

YPYXPXPXYP

XYPYXP

YPYXPXYP

XPXYPYXP

)(

)().|()|(

XP

YPYXPXYP

BAYES RULE

Bayes rule – the basics

)(

)().|()|(

EP

HPHEPEHP

As an illustration, let’s look at the conditional probability of a hypothesis H based on some evidence E

evidenceofyprobabilit

priorlikelihoodposterior

Bayes rule – example

)(

)().|()|(

EP

HPHEPEHP

Consider a vision system used to detect zebra in static images

It has a “stripey area” operator to help it do this (the evidence E)

Let p(h=zebra present) = 0.02 (prior established during training)

Assume the “stripey area” operator is discrete valued (true/false)

Let p(e=true|h=true)=0.8 (it’s a fairly good detector)

Let p(e=true|h=false)=0.1 (there are non-zebra items with stripes in the data set – like the gate)

Given e, we can establish p(h=true|e=true) …

Bayes rule – example

1404.0)|(098.0016.0

016.0)|(

98.0*1.002.0*8.0

02.0*8.0)|(

)().|()().|(

)().|()|(

)(

)().|()|(

ehp

ehp

ehp

hphephphep

hphepehp

Ep

truehptruehtrueepetruehp

Note that this is an increase over the prior = 0.02due to the evidence e

Interpretation

Despite our intuition, our detector does not seem very “good”

Remember, only 1 in 50 images had a zebra That means that 49 out of 50 do not contain a zebra

and the detector is not 100% reliable. Some of these images will be incorrectly determined as having a zebra

Failing to account for “negative” evidence properly is a typical failing of human intuitive reasoning

Moving on …

Human intuition is not very Bayesian (e.g. Kahneman et al., 1982).

Be sure to apply Bayes theory correctly Bayesian networks help us to organise our

thinking clearly Causality and Bayesian networks are related

Bayesian networks

A

ED

B C

Compact representation of the joint probability over a set of variables

Each variable is represented as a node. Each variable can be discrete or continuous

Conditional independence assumptions are encoded using a set of arcs

Set of nodes and arcs is referred to as a graph

No arcs imply nodes are conditionally independent of each other

Different types of graph exist. The one shown is a Directed Acyclic Graph (DAG)

Bayesian networks - terminology

A

ED

B C

A is called a root node and has a prior only

B,D, and E are called leaf nodes A “causes” B and “causes” C. So

value of A determines value of B and C

A is the parent nodes of B and C B and C are child nodes of A To determine E, you need only to

know C. E is conditionally independent of A given C

Encoding conditional independence

A B C

)().|().|(),,(

),(),|(

ce)independen al(condition given But

)().|(),(

),()..|(),,(

APABPBCPCBAP

BCPBACP

BAC

APABPBAP

BAPBACPCBAP

Ni

i

iiN XXPXXXP

1

21 )(parents|(),...,,( FACTOREDREPRESENTATION

Specifying the Conditional Probability Terms (1)

For a discrete node C with discrete parents A and B, the conditional probability term P(C|A,B) can be represented as a value table

a= b= p(c=T|A,B)

red T 0.2

red F 0.1

green T 0.6

green F 0.3

blue T 0.99

blue F 0.05

A

C

B

{red,green,blue} {true,false}

{true,false}

Specifying the Conditional Probability Terms (2)

For a continuous node C with continuous parents A and B, the conditional probability term P(C|A,B) can be represented as a functionA

C

B

A B

p(c|

A,B

)

Specifying the Conditional Probability Terms (3)

For a continuous node C with 1 continuous parent A and and 1 discrete parent B, the conditional probability term P(C|A,B) can be represented as a set of functions (the continuous function is selected according to a “context” determined by B

A

p(c|

A,B

)

A

C

B

{true,false}

Directed Acyclic Graph (DAG)

A

ED

B C

Arcs encode “causal” relationships between nodes

No more than 1 path (regardless of arc direction) between any node and any other node

If we added dotted red arc, we would have a loopy graph

Loopy graphs can be approximated by acyclic ones for inference, but this is outside the scope of this course

Inference and Learning

Inference– Calculating a probability over a set of nodes given the

values of other nodes– Two most useful modes of inference are

PREDICTIVE (from root to leaf) and DIAGNOSTIC (from leaf to root)

Exact and approximate methods– Exact methods exist for Directed Acyclic Graphs

(DAGs)– Approximations exists for other graph types

evidenceofyprobabilit

priorlikelihoodposterior

Summary

Bayes rule allows us to deal with uncertain data

Bayesian networks encode conditional independence. Simple DAGs can be used n causal and diagnostic modes

Next time …

Examples of inference using Bayesian Networks A lot of excellent reference material on Bayesian

reasoning can be found at:

http://www.csse.monash.edu.au/bai