View source for Bayesian numerical analysis

The comparison of algorithms and the analysis of numerical problems in a Bayesian setting, cf. also [[Bayesian approach|Bayesian approach]].

A numerical algorithm is usually developed and applied for inputs sharing some common properties. In a theoretical comparison of algorithms one either selects a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120080/b1200801.png" /> of inputs and defines the cost and the error of an algorithm in a worst-case sense over the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120080/b1200802.png" />. Alternatively, in Bayesian numerical analysis, one puts an [[A priori distribution|a priori distribution]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120080/b1200803.png" /> on the inputs and defines the cost and the error of an algorithm in an average-case sense, i.e., via expectations with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120080/b1200804.png" />. For instance, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120080/b1200805.png" /> denotes the error of a method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120080/b1200806.png" />, applied to an input <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120080/b1200807.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120080/b1200808.png" /></td> </tr></table>

is called the worst-case (maximal) error of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120080/b1200809.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120080/b12008010.png" /> while

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120080/b12008011.png" /></td> </tr></table>

is called the average error of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120080/b12008012.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120080/b12008013.png" />. The notions of optimality of algorithms and complexity of numerical problems are defined analogously in the worst-case setting and the Bayesian setting, see [[Optimization of computational algorithms|Optimization of computational algorithms]] and [[Index transform|Information-based complexity]].

Properties of the inputs, e.g., smoothness of integrands for numerical integration, are expressed by being a member of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120080/b12008014.png" /> or being distributed according to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120080/b12008015.png" />. For problems with inputs from finite-dimensional spaces, the [[Lebesgue measure|Lebesgue measure]] is often used in the definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120080/b12008016.png" />, see [[#References|[a1]]] for [[Linear programming|linear programming]] and [[#References|[a7]]] for solving systems of polynomial equations. For problems with inputs from infinite-dimensional spaces, often a Gaussian measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120080/b12008017.png" /> is used. If the inputs are real-valued functions on some domain, imposing an a priori distribution is essentially equivalent to treating inputs as random functions (cf. also [[Random function|Random function]]).

The Bayesian approach leads to new algorithms and to new insight into numerical problems. A few examples of this are as follows.

===Multivariate integration.===
Under very mild assumptions on the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120080/b12008018.png" />, the cost of optimal methods does not depend exponentially on the number of variables, see [[#References|[a8]]]. On the other hand, often a [[Curse of dimension|curse of dimension]] occurs in the worst-case setting.

===Linear programming.===
Under natural symmetry properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120080/b12008019.png" />, the average cost of the [[Simplex method|simplex method]] is polynomial, while the worst-case (maximal) cost is exponential, see [[#References|[a1]]].

===Global optimization.===
Suitable adaptive (active) methods are superior to all non-adaptive (passive) ones in a Bayesian setting, see [[#References|[a2]]]. In contrast, there is no superiority of adaptive methods in the worst-case setting for any convex and symmetric class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120080/b12008020.png" />. See also [[Optimization of computational algorithms|Optimization of computational algorithms]]; [[Adaptive quadrature|Adaptive quadrature]]; see [[#References|[a6]]] for similar results.

===Zero finding.===
Bisection is worst-case optimal for many classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120080/b12008021.png" /> of (smooth) functions. Thus, the worst-case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120080/b12008022.png" />-complexity is of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120080/b12008023.png" />. For measures that are derived from the [[Wiener measure|Wiener measure]] and concentrated on smooth functions, the average-case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120080/b12008024.png" />-complexity is of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120080/b12008025.png" />. The upper bound is achieved by a hybrid secant-bisection method, see [[#References|[a5]]].

For references, see also [[Index transform|Information-based complexity]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K.H. Borgwardt,   "The simplex method: A probabilistic approach" , Springer  (1987)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.M. Calvin,   "Average performance of a class of adaptive algorithms for global optimization"  ''Ann. Appl. Probab.'' , '''7'''  (1997)  pp. 711–730</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P. Diaconis,   "Bayesian numerical analysis"  S.S. Gupta (ed.)  J.O. Berger (ed.) , ''Statistical Decision Theory and Related Topics IV'' , '''1''' , Springer  (1988)  pp. 163–175</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J.B. Kadane,   G.W. Wasilkowski,   "Average case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120080/b12008026.png" />-complexity in computer science: a Bayesian view"  J.M. Bernardo (ed.) , ''Bayesian Statistics'' , North-Holland  (1988)  pp. 361–374</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  E. Novak,   K. Ritter,   H. Woźniakowski,   "Average case optimality of a hybrid secant-bisection method"  ''Math. Comp.'' , '''64'''  (1995)  pp. 1517–1539</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  K. Ritter,   "Average case analysis of numerical problems" , ''Lecture Notes Math.'' , Springer  (2000)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  M. Shub,   S. Smale,   "Complexity of Bezout's theorem V: polynomial time"  ''Theoret. Comput. Sci.'' , '''133'''  (1994)  pp. 141–164</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  G.W. Wasilkowski,   "Average case complexity of multivariate integration and function approximation: an overview"  ''J. Complexity'' , '''12'''  (1996)  pp. 257–272</TD></TR></table>