Probability distribution of extreme points of a Wiener stochastic process

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.

Proposition

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Let   be a Wiener stochastic process on an interval   with initial value  

By definition of Wiener process, increments have a normal distribution:

 

Let

 

be the cumulative probability distribution function of the minimum value of the   function on interval   conditioned by the value  

It is shown that:[1][3][note 1]

 

Constructive proof

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Case   is an immediate consequence of the minimum definition, in the following it will always be assumed   and also corner case   will be excluded.

Let' s assume   defined in a finite number of points  .

Let   by varying the integer   be a sequence of sets   such that   and   be a dense set in  ,

hence every neighbourhood of each point in   contains an element of one of the sets  .

Let   be a real positive number such that  

Let the event   be defined as:       .

Having excluded corner case  , it is surely  .

Let   be the events defined as:   and let   be the first k among the   which define   .

Since   it is evidently  . Now equation (2.1) will be proved.

(2.1)  

By the   events definition, , hence   . It will now be verified the relation   hence (2.1) will be proved.

The definition of  , the continuity of   and the hypothesis   imply, by the intermediate value theorem,  .

By the continuity of   and the hypothesis that   is dense in   it is deducted that   such that for   it must be   ,

hence   which implies (2.1).

(2.2)  

(2.2) is deducted from (2.1), considering that   implies that the sequence of probabilities   is monotone non decreasing and hence it converges to its supremum. The definition of events   implies   and (2.2) implies  .

In the following it will always be assumed  , so   is well defined.

(2.3)  

In fact, by definition of   it is  , so  .

In a similar way, since by definition of   it is  , (2.4) is valid:

(2.4)  

(2.5) 

The above is explained by the fact that the random variable   has a symmetric probability density compared to its mean which is zero.

By applying in sequence relationships (2.3), (2.5) and (2.4) we get (2.6) :

(2.6)  

With the same procedure used to obtain (2.3), (2.4) and (2.5) taking advantage this time by the relationship   we get (2.7):

(2.7)   

By applying in sequence (2.6) and (2.7) we get:

(2.8)    

From  , considering the continuity of   and the intermediate value theorem we get   ,

which implies  .

Replacing the above in (2.8) and passing to the limits:   and for  , event   converges to  

(2.9)   

 , by substituting   with   in (2.9) we get the equivalent relationship:

(2.10)  

Applying the Bayes' theorem to the joint event  

(2.11)     

Let:  From the above definitions it follows:

 

(2.12) 

Substituting (2.12) into (2.11), we get the equivalent:

(2.13) 

Substituting (2.9) and (2.10) into (2.13):

(2.14)   

It can be observed that in the second member of (2.14) appears the probability distribution of the random variable  , normal with mean   e variance  .

The realizations   and   of the random variable   match respectively the probability densities:

(2.15)  

(2.16)  

Substituting (2.15) e (2.16) into (2.14) and taking the limit for   the thesis is proved:

 

  

  

Bibliography

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  • A versatile stochastic model of a function of unknown and time varying form - Harold J Kushner - Journal of Mathematical Analysis and Applications Volume 5, Issue 1, August 1962, Pages 150-167.
  • The Application of Bayesian Methods for Seeking the Extremum - J. Mockus, J. Tiesis, A. Zilinskas - IFIP Congress 1977, August 8–12 Toronto.

See also

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Notes

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  1. ^ The theorem, as set out and shown for the case of the minimum of the Wiener process, also applies to the maximum.

References

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  1. ^ a b H. J. Kushner, "A New Method of Locating the Maximum Point of an Arbitrary Multipeak Curve in the Presence of Noise", J. Basic Eng 86(1), 97–106 (Mar 01, 1964).
  2. ^ Dario Ballabio, "Una nuova classe di algoritmi stocastici per l'ottimizzazione globale" (A new class of stochastic algorithms for global optimization), University of Milan, Institute of Mathematics, doctoral dissertation presented on July 12th 1978, pp. 29–33.
  3. ^ János D. Pintér, Global Optimization in Action: Continuous and Lipschitz Optimization, 1996 Springer Science & Business Media, page 57.