View source for Bayesian numerical analysis

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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 $P$ of inputs and defines the cost and the error of an algorithm in a worst-case sense over the class $P$. Alternatively, in Bayesian numerical analysis, one puts an [[A priori distribution|a priori distribution]] $\mu$ 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 $\mu$. For instance, if $\epsilon ( p , m )$ denotes the error of a method $m$, applied to an input $p$, then

\begin{equation*} \mathcal{E} _ { \operatorname{wor} } ( P , m ) = \operatorname { sup } _ { p \in P } | \epsilon ( p , m ) | \end{equation*}

is called the worst-case (maximal) error of $m$ on $P$ while

\begin{equation*} \mathcal{E} _ { \text{avg} } ( \mu , m ) = \int | \epsilon ( p , m ) | d \mu ( p ) \end{equation*}

is called the average error of $m$ with respect to $\mu$. 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 $P$ or being distributed according to $\mu$. For problems with inputs from finite-dimensional spaces, the [[Lebesgue measure|Lebesgue measure]] is often used in the definition of $\mu$, 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 $\mu$ 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 $\mu$, 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 $\mu$, 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 $P$. 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 $P$ 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 $\operatorname { log } ( 1 / \epsilon )$. 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 $\operatorname { log } \operatorname { log } ( 1 / \epsilon )$. 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>