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epsi-contaminated global neighborhoods

 

An epsi-contaminated class is characterized by a distribution p() and a real number epsi is in (0,1) [Berger1985]:

  GammaCepsi(p(x)) = {{ r(x) : r(x) = (1-epsi)p(x) + epsiq(x) },

where q() is an arbitrary distribution. We are interested in epsi-contaminated neighborhoods in the space of all Bayesian networks:

GammaCepsi(p(x)) = {{ r(x) : r(x) = (1-epsi) prodi pi + epsiq(x) }.

The posterior expected value for u(x) is:

E[u] = { U(e) }/{ p(e) }

where:

U(e) = sumx is in e ue(x) pe(x)

p(e) = sumx is in e pe(x)

An epsi-contaminated class is a finitely generated convex set of distributions [Cozman1996]. The vertices of this set are unitary point masses on each one of the possible configurations of the network. The maximum and minimum expected values for u(x) occur in these vertices, since we are optimizing a linear function over a convex set.

The upper expectation is:

E[u] = max( { U(e) }/{ p(e) } , { (1-epsi) U(e) + epsiue }/{ (1-epsi) p(e) + epsi } ),

where ue = maxue(x). The two terms inside parenthesis represent the two possibilities for a maximum. The first possibility is that q(x) places all the mass in a configuration for which the evidence has zero probability. The second possibility is that q(x) places all the mass in a configuration for which the evidence has non-zero probability.

The same reasoning leads to the lower expectation:

E[u] = min ( { U(e) }/{ p(e) } , { (1-epsi) U(e) + epsiue }/{ (1-epsi) p(e) + epsi } ),

where ue = minue(x).

Some special cases are important. When u(x) = xq, then E[u] = E[xq], the upper expected value of variable xq. The most important special case is u(x) = deltaa(xq), where deltaa(xq) is one if xq = a and zero otherwise. In this case E[u] = p(xq = a | e), the posterior probability for xq.

The posterior probability bounds are obtained from the previous expressions through simple substitutions:

p(xq = a | e) = { (1-epsi) p(xq = a, e) + epsi }/{ (1-epsi) p(e) + epsi } ,

p(xq = a | e) = { (1-epsi) p(xq = a, e) }/{ (1-epsi) p(e) + epsi } ,

where

p(xq = a, e) = sumx is in {xq, e} p{ xq = a, e}(x).

Notice that standard Bayesian network algorithms can be used to produce the elements of p(xq, e) which are required in this expression [Cannings & Thompson1981, Dechter1996, Zhang & Poole1996].


next up previous
Next: Constant density ratio global Up: Robustness Analysis of Bayesian Previous: Global neighborhoods for Bayesian

© Fabio Cozman[Send Mail?]

Thu Jan 23 15:54:13 EST 1997