Dear Colleagues,
I am in need for a "divergence metric" that would render
Bayesian updating a continuous operator; i.e.,
if D(p,q) is the divergence between distributions p and q,
and p^e is the result of updating p with evidence e:
p^(e)[i] = p[i] P(e|i) / \sum_j p[j] P(e|j)
where P(.|.) doesn't depend on p. Then, I would like to have
(1) D(p^e,q^e) <= K*D(p,q)
where K is a constant (preferable less than one).
In addition, I need that D(.,.) to be a metric in the space
of distributions; i.e,
(2) D(p,q) = 0 iff p = q,
(3) D(p,q) = D(q,p),
(4) D(p,q) <= D(p,r) + D(q,r)
It is easy to see (with a counterexample) that the symmetric
Kullback-Leibler divergence doesn't satisfy (1).
I wonder if such D(.,.) exists.
Thanks in advance
Blai Bonet
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