B-words

Particle Filters in Practice

See also the webpage of the Robotics and State Estimation Lab at the UW and the one of the Robot Soccer Team at UTexas@Austin.

The Maths of Paul the “Psychic” Octopus

David Spiegelhalter writes:

England’s performance in the World-Cup last summer was thankfully overshadowed by the attention given to Paul the Octopus, who was reported as making an unbroken series of correct predictions of match winners. Here we present a mathematical analysis of Paul’s performance in an attempt to answer the question that (briefly) gripped the world: was Paul psychic?    more

The Monty Hall problem

Consider the following problem

Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?

.

(Formulation by Marilyn vos Savant at Whitaker, Craig F. (1990).)

.

So what is the answer? Let’s see a simple solution. The following table shows the three possible arrangements of two goats and one car behind three closed doors and the result of switching or staying after initially picking Door 1

.

Door 1	Door 2	Door 3	Outcome if switching	Outcome if staying
Car	Goat	Goat	Goat	Car
Goat	Car	Goat	Car	Goat
Goat	Goat	Car	Car	Goat

.

Which means that in 2 out of 3 cases we would win the car by switching doors. Hence it would make perfect sense to always switch. To understand why this is happening do take into consideration that the event “staying” is composed by only one element while its complement by two (note that all elements are equiprobable).

.

Frequency of winning when staying/switching doors in a sequence of "Monty-Hall" games

Bayesians vs. Frequentists

Lecture by Michael I. Jordan

Introduction To Bayesian Inference

by

Christopher Bishop