resulting elementary gambles (gamblets) (1) form a hierarchy of nested basis functions leading to the orthogonal decomposition (in the scalar product of the energy norm) of $H^1_0(\Omega)$ (2) enable the sparse compression of the solution space of (3) can be computed and stored in near-linear complexity by solving a nesting of linear systems with uniformly bounded condition numbers (4) enable the computation of the solution of (or its hyperbolic or parabolic analogues) in near-linear complexity. The implementation and complexity of the algorithm are discussed in Section \[secnumimple\] with numerical illustrations.

On the correspondence between statistical inference and numerical approximation {#subsecone3}
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As exposed by Diaconis [@Diaconis:1988], the investigation of the correspondence between statistical inference and numerical approximation can be traced back to Poincar[é]{}’s course in Probability Theory [@Poincare:1896]. It is useful to recall Diaconis’ compelling example [@Diaconis:1988] as an illustration of this conection. Let $f:[0,1]\rightarrow \R$ be a given function and assume that we are interested in the numerical approximation of $\int_0^1 f(t)\,dt$. The Bayesian approach to this quadrature problem is to (1) Put a prior (probability distribution) on continuous functions $\mathcal{C}[0,1]$ (2) Calculate $f$ at $x_1,x_2,\ldots,x_n$ (to obtain the data $(f(x_1),\ldots,f(x_n))$) (3) Compute a posterior (4) Estimate $\int_0^1 f(t)\,dt$ 