13.1: "Inference = Control" in quantum language

It is usually considered that \begin{align} \left\{\begin{array}{ll} \mbox{$\bullet$ statistics is closely related to inference} \\ \mbox{$\bullet$ dynamical system theory is closely related to control} \end{array}\right. \end{align} However, in this chapter, we show that \begin{align} \mbox{ "inference" = "control" } \end{align} In this sense, we conclude that statistics and dynamical system theory are essentially the same.
6.1.1: Inference problem(statistics)

Problem 13.1 [Inference problem and regression analysis] Let $\Omega$ $\equiv$ $\{ \omega_1 , \omega_2 , ... , \omega_{100} \}$ be a set of all students of a certain high school. Define $h : \Omega \to [{}0, 200{}]$ and $w : \Omega \to [{}0, 200{}]$ such that:

\begin{align} & h (\omega_n) = \mbox{ "the height of a student $\omega_n$" } \quad (n= 1,2,..., 100) \nonumber \\ & w (\omega_n) = \mbox{ "the weight of a student $\omega_n$" } \quad (n= 1,2,..., 100) \tag{13.1)} \end{align}

For simplicity, put, $N=5$. For example, see Table 13.1.

Assume that:
 $(a_1):$ The principal of this high school knows the both functions $h$ and $w$. That is, he knows the exact data of the height and weight concerning all students.
Also, assume that:
$(a_2):$ Some day, a certain student helped a drowned boy. But, he left without reporting the name. Thus, all information that the principal knows is as follows:
 (i): he is a student of his high school. (ii): his height [resp. weight] is about 170 cm [resp. about 80 kg].
Now we have the following question:
 $(b):$ Under the above assumption (a$_1$) and (a$_2$), how does the principal infer who is he?

13.1.2; Control problem(dynamical system theory)

Adding the measurement equation $g: {\mathbb R}^3 \to {\mathbb R}$ to th state equation, we have dynamical system theory (13.2)

. That is, \begin{align} \fbox { dynamical system theory } = \left\{\begin{array}{ll} {\rm{(i)}}: \underset{ (\mbox{initial} \omega(0)=\alpha)}{\frac{d \omega (t)}{dt} = v(\omega(t), t{}, e_1(t), \beta)} \; & \cdots \mbox{(state equation)} \\ \\ {\rm{(ii)}}: x(t) = g(\omega(t), t{}, e_2 (t) ) \; & \cdots \mbox{(measurement)} \end{array}\right. \tag{13.2} \end{align}

where $\alpha, \beta$ are parameters, $e_1 (t)$ is noise, $e_2 (t)$ is measurement error.

The following example is the simplest problem concerning inference.

Problem 13.2 [Control problem and regression analysis] We have a rectangular water tank filled with water.

Assume that the height of water at time $t$ is given by the following function $\omega(t)$:

\begin{align} \frac{d\omega}{dt}=\beta_0, \mbox{ then } \omega(t) = \alpha_0 + \beta_0 t, \tag{13.3} \end{align}

where $\alpha_0$ and $\beta_0$ are unknown fixed parameters such that $\alpha_0$ is the height of water filling the tank at the beginning and $\beta_0$ is the increasing height of water per unit time. The measured height $x(t)$ of water at time $t$ is assumed to be represented by

\begin{align} x(t) = \alpha_0 + \beta_0 t + e(t), \end{align}

where $e(t)$ represents a noise (or more precisely, a measurement error) with some suitable conditions. And assume that as follows:

Under this setting, we consider the following problem:

 $(c_1):$ [Control]: Settle the state $(\alpha_0, \beta_0)$ such that measured data (13.4) will be obtained.
or, equivalently,
 $(c_2):$ [Inference]: when measured data (13.4) is obtained, infer the unknown state $(\alpha_0, \beta_0)$.
Note that \begin{align} \mbox{ (${ c}_1$)=(${ c}_2$) } \end{align} from the theoretical point of view. Thus we consider that
 $(d):$ Inference problem and control problem are the same problem. And these are characterized as the reverse problem of measurements.
 $(\sharp):$ the noise $e_1 (t)$ and the measurement error $e_2 (t)$ have the same mathematical structure (i.e., stochastic processes ).