72. Approximate an unknown density without assuming its exact degree
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Part V identified a polynomial operator density under an exact degree assumption. A different question is how to approximate a valid density that is not a polynomial at all.
In this section observations lie in \([0,1]\) with Lebesgue reference measure. This is an explicit change from the preceding \([-1,1]\) convention. Under \(v=(u+1)/2\), the probability measure \(du/2\) becomes \(dv\). A Lipschitz constant with respect to \(u\) becomes twice as large with respect to \(v\).
Write the Bernstein basis and the classical Bernstein–Durrmeyer kernel as
Both variables range over \([0,1]\). The kernel is nonnegative and integrates to one in either variable. Let \(B_a(v)\) be any operator-norm integrable density preserving a common probability cone \(C\), with mass functional \(\lambda\) and normalized base \(K\). Define
where
Proposition 64: Valid positive polynomial approximation
Every \(B_a^{[n]}\) preserves \(C\), has degree at most \(n\), and satisfies exact integrated mass conservation. If
for all actions, then
With identical initial states, the corresponding length-\(T\) laws under a common policy differ in total variation by at most
No finite true-degree hypothesis is needed.
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Proof
Nonnegative integration preserves the closed convex cone. Since \(\int_0^1b_{n,k}=1/(n+1)\) and \(\sum_k b_{n,k}=1\), the kernel is doubly stochastic. Its integral in \(x\) gives \(\int B_a^{[n]}=\int B_a\), proving exact mass conservation.
For the error, subtract \(B_a(x)\) inside the kernel integral, apply the Lipschitz inequality and use \(\|z\|_C=1\). The integrated discrepancy is at most \(L\mathbb E|V-X|\) under the joint probability density \(D_n(x,v)\). This joint law can be constructed by first choosing \(J\) uniformly from \(\{0,\ldots,n\}\), and then drawing \(X,V\) independently, conditionally on \(J=k\), with density \((n+1)b_{n,k}\).
That conditional beta law has variance
The average of \((k+1)(n-k+1)\) over \(k=0,\ldots,n\) is \((n+2)(n+3)/6\), obtained by summing the first two integer powers. Hence
Cauchy–Schwarz gives the claimed integrated bound. The whole-law conclusion is Proposition 59 applied to the two valid instruments on the same domain. \(\square\)
This construction approximates the operator density itself. It does not average a congruence factor and omit its covariance. Its coefficients are positive operators even when their entries in predictive coordinates are signed.
Operationally, it replaces the raw observation \(v\) by a noisy observation \(x\) drawn from \(D_n(x,v)\). It preserves total mass and underlying unconditioned evolution, not the entire original observation law. That changed law is what the bound measures. A prediction policy must be compared as the same function of the reported history.
72.1 A nonpolynomial density and a failed ordinary truncation
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Consider the valid scalar density
It integrates to one, takes values between \(1/2\) and \(3/2\), and has Lipschitz constant two. It is not a single polynomial on \([0,1]\). Proposition 64 nevertheless supplies normalized positive polynomial approximations at every degree, with an explicit approximation budget.
An ordinary orthogonal truncation need not preserve that validity. The valid density \(p(v)=3v^2\) has degree-one \(L^2\) projection
which is negative near zero. The positive construction at degree one instead gives
The two approximations solve different mathematical problems. Neither has permission to call its approximation error zero. The reference checks derive both expressions exactly.