In the mathematical theory of probability, the Wiener process, named after Norbert Wiener, is a stochastic process used in modeling various phenomena, including Brownian motion and fluctuations in financial markets. A formula for the conditional probability distribution of the extremum of the Wiener process and a sketch of its proof appears in work of H. J. Kusher (appendix 3, page 106) published in 1964.[1] a detailed constructive proof appears in work of Dario Ballabio in 1978.[2] This result was developed within a research project about Bayesian optimization algorithms.
In some global optimization problems the analytical definition of the objective function is unknown and it is only possible to get values at fixed points. There are objective functions in which the cost of an evaluation is very high, for example when the evaluation is the result of an experiment or a particularly onerous measurement. In these cases, the search of the global extremum (maximum or minimum) can be carried out using a methodology named "Bayesian optimization", which tend to obtain a priori the best possible result with a predetermined number of evaluations. In summary it is assumed that outside the points in which it has already been evaluated, the objective function has a pattern which can be represented by a stochastic process with appropriate characteristics. The stochastic process is taken as a model of the objective function, assuming that the probability distribution of its extrema gives the best indication about extrema of the objective function. In the simplest case of the one-dimensional optimization, given that the objective function has been evaluated in a number of points, there is the problem to choose in which of the intervals thus identified is more appropriate to invest in a further evaluation. If a Wiener stochastic process is chosen as a model for the objective function, it is possible to calculate the probability distribution of the model extreme points inside each interval, conditioned by the known values at the interval boundaries. The comparison of the obtained distributions provides a criterion for selecting the interval in which the process should be iterated. The probability value of having identified the interval in which falls the global extremum point of the objective function can be used as a stopping criterion. Bayesian optimization is not an efficient method for the accurate search of local extrema so, once the search range has been restricted, depending on the characteristics of the problem, a specific local optimization method can be used.
Let
X(t)
[a,b]
X(a)=Xa.
By definition of Wiener process, increments have a normal distribution:
fora\leqt1<t2\leqb, X(t2)-X(t1)\simN(0,
2(t | |
\sigma | |
2 |
-t1)).
Let
F(z)=\Pr(minaX(t)\leqz\midX(b)=Xb)
be the cumulative probability distribution function of the minimum value of the
X(t)
[a,b]
X(b)=Xb.
F(z)=\begin{cases}1&forz\geqmin\{Xa,Xb\},\\ \exp\left(-2\dfrac{(z-Xb)(z-Xa)}{\sigma2(b-a)}\right)&forz<min(Xa,Xb). \end{cases}
Case
z\geqmin(Xa,Xb)
z<min(Xa,Xb)
minaX(t)=min(Xa,Xb)
Let' s assume
X(t)
tk\in[a,b], 0\leqk\leqn, t0=a
Let
Tn \overset{\underset{def
n
\{Tn\}
Tn\subsetTn+1
+infty | |
cup | |
n=0 |
Tn
[a,b]
hence every neighbourhood of each point in
[a,b]
Tn
Let
\Deltaz
z+\Deltaz<min(Xa,Xb).
E
E \overset{\underset{def
\Longleftrightarrow
(\existst\in[a,b]:X(t)<z+\Deltaz)
Having excluded corner case
minaX(t)=min(Xa,Xb)
P(E)>0
Let
En, n=0,1,2,\ldots
En \overset{\underset{def
\nu
tk\inTn
En
Since
Tn\subsetTn+1
En\subsetEn+1
(2.1)
E=
+infty | |
cup | |
n=0 |
En
By the
En
\foralln En ⇒ E
+infty | |
cup | |
n=0 |
En\subsetE
E
+infty | |
\subsetcup | |
n=0 |
En
The definition of
E
X(t)
z<Xa=X(a)
(\exists\bar{t}\in[a,b]:z<X(\bar{t})<z+\Deltaz)
By the continuity of
X(t)
+infty | |
cup | |
n=0 |
Tn
[a,b]
\exists\bar{n}
t\nu\inT\bar{n}
z<X(t\nu)<z+\Deltaz
hence
E\subsetE\bar{n}\subset
+infty | |
cup | |
n=0 |
En
(2.2)
P(E)=\limnP(En)
(2.2) is deducted from (2.1), considering that
En ⇒ En+1
P(En)
En
\foralln P(En)>0 ⇒ P(En)=P(E\nu)
P(E)=P(E\nu)
In the following it will always be assumed
n\geq\nu
t\nu
(2.3)
P(X(b)\leqslant-Xb+2z)\leqslantP(X(b)-X(t\nu)<-Xb+z)
In fact, by definition of
En
z<X(t\nu)
(X(b)\leqslant-Xb+2z) ⇒ (X(b)-X(t\nu)<-Xb+z)
In a similar way, since by definition of
En
z<X(t\nu)
(2.4)
P(X(b)-X(t\nu)>Xb-z)\leqslantP(X(b)>Xb)
(2.5)
P(X(b)-X(t\nu)<-Xb+z)=P(X(b)-X(t\nu)>Xb-z)
The above is explained by the fact that the random variable
(X(b)-X(t\nu))\thicksimN(\varnothing;
2(b-t | |
\sigma | |
\nu)) |
By applying in sequence relationships (2.3), (2.5) and (2.4) we get (2.6) :
(2.6)
P(X(b)\leqslant-Xb+2z)\leqslantP(X(b)>Xb)
With the same procedure used to obtain (2.3), (2.4) and (2.5) taking advantage this time by the relationship
X(t\nu)<z+\Deltaz
(2.7)
P(X(b)>Xb)\leqslantP(X(b)-X(t\nu)>Xb-z-\Deltaz)
= P(X(b)-X(t\nu)<-Xb+z+\Deltaz)\leqslantP(X(b)<-Xb+2z+2\Deltaz)
By applying in sequence (2.6) and (2.7) we get:
(2.8)
P(X(b)\leqslant-Xb+2z)\leqslantP(X(b)>Xb)
\leqslantP(X(b)<-Xb+2z+2\Deltaz)
From
Xb>z+\Deltaz>z
X(t)
X(b)>Xb>z+\Deltaz>z ⇒ En
which implies
P(X(b)>Xb)=P(En,X(b)>Xb)
Replacing the above in (2.8) and passing to the limits:
\limn En(\Deltaz) → E(\Deltaz)
\Deltaz → 0
E(\Deltaz)
minaX(t)\leqslantz
(2.9)
P(X(b)\leqslant-Xb+2z)=
P(minaX(t)\leqslantz, X(b)>Xb)
\foralldXb>0
(Xb)
(Xb-dXb)
(2.10)
P(X(b)\leqslant-Xb+2z+dXb)=
P(minaX(t)\leqslantz, X(b)>Xb-dXb)
Applying the Bayes' theorem to the joint event
(minaX(t)\leqslantz, Xb-dXb<X(b)\leqslantXb)
(2.11)
P(minaX(t)\leqslantz\midXb-dXb<X(b)\leqslantXb)=
P(minaX(t)\leqslantz, Xb-dXb<X(b)\leqslantXb)
/ P(Xb-dXb<X(b)\leqslantXb)
Let:
B \overset{\underset{def
D=B\cupC ⇒ P(A,D)=P(A,B\cupC)=P(A,B)+P(A,C) ⇒ P(A,C)=P(A,D)-P(A,B)
(2.12)
P(A,C)=P(A,D)-P(A,B)
Substituting (2.12) into (2.11), we get the equivalent:
(2.13)
P(minaX(t)\leqslantz\midXb-dXb<X(b)\leqslantXb)= (P(mina\leqslantX(t)\leqz, X(b)>Xb-dXb)- P(mina\leqslantX(t)\leqz, X(b)>Xb)) / P(Xb-dXb<X(b)\leqslantXb)
Substituting (2.9) and (2.10) into (2.13):
(2.14)
P(minaX(t)\leqslantz\midXb-dXb<X(b)\leqslantXb)=
(P(X(b)\leqslant-Xb+2z+dXb)-P(X(b)\leqslant-Xb+2z)
/ P(Xb-dXb<X(b)\leqslantXb)
It can be observed that in the second member of (2.14) appears the probability distribution of the random variable
X(b)
Xa
\sigma2(b-a)
The realizations
Xb
-Xb+2z
X(b)
(2.15)
P(Xb)dXb=
1 | |
\sigma\sqrt{2\pi(b-a) |
(2.16)
P(-Xb+2z)dXb=
1 | |
\sigma\sqrt{2\pi(b-a) |
Substituting (2.15) e (2.16) into (2.14) and taking the limit for
dXb → 0
F(z)=P(minaX(t)\leqz | X(b)=Xb)=
=
1 | |
\sigma\sqrt{2\pi(b-a) |
\diagup
1 | |
\sigma\sqrt{2\pi(b-a) |
=\expl(-
1 | |
2 |
| |||||||||||||||||||
\sigma2(b-a) |
r)=
\expl(-2
(z-Xb)(z-Xa) | |
\sigma2(b-a) |
r)