In this section, we prepare mathematics, which is used in measurement theory (or in short, MT).

Measurement theory ([2-8]) is, by an analogy of quantum mechanics (or, as a linguistic turn of quantum mechanics), constructed as the scientific theory formulated in a certain -algebra (i.e., a norm closed subalgebra in the operator algebra composed of all bounded operators on a Hilbert space H, cf. [9,10]). MT is composed of two theories (i.e., pure measurement theory (or, in short, PMT] and statistical measurement theory (or, in short, SMT). That is, we see:

(B) MT (measurement theory)

where Axiom 2 is common in PMT and SMT. For completeness, note that measurement theory (B) (i.e., (B₁) and (B₂)) is a kind of language based on the quantum mechanical world view, (cf. [8]). It may be understandable to consider that 

(C) PMT and SMT is related to Fisher’s statistics and Bayesian statistics respectively.

Also, as mentioned in Section 2.6 latter, our concern in this paper is to give an answer to the question “Which is fundamental, PMT or SMT?”.

When, the -algebra composed of all compact operators on a Hilbert space H, the (B) is called quantum measurement theory (or, quantum system theory), which can be regarded as the linguistic aspect of quantum mechanics. Also, when is commutative (that is, when is characterized by, the - algebra composed of all continuous complex-valued functions vanishing at infinity on a locally compact Hausdorff space (cf. [9])), the (B) is called classical measurement theory. Thus, we have the following classification:

In this paper, we mainly devote ourselves to classical MT (i.e., classical PMT and classical SMT).

Now we shall explain the measurement theory (B). Let be a -algebra, and let be the dual Banach space of. That is, { is a continuous linear functional on }, and the norm is defined by

.

The bi-linear functional is also denoted by , or in short. Define the mixed state such that and for all satisfying. And put

A mixed state is called a pure state if it satisfies that for some and implies. Put

which is called a state space. The Riese theorem (cf. [11]) says that ,

Also, it is well known (cf. [9]) that

and

where. The latter implies that can be also identified with (called a spectrum space or maximal ideal space) such as

Here, assume that the -algebra is unital, i.e., it has the identity I. This assumption is not unnatural, since, if, it suffices to reconstruct the such that it includes.

According to the noted idea (cf. [12]) in quantum mechanics, an observable in is defined as follows:

(E₁) [Field] X is a set, (, the power set of X) is a field of X, that is, “”, “”.

(E₂) [Countably additivity] F is a mapping from to satisfying: 1) for every, is a nonnegative element in such that, 2) and, where 0 and I is the 0- element and the identity in respectively. 3): for any countable decomposition of (i.e., such that, ), it holds that

Remark 1. By the Hopf extension theorem (cf. [11]), we have the mathematical probability space (X, , ) where is the smallest -field such that. For the other formulation (i.e., -algebraic formulation), see the appendix in [7].

In what follows, we shall explain PMT in (B₁).

With any system S, a -algebra can be associated in which the pure measurement theory (B₁) of that system can be formulated. A state of the system S is represented by an element and an observable is represented by an observable in. Also, the measurement of the observable O for the system S with the state is denoted by (or more precisely, ). An observer can obtain a measured value by the measurement .

The Axiom^P 1 presented below is a kind of mathematical generalization of Born’s probabilistic interpretation of quantum mechanics. And thus, it is a statement without reality.

Axiom^P 1. [Pure Measurement]. The probability that a measured value obtained by the measurement belongs to a set is given by.

Next, we explain Axiom 2 in (B). Let be a tree, i.e., a partial ordered set such that and implies or. In this paper, we assume that T is finite (cf. Remark 9 in Section 7 later). Assume that there exists an element, called the  root of T, such that () holds. Put . The family is called a causal relation (due to the Heisenberg picture), if it satisfies the following conditions (F₁) and (F₂).

(F₁) With each, a -algebra is associated.

(F₂) For every, a Markov operator is defined (i.e., ,

). And it satisfies that holds for any,.

The family of dual operators is called a dual causal relation (due to the Schrödinger picture). When holds for any, the causal relation is said to be deterministic.

Now Axiom 2 in the measurement theory (B) is presented as follows: 

Axiom 2. [Causality]. The causality is represented by a causal relation.

Next, we have to study how to use the above axioms as follows. That is, we present the following interpretation (G) [= (G₁) – (G₃)], which is characterized as a kind of linguistic turn of so-called Copenhagen interpretation (cf. [7,8]). That is, we propose:

(G₁) Consider the dualism composed of observer and system (= measuring object). And therefore, observer and system must be absolutely separated.

(G₂) Only one measurement is permitted. And thus, the state after a measurement is meaningless since it can not be measured any longer. Also, the causality should be assumed only in the side of system, however, a state never moves. Thus, the Heisenberg picture should be adopted, and thus, the Schrödinger picture should be prohibited.

(G₃) Also, the observer does not have the space-time. Thus, the question: “When and where is a measured value obtained?” is out of measurement theory. And thus, Schrödinger’s cat is out of measurement theory, and so on.

For each, consider a measurement . However, since the (G₂) says that only one measurement is permitted, the measurements should be reconsidered in what follows. Under the commutativity condition such that

we can define the product observable

in such that

Here, is the smallest field including the family. Then, the above is, under the commutativity condition (2), represented by the simultaneous measurement.

Consider a tree with the root. This is also characterized by the map

such that. Let be a causal relation, which is also represented by. Let an observable in the be given for each. Note that is an observable in the.

The pair, is called a sequential causal observable. For each, put. And define the observable

in as follows:

if the commutativity condition holds (i.e., if the product observable exists) for each . Using (3) iteratively, we can finally obtain the observable in. The is called the realizetion (or, realized causal observable) of.

We shall introduce the following notation: it is usual to consider that we do not know the pure state

when we take a measurement . That is because we usually take a measurement in order to know the state. Thus, when we want to emphasize that we do not know the state, is denoted by

. Also, when we know the distribution

of the unknown state, the

is denoted by. The

is called a mixed state. And further, if we know that a mixed state belongs to a compact set

, the is denoted by

.

The Axiom^S 1 presented below is a kind of mathematical generalization of Axiom^P 1.

Axiom^S 1. [Statistical measurement]. The probability that a measured value obtained by the measurement belongs to a set is given by.

Thus, we can propose the statistical measurement theory (B₂), in which Axiom 2 and Interpretation (G) are common.

Let be an observable in a - algebra. Assume that we know that the measured value obtained by a statistical measurement belongs to

. Then, there is a reason to infer that the unknown measured value is distributed under the conditional probability, where

Thus, by a hint of Fisher’s maximum likelihood method, we have the following theorem, which is the most fundamental in this paper.

Theorem 1. [Fisher’s maximum likelihood method in general]. Let be an observable in a -algebra. Let be a compact set. Assume that we know that the measured value obtained by a measurement belongs to. Then, there is a reason to infer that the unknown measured value is distributed under the conditional probability, where

Here, is defined by

Remark 2. Theorem 1 is new throughout our research [2-8], though, in a particular case that, Theorem 1 was proposed in [7] where we devoted ourselves to PMT.

Note that

(H₁) for

, therefore, we see that [PMT] [SMT].

However, we have the following problem:

(H₂) Which is fundamental, PMT or SMT?

Recalling the (C), most readers may consider that PMT is more fundamental than SMT. In fact, throughout our research [2-8], we have believed in the fundamentality of PMT. However, in this paper, we assert that Theorem 1 in SMT is the most fundamental as far as inference. In fact, every result in this paper is regarded as one of the corollaries of Theorem 1. And hence, we shall conclude that SMT is proper as the answer to the problem (A). Also, our proposal has a merit such that the philosophy of statistics is naturally induced by the philosophy of measurement theory (cf. [8]).

We shall devote ourselves to classical case (i.e., ). From here, (or, commutative unital -algebra that includes) is, for simplicity, denoted by. Thus, we put

,

and

.

And, for any mixed state and any observable in, we put:

Also, put (: Borel -field). In order to avoid the confusion between in (6) and, we do not use. Also, for any, we put:

Let be an observable in a commutative -algebra. And let be any observable in. Consider the product observable in. The existence will be shown in Section 7 (Appendix).

Assume that we know that the measured value obtained by a simultaneous measurement belongs to. Then, by (4), we can infer that 

(I) the probability that y belongs to is given by

Thus, we can assert that:

Theorem 2. [Bayes method, cf. [4,5]]. When we know that a measured value obtained by a measurement belongs to, there is a reason to infer that the mixed state after the measurement is equal to, where

Proof. Note that we can regard that

. That is, there exists

such that

Then, Axiom^S 1 says that the probability that a measured value obtained by the measurement belongs to a set is given by, which is equal to in (7). Since is arbitrary, we obtain Theorem 2.

Remark 3. The above (I) is, of course, fundamental. However, in the sense mentioned in the above proof, we admit Theorem 2 as the equivalent statement of the (I). That is, in spite of Interpretation (G₂), we admit the wavefunction collapse such as 

Theorem 2 was, for the first time, proposed in [4,5] without the conscious understanding of Interpretation (G₂). Also, note that(K) in Theorem 2, if, then it clearly holds that.

Also, for our opinion concerning the wavefunction collapse in quantum mechanics, see [7].

Combining Theorem 1 (Fisher’s method) and Theorem 2 (Bayes’ method), we get the following corollary.

Corollary 1. [Fisher-Bayes method (i.e., Regression analysis in a narrow sense)]. When we know that a measured value obtained by a measurement belongs to, there is a reason to infer that the state after the measurement is equal to such that

where the is defined by

Remark 4. As mentioned in the above, note that Corollary 1 is composed of the following two procedure:

In this section, we examine Corollary 1 in a simple example. Readers will find that Corollary 1 can be regarded as regression analysis in a narrow sense.

We have a rectangular water tank filled with water. Assume that the height of water at time t is given by the following function:

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

where represents a noise (or more precisely, a measurement error) with some suitable conditions. And assume that we obtained the measured data of the heights of water at as follows:

Under this setting, we shall study the following problem:

(M) [Inference]: when measured data (10) is obtained, infer the unknown parameter in (9).

In what follows, from the measurement theoretical point of view, we shall answer the problem (M). Let be a series ordered set such that the parent map is defined by .

Put, ,

. For each, consider a continuous map such that

Then, we get the deterministic causal operators hus,

such that

Thus, we have the causal relation as follows.

Put,.

Let be the set of real numbers. Fix. For each, define the normal observable

in such that

Thus, we get the sequential deterministic causal observable

.

Then, the realized causal observable in is, by (3) and (12), obtained as follows:

Putting, we have the measurement

. Recall the (10), that is, the measured value obtained by the measurement is equal to

Define the closed interval such that

for sufficiently large N. Here, Fisher’s method (Theorem 1) says that it suffices to solve the problem.

(N) Find such as

Putting

we have the following problem that is equivalent to (N):

(O) Find such as

Calculating

we get

Thus, we see, by the statement (K), that 

This (i.e.,) is the answer to the problem (M).

Problem 1. Since the above example is quite easy, the validity of Bayes’ theorem in (P) may not be clear. If it is so, instead of the problem (M), we should present the following simple problem. 

(Q) Infer the water level at time 1.

Some may calculate and conclude as follows:

However, this calculation is based on the Schrödinger picture, and thus, the justification of this calculation (18) is not assured. That is because measurement theory (particularly, Interpretation (G₂)) says that the Heisenberg picture should be adopted. Therefore, in order to answer the problem (Q), we must prepare Corollary 2 (i.e., regression analysis in a wide sense) in the following section.

Remark 5. It should be noted that the following two are equivalent:

(R₁) [=(M); Inference]: when measured data (10) is obtained, infer the unknown parameter.

(R₂) [Control]: Settle the parameter such that measured data (10) will be obtained.

That is, we see that

“inference” = “control”.

Hence, from the measurement theoretical point of view, we consider that 

“Statistics” = “Dynamical system theory”though these are superficially different in applications.

Let be the root of a tree T. Let

be a sequential causal observable with the realization in. Thus we have the statistical measurement , where. Assume that we know that the measured value obtained by the measurement belongs to. Then, by (4), we can infer that (S) the probability that y belongs to is given by

Note that we can regard that . That is, there uniquely exists such that

for any observable in . Here, we used the following notation:

Define the observable such that

Then, we can define the Bayes operator

by (20).

Thus, as the generalization of Theorem 2, we have:

Theorem 3. [Causal Bayes’ theorem in classical measurements]. Let be the root of a tree T. Let

be a sequential causal observable with the realization. Thus we have the statistical measurement, where

. Assume that we know that a measured value obtained by the statistical measurement

belongs to. Then, there is a reason to infer that the mixed state

after the statistical measurement

is given by

.

Proof. The proof is similar to the proof of Theorem 2. Thus, we omit it.

Remark 6. In Theorem 3, we see that 

which is the generalization of the (J).

The following example promotes the understanding of Theorem 3.

Example 1. [The simple case such that]. Consider a particular case such that is series ordered set, i.e.,. And consider a causal relation

, that is,

Further consider sequential causal observable

.

Let be its realization. Note, by the Formula (3), that,

Putting, we have the measurement

Let be the posttest state in

(T), that is,. Define

such that

Then, we see that

That is because we see that, for any observable in,

Example 2. [Continued from the above example]. For each, assume that is deterministic, that is, there exists a continuous map satisfying (12). And, putting, consider the measurement

.

Then, we see, by (22), that, for any in,

Thus, we see that

Further we easily see that

Now we can present Corollary 2 (i.e., regression analysis in a wide sense) as follows. 

Corollary 2. [Causal Fisher-Bayes method (i.e., Regression analysis in a wide sense)]. Let be the root of a tree T. Let

be a sequential causal observable with the realization Assume the statistical measurement. And assume that we know that a measured value obtained by the measurement belongs to. Then, there is a reason to infer that the mixed state after the measurement

is given by. Here, the is defined by

Remark 7. Note that Fisher maximum likelihood method and Bayes’ theorem are hidden in Corollary 2. That is, Corollary 2 includes the following procedure:

which is the generalization of the (L).

Answer 1. [Answer to Problem 1 (Q)]. Now we can answer Problem 1 (Q) as follows. The (17) says that

. Thus, using (23), we see that

. Also, note that (17) and (23) are consequences of Corollary 2. Hence, the calculation (18) is justified by Corollary 2.

Remark 8. As mentioned in Section 1, in our research [2-8], we have been concerned with the problem (A). Particularly, in [6], we discussed Corollary 2 in the commutative -algebra. However, this was somewhat shallow, since “max” is not proper in but. Now we believe that fundamental statements concerning statistics should be always asserted in the framework of. Also, note that Corollary 2 is the natural generalization of Theorem 6.3 in [5].