\documentstyle[IEEEtran]{article}
\begin{document}
\tolerance 10000
\title{Auxiliary Problems of Statistical Data Processing: Interval Approach}
\author{V. P. Kuznetsov}

\pagestyle{myheadings}

\markright{APIC'95, El Paso,
Extended Abstracts,
A Supplement to the international journal of {\rm Reliable
Computing}\ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ }

\maketitle

\begin{abstract}
In \cite{1}, we have considered interval methods of solving the {\it basic}
data statistical data processing problems, in which:
\begin{itemize}

\item the input statistical information consists of finitely many 
estimates of generalized moments (i.e., of finitely many inequalities
of the type $E(f_i)\in {\bf a}_i$); and

\item the desired output is an (interval) estimate for
a given statistical characteristic $E(f)$. 
\end{itemize}

In many real-life situations, we have statistical problems
of different type:
\begin{itemize}
\item we may have an {\it additional statistical
information}; and

\item we may have an {\it objective} that is different from
finding an estimate for a given statistical characteristics.
\end{itemize}
In this paper, we describe how interval methods can be applied to 
these auxiliary statistical problems. 
\end{abstract} 
  
\auffil{The author is with the Dept. of Prognosis, 
Institute of Complex Systems, Moscow-Murmansk, Russia.
The author is thankful to V. Kreinovich 
for his help in writing and editing this text.}


\section{Sometimes, We Encounter Problems That Are Different
From Basic Statistical Data Processing} 
Let us describe possible auxiliary problems
(their solutions will be outlined in the forthcoming sections). 

\subsection{Problems in which we have additional statistical information} 
In the basic problems from \cite{1}, we consider the case when we have one or
several variables $x_i$ that are measured several times, and, as a
result of these measurements, we get the interval values for finitely
many characteristics. While doing it, we make two implicit
assumptions:
\begin{itemize}
\item First, this information is usually formulated in terms
of several variables $x_i$, and nothing is known  a priori about the 
relationship between the different variables $x_i$. 

\item Second, that the actual values $x_i$ of the physical
quantities are not changing from one observations to another (or, to
be more precise, that the changes in $x_i$ are negligible). In
mathematical terms, this means that $x_i$ is {\it not} depending on
time $t$. 
\end{itemize}
In real life, we can have an additional information with
respect to both possible dependencies:
\begin{itemize}
\item First, we can have some additional a priori
information about the relationship between $x_i$.

\item Second, it is possible that the values of $x_i$ do
change essentially with time. 
\end{itemize}

\subsection{Problems in which we have a different objective} As a
basic problem, we considered the problem of estimating the value of a
statistical characteristics (namely, of a generalized interval). 
Instead of that objective, we may have different ones:
\begin{itemize}

\item We may want to estimate the value of a more
complicated statistical characteristics, or

\item Instead of solving an {\it analysis} problem, we may
want to solve a {\it synthesis} problem, i.e., use the given
information to make a certain {\it decision}. \end{itemize}

\section{Case of Additional Statistical Information:
Dependency Between $x_i$}

\subsection{Formulation of the problem}
As we have already mentioned, in the description of a basic
problem, we assumed that we only have the information of the type
$E(f_i)\in{\bf a}_i=[a^-_i,a^+_i]$. This information is
obtained by {\it generalizing} the empirical information: 
\begin{itemize}
\item We observe that the average value of $f_i$
(estimated from a finite sample) is in between $a^-_i$ and $a^+_i$; and
\item if this finite sample is sufficiently large to make
statistical predictions correct, we can then conclude that, crudely
speaking, a similar inequality is true for an {\it arbitrary} sample,
i.e., in statistical terms, that $E(f_i)\in{\bf a}_i$. 
\end{itemize}
In some cases, we can generalize this information even
further: namely, we can, e.g., observe that whenever we estimate the
characteristics $f_i(x_i)$ that correspond to two different variables,
then $E(f_i\cdot f_j)=E(f_i)E(f_j)$. In statistical terms, this means
that all experimental results are consistent with the hypothesis that
the variables $x_i$ and $x_j$ are {\it statistically independent.} We
can therefore generalize this equality $E(f_i\cdot f_j)=E(f_i)E(f_j)$
(that is confirmed for finitely many $f_i$ that have actually been
tested) to {\it arbitrary} functions of $x_i$ and $x_j$, i.e., in
other words, we can {\it conclude} that the variables $x_i$ are
independent.

In these cases, in addition to the partial statistical
information 
$$(m_i,f_{i1},...,f_{i m_i},{\bf a}_{i1},...,{\bf
a}_{m_i})$$ about each of the variables $x_i, 1\le i\le n$, 
we know that these variables are {\it independent}. How to solve
statistical data processing problem in this case? 

\subsection{Idea} Independence means
that $$E(f(x_i)f'(x_{i'}))=E(f(x_i))E(f'(x_{i'}))$$ whenever $i\ne i'$. 
Therefore, if we
know that $E(f(x_i))\in {\bf a}$, and that $E(f'(x_{i'}))\in {\bf
a}'$, then we can conclude that 
$$E(f(x_i)f'(x_{i'}))=E(f(x_i)E(f'(x_{i'}))\in {\bf a}\cdot{\bf a}'$$
(here, $\cdot$ indicates the product of the two intervals in the sense
of interval computations). 
In view of the above idea, if we
 want to estimate the values of some characteristics $g:R^n\to R$,
we can use the following statistical information about $\vec
x=(x_1,...,x_n)$:
\begin{itemize}

\item $E(f_{ij}(x_i))\in {\bf a}_{ij}$ for all $i\le n$ and
$j\le n_i$;

\item $E(f_{i_1j_1}(x_{i_1})f_{i_2 j_2}(x_{i_2}))\in {\bf
a}_{i_1j_1}\cdot {\bf a}_{i_2j_2}$;

\item ...

\item $E(f_{1 j_1}(x_1)...f_{n j_n}(x_n))
\in {\bf a}_{1 j_1}\cdot ...\cdot {\bf a}_{n j_n}$.
\end{itemize}

\noindent{\bf METHODOLOGY 3.} (solving the basic problem of
statistical data processing for the case when the error of input
variables are independent)

{\bf Input:}
\begin{itemize}
\item {\tt an integer $n$ (called the {\it number of input
variables});}

\item{\tt for every $i=1,...,n$, a partial statistical
information $(m_i,f_{i1},...,f_{im},{\bf a}_{i1},...,{\bf a}_{im})$ on a set
$R$ ($f_{ij}$ are functions of one variable, and ${\bf a}_{ij}$
are intervals); and }

\item{\tt a function $g:R^n\to R$.}
\end{itemize}

{\bf Objective:} {\tt To estimate the interval}

\noindent{\tt ${\bf a}=[a^-,a^+]$ of possible values of}

\noindent{\tt $E(g(x_1,...,x_n))$
for all distributions in which all $x_i$ are independent, and each
partial distribution is consistent with the given statistical
information.}

{\bf Objective in precise mathematical terms:} {\tt To
estimate the interval of posible values of the mathematical
expectation $E(g(x))$ for all
probability distributions for which all $x_i$ are independent, and 
$E(f_{ij}(x_i))\in {\bf a}_{ij}$ for all $i$ and $j$.} 

{\bf Methodology:} 
\begin{itemize}
\item {\tt Formulate the basic statistical problem (in the sense
of \cite{1}) as follows: In this problem, $X=R^n$, and the
partial statistical information will consists of the following parts:}
\begin{itemize}
\item[$\bullet$]$E(f_{ij}(x_i))\in {\bf a}_{ij}$ for all $i\le n$ and
$j\le n_i$;

\item[$\bullet$]$E(f_{i_1j_1}(x_{i_1})f_{i_2 j_2}(x_{i_2}))\in {\bf
a}_{i_1j_1}\cdot {\bf a}_{i_2j_2}$;

\item[$\bullet$] ...

\item[$\bullet$]$E(f_{1j_1}(x_1)...f_{n j_n}(x_n))
\in {\bf a}_{1 j_1}\cdot ...\cdot {\bf a}_{n j_n}$.
\end{itemize}
\item[]{\tt Here, we use interval computations to compute the product
of the intervals.}

\item{\tt Solve the resulting problem, and return its
result as an estimate for the solution of the given problem.}
\end{itemize}

\noindent {\it Comment 1.} 
This method was proposed in Ch. 2 of \cite{Kuznetsov 1991}.
\smallskip

\noindent {\it Comment 2.} This method uses interval computation techniques. 
\smallskip

\noindent {\it Comment 3.} 
The resulting inequalities do not completely
describe the fact that the variables $x_i$ are independent. For
example, if we have no conditions on $x_i$ at all, we will thus get no
conditions on $R^n$, but definitely not all probability 
distributions on $R^n$ correspond to the independent case. We know of
no other general method that would estimate the values of an arbitrary
statistical characteristics under the condition of independence. The
reason is that $n$ independent probability distributions can be
described by $n$ probability measures (or density functions
$\rho_i$). Statistical information about each variable $x_i$ can be
represented (as above) as a linear inequality in terms of $\rho_i$.
However, the value of the desired characteristics $E(g(x_1,...,x_n))$
is expressible by a {\it non-linear} formula $E(f)=\int
g(x_1,...,x_n)\rho(x_1,...,x_n)\,dx_1...dx_n=\int
g(x_1,...,x_n)\rho_1(x_1)\cdot ...\cdot \rho_n(x_n)\,dx_1...dx_n$. To
be more precise, this expression is linear in each of the variables
$\rho_i(x_i)$, so, it is {\it multi-linear}. So,
instead of a {\it linear programming problem}, we get a {\it
multi-linear} optimization problem, that is more difficult to solve. 
\smallskip

\noindent {\it Comment 4.} One important case of this problem is
when we have partial information about several independent variables
$v_i$ (e.g., we know the intervals to which their moments belong), and
we are interested in the characteristics of the sum $v_1+...+v_n$.
This case is important because under some reasonable assumptions, 
when $n\to\infty$, the resulting distribution tends to Gaussian. The
corresponding {\it central limit theorem} is used in practice to
justify the fact that when the error consists of many small
independent components, then the resulting error distribution is close
to Gaussian. Analogues of central limit theorem for the case of
different partial information are considered in Ch. 3 of
\cite{Kuznetsov 1991}.

\section{Case of Additional Statistical Information:
Dependency of the Quantities $x_i$ on Time $t$}

In the previous subsections, we considered the case when we
are interested in the values of one or several physical quantities. In
these case, the set $X$ on which all our probability measures are
defined coincides either with $R$, or with $R^n$ for $n\ge 2$). 
In these situations, even if we repeat measurements several times, at
different moments of time, we are still measuring the same physical
quantities. In other words, in these cases, even if in reality the
actual values of the measured characteristics change with time, we can
safely neglect these changes.

In some real-life cases, however, the change of the values of the desired
characteristics from one measurement to another is too large to be
neglected. In these cases, the values that we are measuring can no
longer be described by one or few values; the adequate description for
these characteristics $x_i$ is by {\it functions} $x_i(t)$ 
that describe how the
values of these characteristics $x_i$ change with time $t$.

This new {\it dynamic} feature leads to a possibility of a
new type of statistical information. Indeed:
\begin{itemize}

\item In {\it static} case, the only possible type of information
consisted of an inequality $E(f_i)\in {\bf a}_i$. This information is
obtained by {\it generalizing} the empirical information: 
\begin{itemize}
\item[$\bullet$]We observe that the average value of $f_i$
(estimated from a finite sample) is in between $a^-_i$ and $a^+_i$; and

\item[$\bullet$]if this finite sample is sufficiently large to make
statistical predictions correct, we can then conclude that, crudely
speaking, a similar inequality is true for an {\it arbitrary} sample,
i.e., in statistical terms, that $E(f_i)\in{\bf a}_i$. 
\end{itemize}
\item In {\it dynamic} case, we can similarly get estimates,
e.g., for $E(x(t))$ for different values $t_1,...,t_n$ of the time 
$t$. As a result, we conclude that $E(x(t_i))\in {\bf a}_i=[a^-,a^+_i]$.
If the resulting values $a^\pm_i$ are irregularly depending on $i$, then
these $n$ inequality are all the information that we can get. However,
in many real-life cases, the bounds $a^\pm_i$ that correspond to the
moment of time $t_i$ change rather regularly
with time $t_i$; this regularity means that we can express the
dependency of $a^\pm_i$ on $t_i$ by a simple formula:
$a^\pm_i=f^\pm(t_i)$. Therefore, as a result of these estimates, we
conclude that for the values $t_i$, 
$E(x(t_i))\in [f^-(t_i),f^+(t_i)]$. Now, if this is true for
sufficiently many moments of time $t_i$, then, similarly to the
above-described generalization step, we can make a {\it second
generalization}, and conclude that a similar inequality $E(x(t))\in
[f^-(t),f^+(t)]$ is true for all $t$. 
\end{itemize}
As a result, we have a statistical information that consists
of {\it continuum many} inequalities: in the above example, inequalities
that correspond to all possible values of $t$.

Other examples of this type of statistical information include
conditions on correlation $E(x(t)x(t'))$, on moments $E((x(t))^a)$, etc.
Methods and results of handling this information are described in Ch. 4
of \cite{Kuznetsov 1991}.

\section{Case of a Different Objective: Complicated
Statistical Characteristics}

\subsection{Formulation of the problem}
There are cases when we are interested in possible values of a
statistical characteristics that is more complicated than a
generalized moment $E(g)$.
For example, suppose that we have some information 
about the general error distribution $\mu$. Now, we have an auxiliary
device that enables us to catch errors that exceed a certain number
$\Delta_0$. As a result of applying this device, we get only errors
that are $\le \Delta_0$. What are the statistical characteristics of
the resulting ``curtailed'' error distribution? This new distribution
can be characterized by the {\it conditional} mathematical
expectations $E(g|B)$ under the condition $B$ (in our case, $B$ stands
for $|v|\le\Delta_0$). This condition mathematical expectation is
defined as $E(g\cdot\chi_B)/E(\chi_B)$. How to find the interval of
possible values of this fraction? In general, 
how to find the interval of possible values for the {\it fraction}
$E(N)/E(D)$ of two generalized moments?

\subsection{First approach: naive interval computations} 
One way to do solve this problem 
is to compute (or estimate) the intervals of
possible values of $E(N)$ and $E(D)$, and then apply interval
division to the resulting intervals. 
\newpage

\noindent{\bf METHODOLOGY 4.} (computing conditional statistical
characteristics)

{\bf Input:}
\begin{itemize}
\item{\tt a set $X$;}

\item{\tt a partial statistical information 
$(m,f_1,...,f_m,{\bf a}_1,...,{\bf a}_m)$ on a set $X$; and}

\item{\tt a function $g:X\to R$:}

\item{\tt a set $B\subset X$.}
\end{itemize}

\noindent{\bf Objective:} {\tt To estimate the interval}

\noindent{\tt  ${\bf a}=[a^-,a^+]$ 
of possible values of the}

\noindent{\tt conditional characteristic $E(g(x)|B)$
for all distributions that are consistent with the given statistical
information.}

\noindent{\bf Objective in precise mathematical terms:} {\tt To
estimate the interval of posible values of the conditional mathematical
expectation $E(g(x)|B)$ for all
probability distributions for which $E(f_i(x))\in {\bf a}_i$ for
$i=1,...,m$.} 

\noindent{\bf Methodology:} 
\begin{itemize}

\item{\tt Formulate and solve the following two auxiliary
statistical problems:}
\begin{itemize}

\item[$\bullet$]{\tt For given partial statistical information 
$(m,f_1,...,f_m,{\bf a}_1,...,{\bf a}_m)$,
estimate the interval of possible values of
$E(g\cdot\chi_B)$. Denote the resulting interval by $\bf N$.}

\item[$\bullet$]{\tt For given partial statistical information 
$(m,f_1,...,f_m,{\bf a}_1,...,{\bf a}_m)$,
estimate the interval of possible values of
$E(\chi_B)$ ($=p(B)$). Denote the resulting interval by $\bf D$.}
\end{itemize}
\item{\tt Divide $\bf N$ by $\bf D$ (using interval
computations), and return the result of this division as the desired
interval.}
\end{itemize}

\noindent{\it Comment.} This idea is similar to 
naive interval computations, so no wonder that its result is 
often an overestimate and therefore, a new method is needed to get a
better estimate.

\subsection{More complicated techniques} 
A general methodology for computing 
$$E(f\cdot\chi_B)/E(\chi_B)$$ is proposed in \cite{Kuznetsov 1991}, Section 1.6.
This methodology is based on the following idea:
\begin{itemize}

\item First, based on the initial statistical information, 
we find the interval $[p^-,p^+]$ of possible values of $p(B)=E(\chi_B)$. 

\item Now, for every $p\in [p^-,p^+]$, we can add the condition
$p(B)=p$ (i.e., $E(\chi_B)\in [p,p]$) to the initial set of conditions
and find the interval $[a^-(p),a^+(p)]$ 
of possible values of $E(g\cdot\chi_B)$. 

\item For every $p$, the ratio 
$E(f\cdot\chi_B)/E(\chi_B)$ can thus take any value from $a^-(p)/p$ to
$a^+(p)/p$. Therefore, the desired interval can be described as
$[a^-,a^+]$, where $a^-=\min(a^-(p)/p)$, $a^+=\max(a^+(p)/p)$, and $p$
runs from $p^-$ to $p^+$.
\end{itemize}

\noindent{\bf METHODOLOGY 5.} (computing conditional statistical
characteristics)

{\bf Input:}
\begin{itemize}

\item{\tt a set $X$;}

\item{\tt a partial statistical information 
$(m,f_1,...,f_m,{\bf a}_1,...,{\bf a}_m)$ on a set $X$; and}

\item{\tt a function $g:X\to R$:}

\item{\tt a set $B\subset X$.}
\end{itemize}

{\bf Objective:} {\tt To estimate the interval}

\noindent{\tt ${\bf a}=[a^-,a^+]$ 
of possible values of the}

\noindent{\tt conditional characteristic $E(g(x)|B)$
for all distributions that are consistent with the given statistical
information.}

{\bf Objective in precise mathematical terms:} {\tt To
estimate the interval of posible values of the conditional mathematical
expectation $E(g(x)|B)$ for all
probability distributions for which $$E(f_i(x))\in {\bf a}_i$$ for
$i=1,...,m$.} 

{\bf Methodology:} 
\begin{itemize}

\item{\tt Formulate and solve the following auxiliary
statistical problem:}
\begin{itemize}

\item[]
{\tt For given partial statistical information 
$(m,f_1,...,f_m,{\bf a}_1,...,{\bf a}_m)$,
estimate the interval of possible values of
$E(\chi_B)$ ($=p(B)$).}
\end{itemize}
\item[]{\tt Denote the resulting interval by $[p^-,p^+]$.}

\item{\tt For every $p$ from the interval $[p^-,p^+]$, formulate and solve
the following auxiliary statistical problem:}
\begin{itemize}
\item[]{\tt 
For given statistical information 
$(m,f_1,...,f_m, f_{m+1},{\bf a}_1,...,{\bf a}_m, {\bf a}_{m+1})$,
where $$f_{m+1}=\chi_B,\ {\bf a}_{m+1}=[p,p],$$ 
estimate the interval of possible values of
$E(g\cdot \chi_B)$.}
\end{itemize}
\item[]{\tt Denote the resulting interval by $[a^-(p),a^+(p)]$.}

\item{\tt Estimate 
$$a^-=\min(a^-(p)/p), \ a^+=\max(a^+(p)/p)$$ (where $p$
runs from $p^-$ to $p^+$), and return $[a^-,a^+]$ as the desired interval.}
\end{itemize}

\noindent This methodology is based on our ability to solve
statistical data processing problems (with information of the type
$E(f_i)\in {\bf a}_i$). Since there is no general algorithm to solve such
problems, we cannot expect this method to always work. However, in
some cases, it does lead to an algorithm \cite{Kuznetsov 1991}, Section 1.6:
\begin{itemize}
\item Case when $g=\chi_A$, i.e., when we are interested in {\it
conditional probabilities.} 

\item Case when $g(x)=|x|$, and $B=[a,b]$, i.e., when we
know the average value of $|x|$, and we want to estimate a similar
average in case we restrict ourselves only to values form the interval
$[a,b]$ (for $[a,b]=[-\Delta_0,\Delta_0]$, this is exactly the case
that was our original motivation for considering conditional
statistical characteristics). 
\end{itemize}

\subsection{Case study: expert systems} 
The first case (when we are interested in conditional probabilities) 
is important for expert systems. One of the main parts of an
expert system is its {\it knowledge base}, that contains 
experts' statements. One way to describe the
experts' uncertainty in their own statements is to assign {\it
probabilities}. Suppose now that we know the probability of a
statement $A$, and that later on, some fact $B$ turned out to be true.
This additional information can change the experts' degrees of belief
in their statements: instead of the original probability 
$p(A)$, we can now characterized the
new degree of belief by a {\it conditional} probability $p(A|B)$. 

The necessity to update expert systems leads to another
statistical problem that is even
more complicated than the problem of estimating the conditional
probabilities: Namely, 
\begin{itemize}

\item Conditional probabilities appear when we add to the
knowledge base a new fact $B$ that is known to be true.

\item However, in many cases, the knowledge that we are
adding to knowledge base also comes from an expert, and therefore,
comes with a certain probability. In other words, we add $B$, but $B$
is true only with a certain probability $\gamma$. 
\end{itemize}
In \cite{Kuznetsov 1991}, Section 1.6, the update problem is
formulated and solved for this case as well. 

\subsection{Special case: how to estimate the actual value of the
desired quantity based on the results of several measurement with
random errors}
Suppose that we measure the value $x$ of the desired physical
quantity in the presence of an additive noise $\gamma$. Then, after $n$
measurements, we get $n$ values $\tilde x^{(1)}=x+\gamma^{(1)}$, ..., 
$\tilde x^{(n)}=x+\gamma^{(n)}$,
where $\gamma^{(i)}$ are the corresponding (unknown) noise values. It is
usually assumed that the variables $\gamma^{(i)}$ that correspond to
different measurements are independent. 

The statistical characteristics of the noise are usually not known
precisely; instead, we have a (partial) 
statistical information about the noise $\gamma$. Since the information
about $\gamma$ is statistical, we cannot uniquely determine $x$:
instead, we can only have {\it statistical} information about the actual
value of $x$. In order to describe this information, we must express the
information contained in the measurement results $\tilde x^{(i)}$ in terms of 
the random variables $\gamma^{(i)}$. 

We have a description ($\tilde x^{(i)}=x+\gamma^{(i)}$), 
but this description contains an
additional (unwelcome) variable $x$. So, to get the desired description,
we must reformulate it without $x$. This can be done as follows:
\smallskip

\noindent{\bf PROPOSITION 1.} {\it Let $n$ numbers 
$\tilde x^{(1)},..., \tilde x_n$ be given.
Then, for an arbitrary tuple $(\gamma^{(1)},...,\gamma^{(n)})$, 
the following two conditions are equivalent to each other:}
\begin{itemize}

\item{\it there exists a real number $x$ such that for every
$i$, $x^{(i)}=x+\gamma^{(i)}$;}

\item{\it for every $i$ and $j$, $\tilde
x^{(i)}-\gamma^{(i)}=x^{(j)}-\gamma^{(j)}$.}
\end{itemize}

Due to this proposition, the 
mathematical expectation of $x$ can be
described as a conditional mathematical expectation 
$$\bar x=E(x^{(1)}-\gamma^{(1)}\,|\,
x^{(i)}-\gamma^{(i)}=x^{(j)}-\gamma^{(j)})$$ under the condition that
$x^{(i)}-\gamma^{(i)}=x^{(j)}-\gamma^{(j)}$ for all $i$ and $j$; 
the standard deviation
of $x$ can be described as $\sqrt{E((x^{(1)}-\gamma^{(1)}-
\bar x)^2\,|\, ...)}$ under
the same conditions, etc. 
\smallskip

\noindent{\bf METHODOLOGY 6.} (estimating the actual value of the
desired quantity based on the results of several measurements with
random aditive errors). 

{\bf Input:}
\begin{itemize}

\item{\tt an integer $n$ (called the {\it number of
measurements});}

\item{\tt $n$ real numbers $\tilde x^{(1)},...,\tilde x^{(n)}$
(called {\it measurement results});}

\item{\tt for every $i=1,...,n$, a partial statistical
information 
$(m_i,f_{i1},...,f_{im},{\bf a}_{i1},...,{\bf a}_{im})$ on a set
$R$ ($f_{ij}$ are functions of one variable, and ${\bf a}_{ij}$
are intervals);}

\item{\tt a function $g:R\to R$.}
\end{itemize}

{\bf Objective:} {\tt To estimate the interval}

\noindent{\tt ${\bf a}=[a^-,a^+]$ of possible values of $E(g(x))$
for all distributions $\gamma^{(i)}$ 
in which all $\gamma^{(i)}$ are independent, $\tilde
x^{(i)}=x+\gamma^{(i)}$, and each
partial distribution is consistent with the given statistical
information.}

{\bf Objective in precise mathematical terms:} {\tt To
estimate the interval of possible values of the mathematical
expectation $E(g(x))$ for all distributions $\gamma^{(i)}$ in which all
$\gamma^{(i)}$ are independent, $\tilde
x^{(i)}=x+\gamma^{(i)}$, and 
$E(f_{ij}(\gamma^{(i)}))\in {\bf a}_{ij}$ for all $i$ and $j$.} 

{\bf Methodology:} 
\begin{itemize}

\item{\tt Formulate the statistical problem (in the sense
of \cite{1}) as follows: In this problem, $X=R^n$, and the
partial statistical information will consists of the following parts:}
\begin{itemize}
\item[$\bullet$]$E(f_{ij}(\gamma^{(i)}))\in 
{\bf a}_{ij}$ for all $i\le n$ and $j\le n_i$;

\item[$\bullet$]{\tt for all $i$ and $j$,
$$\gamma^{(i)}-\gamma^{(j)}=\tilde x^{(j)}-\tilde x^{(i)};$$}

\item[$\bullet$]{\tt all variables $\gamma^{(i)}$ are independent;}
\end{itemize}
\item[]{\tt and the goal is to estimate 
$E(g(x^{(1)}-\gamma^{(1)}))$.} 

\item{\tt Solve the formulated problem, and return 
the resulting estimate as the desired solution.}
\end{itemize}

In \cite{Kuznetsov 1991}, Ch. 6, different cases of this problem are
considered (e.g., estimating $x$ when we know several moments).
Also, in that chapter, several 
more complicated noises are considered, including the multiplicative noise 
$\tilde x^{(i)}=x\cdot\gamma^{(i)}$. In this case, the possibility to
describe the measurement results in terms of the noise values is
explained in the following proposition:
\smallskip 

\noindent{\bf PROPOSITION 2.} {\it Let $n$ numbers 
$\tilde x^{(1)},..., \tilde x_n$ be given.
Then, for an arbitrary tuple $(\gamma^{(1)},...,\gamma^{(n)})$, 
the following two conditions are equivalent to each other:}
\begin{itemize}

\item{\it there exists a real number $x$ such that for every
$i$, $x^{(i)}=x\cdot\gamma^{(i)}$;}

\item{\it for every $i$ and $j$, $\tilde
x^{(i)}/\gamma^{(i)}=x^{(j)}/\gamma^{(j)}$.}
\end{itemize}

Due to this proposition, e.g., the mathematical
expectation of $x$ is equal to the conditional mathematical expectation
of $\tilde x^{(i)}/\gamma^{(i)}$ under the condition that 
$\tilde x^{(i)}/\gamma^{(i)}=\tilde x^{(j)}/\gamma^{(j)}$, etc. 
\smallskip

\noindent{\bf METHODOLOGY 7.} (estimating the actual value of the
desired quantity based on the results of several measurements with
random multiplicative errors). 

{\bf Input:}
\begin{itemize}

\item{\tt an integer $n$ (called the {\it number of
measurements});}

\item{\tt $n$ real numbers $\tilde x^{(1)},...,\tilde x^{(n)}$
(called {\it measurement results});}

\item{\tt for every $i=1,...,n$, a partial statistical
information 
$(m_i,f_{i1},...,f_{im},{\bf a}_{i1},...,{\bf a}_{im})$ on a set
$R$ ($f_{ij}$ are functions of one variable, and ${\bf a}_{ij}$
are intervals);}

\item{\tt a function $g:R\to R$.}
\end{itemize}

{\bf Objective:} {\tt To estimate the interval ${\bf
a}=[a^-,a^+]$ of possible values of $E(g(x))$
for all distributions $\gamma^{(i)}$ 
in which all $\gamma^{(i)}$ are independent, $\tilde
x_i=x\cdot \gamma^{(i)}$, and each
partial distribution is consistent with the given statistical
information.}

{\bf Objective in precise mathematical terms:} {\tt To
estimate the interval of posible values of the mathematical
expectation $E(g(x))$ for all distributions $\gamma^{(i)}$ in which all
$\gamma^{(i)}$ are independent, $\tilde
x^{(i)}=x\cdot \gamma^{(i)}$, and 
$E(f_{ij}(\gamma^{(i)}))\in {\bf a}_{ij}$ for all $i$ and $j$.} 

{\bf Methodology:} 
\begin{itemize}

\item{\tt Formulate the statistical problem (in the sense
of \cite{1}) as follows: In this problem, $X=R^n$, and the
partial statistical information will consists of the following parts:}
\begin{itemize}

\item[$\bullet$]$E(f_{ij}(\gamma^{(i)}))\in 
{\bf a}_{ij}$ for all $i\le n$ and $j\le n_i$;

\item[$\bullet$]{\tt for all $i$ and $j$,
$\gamma^{(i)}/\gamma^{(j)}=\tilde x^{(j)}/\tilde x^{(i)}$;}

\item[$\bullet$]{\tt all variables $\gamma^{(i)}$ are independent;}
\end{itemize}
\item[]{\tt and the goal is to estimate 
$E(g(x^{(1)}/\gamma^{(1)}))$.}

\item{\tt Solve the formulated problem, and return 
the resulting estimate as the desired solution.}
\end{itemize}

\noindent{\it Comment.} 
In all these cases, for each characteristic $E(g(x))$ of $x$, we get an 
{\it interval} of possible values.

\section{Case of a Different Objective: Decision Making}

\subsection{First case of decision making: we know the objective
function, and we know the exact values of all the parameters of the
system}
Usually, we know what objective we want to achieve, i.e., what
numerical criterion $J$ we want to maximize, and how the value of this
criterion depends on our decision $u$ and on the parameters $\vec
x=(x_1,...,x_n)$ that describe the decision-making situation. In other
words, we know the function $J(u,x_1,...,x_n)$.

In the rare cases when 
we know the exact values of all the parameters $\vec x=(x_1,...,x_n)$,
the choice of a decision $u$ can be described as an optimization problem
$J(u,\vec x)\to\max$.

\subsection{Second case of decision making: we know the objective
function, but we do not know the exact values of the 
parameters of the system}
In real-life situations, however, we do not know the values $x_i$
precisely. Instead, we have a (partial) statistical information about
the values of $x_i$. In many such cases, a reasonable choice is to find
a decision $u$ that maximizes the {\it expected} value of $J$, i.e., for
which $E(J(u,\vec x))\to\max$. This choice makes perfect sense if we
face a series of repeating similar optimization problems (e.g., in
business or manufacturing), then, we want to choose solutions that are
in the average most profitable.

Since we only have {\it partial} statistical information about $\vec x$,
choosing a decision $u$ does not determine the mathematical expectation
$E(J(u,\vec x))$ uniquely. Instead, we have an {\it interval}
$[J^-(u),J^+(u)]$ of
possible values of $E(J)$. Therefore, we face a problem of choosing a
decision when only the {\it interval} containing an objective function is
known. In Ch. 5 of 
\cite{Kuznetsov 1991}, interval decision-making techniques (including
pessimism-optimism) are applied to the above-described statistical
decision making problem.

Decision is usually characterized by one of several numerical
parameters: e.g., how many items of a certain sort to produce, how many
to transport, etc. In mathematical terms, the set of all possible
decisions is usually {\it finite-dimensional}. 
However, there are special situations when the
decision is either simpler or more complicated than simply 
choosing a few parameters. 
In \cite{Kuznetsov 1991}, two such situations are considered:  
\begin{itemize}
\item{\it Hypothesis testing}, when we have a {\it finite} list of
hypotheses, and a decision consists of choosing one of them 
\cite{Kuznetsov 1991}, Ch. 7).
Here, the set of all possible decisions is finite and therefore,
0--dimensional. 

\item{\it Signal reconstruction}, when we try to reconstruct
the original signal from the noisy one that we have measured. In this
case, a decision consists of choosing a reconstructed signal $x(t)$.
Here, the set of decisions coincides with the set of all functions and
is, therefore, {\it infinite-dimensional}. Corresponding algorithms are
described in Ch. 5 of \cite{Kuznetsov 1991}. 
\end{itemize}

\section{Proof of Proposition 1.} 
If $x^{(i)}=x-\gamma^{(i)}$, then,
subtracting $i-$th equality from the $j-$th, we get the second
condition; vice versa, if the second condition is true, then we can take
this common difference as $x$. Q.E.D.



\begin{thebibliography}{99}


\bibitem{Kuznetsov 1991} 
V. P. Kuznetsov, {\it Interval statistical models}, Moscow, Radio i
Svyaz Publ., 1991 (in Russian). 

\bibitem{1}V. P. Kuznetsov, ``Interval Methods For Processing Statistical
Characteristics'', {\it These Proceedings}. 

\end{thebibliography}
\end{document}