**Abbreviations**

The following abbreviations are used in this manuscript:


## **Appendix A**

**Proof of Lemma 1.** Conditions (v)—(viii) imply fulfillment of the inequalities:

$$\mathcal{L}(y|q) \lessapprox \sup\_{\mathbf{x} \in \mathbb{R}^n} \nu(q, \mathbf{x}|y) \lessapprox \frac{1}{\lambda\_0^{k/2}} \max\_{\mathbf{x} \in \mathbb{R}^n} \phi\_V(\mathbf{x}) \overset{\triangle}{=} M < \infty.$$

Furthermore, for <sup>∀</sup> (0 <sup>&</sup>lt; <sup>&</sup>lt; 1) there exists a compact set *<sup>S</sup>*() ∈ B(R*n*), such that *<sup>S</sup>*() Ψ(*dx*|*q*) - <sup>1</sup> − , and by the Weierstrass theorem *<sup>m</sup>*(*y*) min(*q*,*x*)∈C×*S*() *<sup>ν</sup>*(*q*, *<sup>x</sup>*|*y*) > 0. Each measure *<sup>F</sup>* <sup>∈</sup> <sup>F</sup> can be associated with the measure *<sup>μ</sup>F*(*dq*|*y*) <sup>L</sup>(*y*|*q*)*F*(*dq*). Obviously, *<sup>μ</sup><sup>F</sup> <sup>F</sup>*, and *<sup>μ</sup><sup>F</sup>* is finite, i.e., 0 < *<sup>m</sup>*(*y*) <sup>C</sup> *<sup>μ</sup>F*(*dq*|*y*) *<sup>M</sup>* <sup>&</sup>lt; <sup>∞</sup>. Hence, <sup>∀</sup> *<sup>y</sup>* <sup>∈</sup> <sup>R</sup>*<sup>k</sup>* and <sup>∀</sup> *<sup>F</sup>* <sup>∈</sup> <sup>F</sup>. The measure *<sup>F</sup>* (*F*, *dq*|*y*) (15) is probabilistic; moreover *F F*. The measure *F* (*F* , *dq*|*y*) (16) is also a probabilistic one defined on (C, B(C)), *F F*, and the denominator in (16) has the following lower and upper bounds:

$$0 < \frac{1}{M} \lesssim \int\_{\mathcal{C}} \mathcal{L}^{-1}(y|q) F(dq) \lesssim \frac{1}{m(y)} < \infty.$$

From (15) and (16) it follows that *F* ∼ *F* , and the corresponding measure transformations are mutually inverse, i.e., <sup>∀</sup> *<sup>F</sup>* <sup>∈</sup> <sup>F</sup> the identity *<sup>F</sup>* (*<sup>F</sup>* (*F*)) ≡ *F* (*F* (*F*)) ≡ *F* holds, and, moreover, {*F* (*F*) : *<sup>F</sup>* <sup>∈</sup> <sup>F</sup>} <sup>=</sup> {*<sup>F</sup>* (*F*) : *<sup>F</sup>* <sup>∈</sup> <sup>F</sup>} <sup>=</sup> <sup>F</sup>. Assertion (1) of Lemma <sup>1</sup> is proven.

The set F *<sup>L</sup>* is <sup>∗</sup>-weakly closed, because the set <sup>F</sup> *<sup>L</sup>* is, and the function L(*y*|*q*) is nonnegative, continuous and bounded in *q* ∈ C.

Let *F* <sup>1</sup>, *F* <sup>2</sup> <sup>∈</sup> <sup>F</sup> *<sup>L</sup>* be two arbitrary distributions from <sup>F</sup> *<sup>L</sup>*, and *F <sup>α</sup> αF* <sup>1</sup> + (1 − *α*)*F* <sup>2</sup> be its convex linear combination with a fixed parameter *α* ∈ [0, 1]. We should prove that *F <sup>α</sup>* <sup>∈</sup> <sup>F</sup> *<sup>L</sup>*. By the definition of <sup>F</sup> *<sup>L</sup>* there exist distributions *<sup>F</sup>*1, *<sup>F</sup>*<sup>2</sup> <sup>∈</sup> <sup>F</sup>*<sup>L</sup>* such that *<sup>F</sup>* <sup>1</sup> = *F* (*F*1) and *F* <sup>2</sup> = *F* (*F*2). Furthermore, for the convex combination *F<sup>β</sup>* = *βF*<sup>1</sup> + (1 − *β*)*F*<sup>2</sup> with

$$\beta \stackrel{\triangle}{=} \frac{\alpha \mathfrak{L}(F\_2|y)}{\alpha \mathfrak{L}(F\_2|y) + (1-\alpha)\mathfrak{L}(F\_1|y)} \in [0,1],$$

we can verify easily that *F <sup>α</sup>* = *F* (*Fβ*), i.e., *F <sup>α</sup>* <sup>∈</sup> <sup>F</sup> *<sup>L</sup>*. Assertion (2) of Lemma 1 is proven.

## **Appendix B**

**Proof of Theorem 1.** The set H(*y*) = R by condition (ix); thus it is convex and closed. The set F *<sup>L</sup>* is convex and ∗-weakly closed due to Lemma 1. From this fact and (20) it follows that <sup>W</sup>(F*L*|*y*) is also a convex closed set. Moreover, it is bounded due to condition (viii). The function **J** (22) is strictly convex in *η* and concave (affine) in *w*. These conditions are sufficient for the existence of a saddle point [40]. It should be noted that both the set <sup>H</sup>(*y*) <sup>×</sup> <sup>W</sup>(F*L*|*y*) and the saddle point (**h**(*y*), **<sup>w</sup>** (*y*)) depend on the observed sample *<sup>y</sup>*. For the saddle point the following equalities are true:

$$\mathbf{J}(\widehat{\mathbf{h}}(y), \widehat{\mathbf{w}}(y)) = \min\_{\eta \in \mathbb{H}(y)} \max\_{w \in \mathcal{W}(\mathbb{F}\_L|y)} \mathbf{J}(\eta, w) = \max\_{w \in \mathcal{W}(\mathbb{F}\_L|y)} \min\_{\eta \in \mathbb{H}(y)} \mathbf{J}(\eta, w) = \max\_{w \in \mathcal{W}(\mathbb{F}\_L|y)} \mathbf{J}^\*(w),$$

i.e., **<sup>w</sup>** (*y*) <sup>∈</sup> Argmax *<sup>w</sup>*∈W(F*L*|*y*) **J**∗(*w*).

Now we prove the uniqueness of the saddle point **<sup>w</sup>** (*y*). Let *<sup>w</sup>* (*y*) = col(*w* <sup>1</sup>(*y*), *w* <sup>2</sup>(*y*)) and *w* (*y*) = col(*w* <sup>1</sup> (*y*), *w* <sup>2</sup> (*y*)) be two different saddle points, and J (*y*) **J**∗(*w* (*y*)) = **J**∗(*w* (*y*)) and *w* (*y*) *αw* (*y*)+(1 − *α*)*w* (*y*) be arbitrary convex combinations of the chosen points (0 < *α* < 1). After elementary algebraic transformations we have:

$$\mathbf{J}\_\*(w^{\prime\prime\prime}(y)) = \mathcal{I}(y) + a(1-a) \|w^{\prime}(y)\_2 - w\_2^{\prime\prime}(y)\|^2 > \mathcal{I}(y)\_2$$

which contradicts our assumption that *w* (*y*) and *w* (*y*) are two different solutions to the finite-dimensional dual problem. Theorem 1 is proven.
