IPMU final Half-day 5

Serafín Moral, University of Granada,

did the keynote on desirable gambles: "During the last few years I did lots of data mining, but I have not found the gold yet, so instead I will explain to you the last three chapters of Peter Walley's seminal 1991 book that you did not bother to read or understand."
Note: I am paraphrasing, but he is 100% right on the mark there - to me at last. BTW the book is available again, it seems they have done a reprint.

Sets of desirable gambles (SDG) and partial preference ordering are the simplest and the most general mathematical model of uncertainty (Walley, 2000).

Definitions
Let X be a variable taking values in Ω finite
A gamble is a reward/loss function f: Ω -> R
D is a SDG iff
A1: if f(ω) > 0 for any ω, then f is in D
A2: for any f in D, for any a > 0, the gamble a.f is in D
A3: if f₁ and f₂ are in D, then f₁+ f₂ is in D

A SDG D avoids partial loss iff 0 is not in D
A SDG D is coherent iff it is closed and it avoids partial loss

D is a set of almost desirable gambles (SADG) iff
D is a SDG, and
A4: if f+ε in D for any ε>0, then f is in D

D avoids sure loss iff (look in the book !)

There are SADG that are not SDG.

An SDG D defines a credal set K = {P, E_P( f) >= 0, for any f in D}

If D' is a SADG associated with an SDG D, then they define the same K

A credal set K defines a SADG { f: E_P( f) >=0, for any P in K}

Which shows that credal sets are less general than SDG.

A SDG D defines a lower prevision WTA and upper prevision WTP:
WTA = sup {a: f-a in D}
WTP = inf {a: - f+a in D}

Conditioning
There are two notions of Conditioning a SDG D with respect to the subset B of W.

One way is to consider the SDG
D_B = { f: f.1_B in D} U { f: f>0} where 1_B is the indicator function of B.

Note that there is no restriction on conditioning on subsets of probability zero.

The other way is to use the credal set (loosing information)
D"={ f: E_P( f)>=0 for any P in K, and(?) such that there is P in K, E_P( f) > 0}

The corresponding credal set is the K"={P(.|B), P in K, P(B)>0}

Implementation

A SADG can be represented as the closure of a finite set of gambles.

Generally a SDG cannot, but there is an ε-set representation:
{ f₁ + ε 1_B1, ... , f_n + ε 1_Bn}

This is enough for today, the rest was about conditional probabilities representation, consistency checking with LP (Linear Programming), inference (given D, is f desirable ?), combination and marginalization. There are two ways to define independance, epistemic and stochastic, and that seems to be a deep problem for graphical computations.

Other talks

Hwang wants your smartphone to know what you are doing (presumably to push you advertisements...). So he needs to fusion lots of data GPS, call/sms logs, device state logs, media player actions, acceleration (?), weather (taken from web)... Lucky us there is not yet enough CPU power to do so.
Fayad works for (PSA|Renault) to detect AND recognize pedestrians at carpettime minus 2s. They fusion laser, mono cam, stereo cam and radar images to track targets. I don't want to be involved in the front-end testing !.
Daniel looked at DSmT theory, in spite of the authors character. I take out two messages: 1/ The hyperpower set can be seen as a part of the power power set P(P(W)). 2/ It does not have the 'complement' operation, so singletons are not adressable.