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Micro-Macro Duality in “Central Limit Theorem” in QFT

(,a) dQ(a) and

e(,b) dP (b) do not depend on a 5 A and b 5 B respectively. Indeed,

observe that

Z

Z

dP (a | b) dP (b) =

e f2 (a,b)(,b) dQ(a) dP (b)

dQ(a) =

B

so that

B

Z

e f2 (a,b)(,b) dP (b) = 1

B

If the Radon-NikodymRderivative e(,b) dP (b)/d(b) is constant, then

e(,b) = [dP (b)/d(b)] B e f2 (a,b) d(b), and

dP (a | b) =

e f2 (a,b) dQ(a) d(b)

dP (b | a) dQ(a)

R

=

dP (b)

dP (b) B e f2 (a,b) d(b)

Therefore, transition kernel dP (b | a) takes the following simple form

Z

dP (b | a) = e f2 (a,b)0 () d(b) ,

0 () = ln

e f2 (a,b) d(b) (12)

B

In particular, one can show that these simplied expressions can be

used in the case when

= (

, +) is a locally compact group with invariant

measure , and the utility function is translation invariant: f2 (a+c, b+c) =

f2 (a, b). An important example is when

is a vector space, and f2 (a, b)

depends only on the dierence a b 15 .

105

4. Optimal Search of DNA Sequences

Fisher’s geometric model of adaptation in Euclidean space can be generalized to any topological space, in which convergence of some sequence {$ n }

to an optimal point > corresponds to increasing values f ($n ) of some objective function (e.g. tness values). We shall rst consider the case of a

metric space, and then specialize the results to a Hamming space.

4.1. Stochastic search in metric spaces

Let (

, d) be a metric space. A search in

is the problem of constructing a

sequence {$ n } converging to some target point > 5

, and it corresponds to

an optimization problem of maximizing some objective function f :

$ R

such that sup f ($) = f (>). If the objective function is su!ciently smooth

in the neighbourhood of >, then f ($) can be replaced in this neighbourhood

by negative distance d(>, $). The di!culty is that the objective function

may be not smooth (i.e. rugged), its values may be unknown or there can

be some cost associated with its computation.

A stochastic search is the problem of constructing a sequence {p(t)} of

probability measures on

, and then evaluating the objective function f

on a random sample of points $ from p(t). The expected values E{f }(t)

converge to sup f ($) if the sequence {p(t)} converges to the Dirac measure

> (d$).

Transformations from p(t) := dPt ($) to p(t + 1) := dPt+1 ($) can be

realized

by a Markov process dP (b | a) on

such that p(t + 1) = T p(t) =

R

dP

(b

| a) dPt (a). This corresponds to ‘moving’ points a in the sample

A

from p(t) to some new points b in the sample from p(t + 1). We shall

refer to this random process as mutation of points a to points b, and the

distance d(a, b) = r will be called the mutation radius. This process is shown

schematically on Figure 1, and it is a generalization of Fisher’s geometric

theory of adaptation1 from Euclidean space Rl of l traits to a general metric

space (

, d). Thus, adaptation corresponds to mutation of point a in the

sphere S(>, n) := {$ : d(>, $) = n} of radius n to point b 5 S(>, m) such

that m < n.

. The problem of optimization if this Markov process can be formulated

using the following considerations:

(1) The transition kernels should maximize the expected value of the

dierence f (b) f (a). Note that if f ($) is su!ciently smooth, then

it can be replaced by d(>, $), and the transition kernel should

maximize the expected dierence d(>, a) d(>, b).

106

(2) The transition kernel must satisfy a constraint on mutual information IS {a, b}. Indeed, suppose that the value of information divergence of T p(t) from p(t) is > 0. Then it is clear from the

relation IS {a, b} = of Proposition 7 that IS {a, b} <

sup IS {a, b}. In fact, the supremum IS {a, b} is achieved if and only

if dP (b | a) corresponds to an injective mapping between a and b,

so that T p(t) = p(t) and = 0.

Thus, the optimization problem is a special case of variational problem (8):

maximize Ew {d(>, a) d(>, b)} subject to IS {a, b}

(13)

Its solutions are exponential transition kernels dP (b | a) = exp{ [d(>, a)

d(>, b)] (, a)}. Moreover, the factors exp{ d(>, a)} can be removed,

so that the kernels take the following form:

Z

e

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Micro-Macro Duality in “Central Limit Theorem” in QFT