Micro-Macro Duality in “Central Limit Theorem” in QFT
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(,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 .
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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).
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(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