**3. The Main Result**

As is known, the CE is determined in a non-unique way, hence we should specify a version of the CE so as to use it in further inferences. If the distribution *F* of the vector *γ* is known, then the CE of an integrable random value *<sup>h</sup>*(*γ*, *<sup>X</sup>*) : C × <sup>R</sup>*<sup>m</sup>* <sup>→</sup> <sup>R</sup> can be calculated by the abstract variant of the Bayes formula:

$$\hat{h}^{F}(\boldsymbol{y}) = \frac{\int\_{\mathcal{C}\times\mathbb{R}^{n}} h(\boldsymbol{q},\boldsymbol{x}) |\det(\boldsymbol{B}(\boldsymbol{q},\boldsymbol{x}))|^{-1} \boldsymbol{\Phi}\_{V}(\boldsymbol{B}^{-1}(\boldsymbol{q},\boldsymbol{x})(\boldsymbol{y} - \boldsymbol{A}(\boldsymbol{q},\boldsymbol{x}))) \Psi(d\boldsymbol{x}|\boldsymbol{q}) \mathcal{F}(d\boldsymbol{q})}{\int\_{\mathcal{C}\times\mathbb{R}^{n}} |\det(\boldsymbol{B}(\boldsymbol{q}',\boldsymbol{x}'))|^{-1} \boldsymbol{\Phi}\_{V}(\boldsymbol{B}^{-1}(\boldsymbol{q}',\boldsymbol{x}')(\boldsymbol{y} - \boldsymbol{A}(\boldsymbol{q}',\boldsymbol{x}'))) \Psi(d\boldsymbol{x}'|\boldsymbol{q}') \mathcal{F}(d\boldsymbol{q}')},\tag{11}$$

i.e., <sup>E</sup>*F*{*h*(*γ*, *<sup>X</sup>*)|*<sup>Y</sup>* <sup>=</sup> *<sup>y</sup>*} <sup>=</sup> *hF*(*y*) (11) <sup>P</sup>*<sup>F</sup>* <sup>−</sup> a.s. Below in the presentation we use the CE version, defined by (11). It should also be noted that if *h*(·) is the desired minimax estimator, then the inclusion (8) must be satisfied point-wise for any sample *<sup>y</sup>* <sup>∈</sup> <sup>R</sup>*k*.

Further in the paper the function:

$$J\_\*(F|y) \triangleq \min\_{\overline{h} \in \mathbb{H}} J(\overline{h}, F|y) = J(\widehat{h}^F, F|Y) = \|\widehat{h}\|^2 \|\overline{y}\|^2 (y) - \|\widehat{h}^F(y)\|^2 \tag{12}$$

is called *the dual criterion* for *J*∗ (7). All CEs in (12) are calculated by (11).

Using (3) for the calculation of L, the notation:

$$\nu(q, \mathfrak{x}|\underline{y}) \triangleq |\det B(q, \mathfrak{x})|^{-1} \phi\_V(B(q, \mathfrak{x})^{-1}(\underline{y} - A(q, \mathfrak{x}))),\tag{13}$$

and the CE version (11), the loss function (6) can be rewritten in the form:

$$J(\overline{h}, F|y) = \frac{\int\_{\mathcal{C} \times \mathbb{R}^n} \|h(q, \mathbf{x}) - \overline{h}(y)\|^2 \nu(q, \mathbf{x}|y) \Psi(dx|y) F(dq)}{\int\_{\mathcal{C}} \mathcal{L}(y|q') F(dq')}. \tag{14}$$

As can be seen from (14), the function *J*(*h*, *F*|*y*) is neither convex nor concave in *F*, which complicates the solution to the estimation problem (8). Moreover, the argument *F* lies in the abstract infinite-dimensional space of the probability measures. Nevertheless, the problem can be reduced to a standard finite-dimensional minimax problem with a convex–concave criterion.

First, we introduce a new reference measure *F* and verify that the loss function (14) represents a functional, which is linear in *F* .

Let:

$$F'(F, dq \vert y) \stackrel{\triangle}{=} \frac{\mathcal{L}(y \vert q) F(dq)}{\int\_{\mathcal{C}} \mathcal{L}(q' \vert y) F(dq')}.\tag{15}$$

**Lemma 1.** *If conditions (i)–(ix) are satisfied, then the following assertions are true.*

*1. F* (*F*, *dq*|*y*) *is a probability measure for* <sup>∀</sup> *<sup>y</sup>* <sup>∈</sup> <sup>R</sup>*k, and <sup>F</sup>* (*F*, ·|*y*) ∼ *F*(·)*. The transformation (15) is a bijection of* F *into itself, and its inversion F has the form:*

$$F^{\prime\prime}(F^{\prime}, dq | y) \stackrel{\triangle}{=} \frac{\mathcal{L}^{-1}(y | q) F^{\prime}(dq)}{\int\_{\mathcal{C}} \mathcal{L}^{-1}(q^{\prime} | y) F^{\prime}(dq^{\prime})}.\tag{16}$$

*2. The set* F *<sup>L</sup> of all distributions obtained from* <sup>F</sup>*<sup>L</sup> by the transformation (15):*

$$\mathbb{F}\_L' \triangleq \{ \overline{F}(\cdot) \, : \, \exists \, F \in \mathbb{F}\_{L\prime} \, \overline{F}(\cdot) = F^{\prime}(F\_{\prime} \cdot | y) \} \tag{17}$$

*is convex and* ∗*-weakly closed.*

The proof of Lemma 1 is given in Appendix A.

Applying the Fubini theorem and keeping in mind (11) and (15), we can rewrite the loss function (14) in the form:

$$J(\overline{h}, F|y) = \frac{\int\_{\mathcal{C}} \frac{\int\_{\mathbb{R}^n} \|h(q, \mathbf{x}) - \overline{h}(y)\|^2 \nu(q, \mathbf{x}|y) \Psi(dx|y)}{\mathcal{L}(y|q)} \mathcal{L}(y|q) F(dq)}{\int\_{\mathcal{C}} \mathcal{L}(q'|y) F(dq')}$$

$$= \int\_{\mathcal{C}} \frac{\int\_{\mathbb{R}^n} \|h(q, \mathbf{x}) - \overline{h}(y)\|^2 \nu(q, \mathbf{x}|y) \Psi(dx|y)}{\mathcal{L}(y|q)} F'(\mathcal{F}, dq|y) = J(\overline{h}, F'|y). \tag{18}$$

To reduce the initial problem to some finite-dimensional equivalent, we also introduce the vectors:

$$\begin{split} w(y|q) &\triangleq \text{col}(w\_1(y|q), w\_2(y|q)) \in \mathbb{R}^{\ell+1} :\\ w\_1(y|q) &\triangleq \mathbb{E}\_{\Gamma} \left\{ \|h(\gamma, X)\|^2 | Y = y, \,\,\gamma = q \right\} = \frac{\int\_{\mathbb{R}^n} \|h(q, x)\|^2 \nu(q, x) \Psi(dx|y)}{\mathcal{L}(y|q)},\\ w\_2(y|q) &\triangleq \mathbb{E}\_{\Gamma} \{h(\gamma, X) | Y = y, \,\,\gamma = q\} = \frac{\int\_{\mathbb{R}^n} h(q, x) \nu(q, x) \Psi(dx|y)}{\mathcal{L}(y|q)};\\ w(F|y) &\triangleq \text{col}(w\_1(F|y), w\_2(F|y)) \in \mathbb{R}^{\ell+1} :\\ w\_1(F|y) &\triangleq \mathbb{E}\_{\Gamma} \left\{ \|h(\gamma, X)\|^2 | Y = y \right\} = \int\_{\mathcal{C}} w\_1(y|q) F'(F, dq | y),\\ w\_2(F|y) &\triangleq \mathbb{E}\_{\Gamma} \{h(\gamma, X) | Y = y\} = \int\_{\mathcal{C}} w\_2(y|q) F'(F, dq | y),\end{split} \tag{20}$$

C

and their collections generated by the subsets <sup>C</sup> and <sup>F</sup>*L*:

$$\begin{array}{l}\text{W}(\mathcal{C}|y) \triangleq \{w(y|q) : q \in \mathcal{C}\},\\\text{W}(\mathbb{F}\_{L}|y) \triangleq \{w(F|y) : F \in \mathbb{F}\_{L}\}.\end{array} \tag{21}$$

Here and below the notation H(*y*) also stands for the whole set of the estimate values *<sup>h</sup>* <sup>∈</sup> <sup>H</sup> calculated for the fixed argument *<sup>y</sup>*.

The set <sup>W</sup>(F*L*|*y*) ∈ B(R+1) is compact; moreover (see [37]), the inclusion <sup>W</sup>(F*L*|*y*) <sup>⊆</sup> conv(W(C|*y*)) holds.

On the set <sup>R</sup> <sup>×</sup> <sup>R</sup>+<sup>1</sup> we prepare the new loss function:

$$\mathbf{J}(\boldsymbol{\eta},\boldsymbol{w}) \triangleq \boldsymbol{w}\_1 - \mathbf{2}\langle \boldsymbol{\eta},\boldsymbol{w}\_2 \rangle + \|\boldsymbol{\eta}\|^2 = \boldsymbol{w}\_1 - \|\boldsymbol{w}\_2\|^2 + \|\boldsymbol{\eta} - \boldsymbol{w}\_2\|^2. \tag{22}$$

It is easy to verify that the loss function (18) can be expressed via (22):

$$J(\overline{h}, F|y) = \int\_{\mathcal{C}} \mathbf{J}(\overline{h}(y), w(y|q)) F'(F, dq|y) = \mathbf{J}(\overline{h}(y), w(F|y)).$$

The corresponding guaranteeing criterion takes the form:

$$\mathbf{J}^\*(\eta|y) \triangleq \sup\_{w \in \mathcal{W}(\mathbb{F}\_L|y)} \mathbf{J}(\eta, w), \tag{23}$$

and its dual can be determined as:

$$\mathbf{J}\_{\*}(w) \stackrel{\triangle}{=} \min\_{\eta \in \mathbb{H}(y)} \mathbf{J}(\eta, w) = f(w\_{2\*} w) = w\_{1} - \|w\_{2}\|^{2}. \tag{24}$$

*The finite-dimensional minimax problem* is to find:

$$\widehat{\mathbf{h}}(\boldsymbol{y}) \in \underset{\boldsymbol{\eta} \in \mathbb{H}(\boldsymbol{y})}{\operatorname{Argmin}} \mathbf{J}^\*(\boldsymbol{\eta}|\boldsymbol{y}).\tag{25}$$

From the definitions of <sup>W</sup>(F*L*|*y*), <sup>H</sup>(*y*) and criterion (23) it follows that the problem (25) is equivalent to the initial minimax estimation problem (8):

$$\min\_{\overline{h}\in\mathbb{H}} J^\*(\overline{h}|y) = \min\_{\eta\in\mathbb{H}(y)} \mathbf{J}^\*(\eta|y) \triangleq \mathcal{I}(y),\tag{26}$$

$$\underset{\overline{h}\in\mathbb{H}}{\text{Argmin}}\, f^\*(\overline{h}|y)\Big|\_{y}\triangleq\{\widehat{h}(y):\ J^\*(\widehat{h}|y)=\mathcal{J}(y)\}=\underset{\eta\in\mathbb{H}(y)}{\text{Argmin}}\,\mathbf{J}^\*(\eta|y)\tag{27}$$

for <sup>∀</sup> *<sup>y</sup>* <sup>∈</sup> <sup>R</sup>*k*.

The following theorem characterizes the solution to the finite-dimensional minimax problem in terms of a saddle point of the loss function **J**.

**Theorem 1.** *For* <sup>∀</sup> *<sup>y</sup>* <sup>∈</sup> <sup>R</sup>*k, the loss function* **<sup>J</sup>**(*η*, *<sup>w</sup>*) *(22) has the unique saddle point* (**h**(*y*), **<sup>w</sup>** (*y*)) *on the set* <sup>H</sup>(*y*) <sup>×</sup> <sup>W</sup>(F*L*|*y*)*. The second block subvector* **<sup>w</sup>** (*y*) = col(**w** <sup>1</sup>(*y*), **<sup>w</sup>** <sup>2</sup>(*y*)) <sup>∈</sup> <sup>W</sup>(F*L*|*y*) *of the saddle point is the unique solution to the finite-dimensional dual problem:*

$$\{\hat{\mathbf{w}}(y)\} = \underset{w \in \mathcal{W}(\mathbb{F}\_L|y)}{\text{Argmax}} \, \mathbf{J}\_\*(w),\tag{28}$$

*and* **<sup>h</sup>**(*y*) = **<sup>w</sup>** <sup>2</sup>(*y*) *is the second sub-vector of this optimum* **<sup>w</sup>** (*y*)*.*

The proof of Theorem 1 is given in Appendix B.

By the definition of <sup>W</sup>(F*L*|*y*), for any vector **<sup>w</sup>** (*y*) there exists at least one distribution *<sup>F</sup>* such that:

$$\hat{\mathbf{w}}\_1(y) = \mathbb{E}\_{\tilde{F}}\{ \|h(\gamma, X)\|^2 | Y = y \}, \quad \hat{\mathbf{w}}\_2(y) = \mathbb{E}\_{\tilde{F}}\{ h(\gamma, X) | Y = y \}. \tag{29}$$

*<sup>F</sup>* is an LFD, and the whole set of the distributions satisfying (29) is denoted by <sup>F</sup>*L*. Theorem 1 allows to obtain a solution to the initial minimax estimation problem. The result is formulated as:

**Corollary 1.** *The estimator* **<sup>w</sup>** <sup>2</sup>(*y*) *introduced in Theorem <sup>1</sup> is a solution to the minimax estimation problem (8), i.e., h*(*y*) = **<sup>w</sup>** <sup>2</sup>(*y*) *point-wise. The set* {(*h*, *<sup>F</sup>*)}*F*∈F*<sup>L</sup> presents the saddle points of the loss function <sup>J</sup> (6) on the set* <sup>H</sup> <sup>×</sup> <sup>F</sup>*L. The estimator* **<sup>h</sup>**(*y*) *is invariant to the LFD choice: if <sup>F</sup> and <sup>F</sup> are different LFDs then* <sup>E</sup>*F*{*h*(*γ*, *<sup>X</sup>*)|*<sup>Y</sup>* <sup>=</sup> *<sup>y</sup>*} <sup>=</sup> <sup>E</sup>*F*{*h*(*γ*, *<sup>X</sup>*)|*<sup>Y</sup>* <sup>=</sup> *<sup>y</sup>*} <sup>=</sup> **<sup>w</sup>** <sup>2</sup>(*y*)*.*

The following assertion characterizes the key property of the LFD set <sup>F</sup>*L*.

**Corollary 2.** *There exists a variant of the LFD <sup>F</sup><sup>L</sup>* <sup>∈</sup> <sup>F</sup> *concentrated at most at* dim(W(F*L*|*y*)) <sup>+</sup> <sup>1</sup> *points of the set* C*.*

The proof of Corollary 2 is given in Appendix C.

The presence of the discrete version of LFD is a remarkable fact. Let us remind the reader that initially, we suppose that the uncertain vector *γ* lies in the set C. The deterministic hypothesis concerning *γ* hopelessly obstructed the solution to the minimax estimation problem. To overcome this obstacle we provide the randomness of *γ*: the vector keeps constant during the individual observation experiment, and the stochastic nature of *γ* appears from experiment to experiment only. The existence of a discrete LFD returns us partially to the primordial situation. The point is that there exists a set of *γ* values that are the most difficult for estimation. Tuning to these parameters we can obtain estimates of *γ* with the guaranteed quality.

Theorem 1 and Corollary 1 simplify the solution to the initial problem (8), reducing it to the maximization of the finite-dimensional quadratic function (28) over the convex compact set.
