Bayes Theorem

Bayes Theorem is \[ P(G|O)=\frac{P(O|G)P(G)}{P(O)} \] Gives the probability of event G given observation O based on prior information (all the terms on the right of the =)

• Bayes Theorem has been used in robotics, medicine, psychology, neuroscience, statistics, ...

Bayes Theorem example. Gender classification

If observation ($O$) is whether a person has a beard (B) and we wish to guess the gender ($G$) then

\[P(G=male\,\,|\,\, O=beard)=\frac{P(O=beard\,\,|\,\,G=male)P(G=male)}{P(O=beard)}\] \[P(G=female\,\,|\,\, O=beard)=\frac{P(O=beard\,\,|\,\,G=female)P(G=female)}{P(O=beard)}\]

A little short cut, $P(O=beard)=0.5\left(P(O=beard\,\,|\,\,G=female)+P(O=beard\,\,|\,\,G=male)\right)$

Bayes Theorem: Multiple observations example

If we make several observations then we can extend Bayes Theorem so

We want to determine the event `Gender=female', i.e. $G=F$

Let

• observation1 ($O_1$) give whether a person is wearing a dress (D)
• observation2 ($O_2$) say whether a person has long hair (L)

then \[ P(G=F\,|\, O_1=D\,\, \hbox{and}\,\, O_2 = L)\approx \frac{P(O_1=D\,\,|\,\,G=F)P(O_2=L\,|\,G=F)P(G=F)}{P(O_1=D)P(O_2=L)} \]

Why conditional probabilities are important

Through experience we as humans have learned a mechanism something like Bayesian priors to allow us to make gender judgements on

• wizards (such as Gandalf)
• cultural traditions (e.g. the Greek Evzones)
• drag persona such as Conchita Wurst
• cross dressers such as Grayson Perry

Is this how brains work and if so, should we make robots think like this?

Mini Golf Robot MEng 2013

Interaction with golf robot

Extra information

• The real reason for brains, TEDtalk by Daniel Wolpert, Cambridge University Engineering Department
• How does my brain work More TED talks
• Introductory Slides for first year robotics

Some Robots from the University of Reading

More Robots from UoR

Clearly:

Prediction in humans has a consequence, the abililty to ignore information not relevant to the prediction.

For an example, run this video.