A Measurement Theoretical Foundation of Statistics

It is a matter of course that Kolmogorov’s probability theory is a very useful mathematical tool for the analysis of statistics. However, this fact never means that statistics is based on Kolmogorov’s probability theory, since it is not guaranteed that mathematics and our world are connected. In order that mathematics asserts some statements concerning our world, a certain theory (so called “world view”) mediates between mathematics and our world. Recently we propose measurement theory (i.e., the theory of the quantum mechanical world view), which is characterized as the linguistic turn of quantum mechanics. In this paper, we assert that statistics is based on measurement theory. And, for example, we show, from the pure theoretical point of view (i.e., from the measurement theoretical point of view), that regression analysis can not be justified without Bayes’ theorem. This may imply that even the conventional classification of (Fisher’s) statistics and Bayesian statistics should be reconsidered.


Introduction
For example, consider Newtonian mechanics.It is natural to understand that Newton mechanics is based on Newton's three laws of motion, though the mathematical theory of differential equations is a useful tool for the analysis of Newtonian mechanics.That is because any mathematical theory is a closed logical system derived from set theory, and thus, it is not qualified to assert statements concerning our world without laws.If it is so, and, if Kolmogorov's probability theory [1] is a mathematical theory, we think that the foundation of statistics does not yet established.Thus, the following problem is natural: (A) What kind of law is statistics based on?Or, propose a foundation of statistics!
The purpose of this paper is to answer this problem.
Although in a series of our research [2][3][4][5][6][7][8] we have been concerned with this problem (A), in this paper we give a decisive answer to the problem (A) in the light of our final version [7,8] of measurement theory.Here, as mentioned in Section 2 later, measurement theory (i.e., the theory of the quantum mechanical world view) is characterized as the linguistic turn of quantum mechanics.Hence, note that measurement theory is not physics but a kind of language, and thus, the "law" in (A) is called "axiom" in this paper.

Measurement Theory (Axioms and
Interpretation)

Mathematical Preparations
In this section, we prepare mathematics, which is used in measurement theory (or in short, MT).Measurement theory ( [2][3][4][5][6][7][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: where Axiom 2 is common in PMT and SMT.For com-pleteness, note that measurement theory (B) (i.e., (B 1 ) and (B 2 )) 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, { The bi-linear functional   which is called a state space.The Riese theorem (cf.[11]) says that

S
Also, it is well known (cf.[9]) 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 . According to the noted idea (cf.[12]) in quantum mechanics, an observable in  is defined as follows: Countably additivity] F is a mapping from to satisfying: 1) for every , is a nonnegative element in such that

 
F X I  , where 0 and I is the 0element and the identity in A respectively.3): for any ., in the sense of weak convergence).
Remark 1.By the Hopf extension theorem (cf.[11]), we have the mathematical probability space (X,  , where  is the smallest  -field such that F   .For the other formulation (i.e., -algebraic formulation), see the appendix in [7].* W

Pure Measurement Theory in (B 1 )
In what follows, we shall explain PMT in (B 1 ).
With any system S, a * -algebra  can be associated in which the pure measurement theory (B 1 ) 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 . 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

 
x X  by the measurement .

  [ ] ,
A M O S  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.

Interpretation
Next, we have to study how to use the above axioms as follows.That is, we present the following interpretation (G) [= (G 1 ) -(G 3 )], which is characterized as a kind of linguistic turn of so-called Copenhagen interpretation (cf.[7,8]).That is, we propose: (G 1 ) Consider the dualism composed of observer and system (= measuring object).And therefore, observer and system must be absolutely separated.
(G 2 ) 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.
(G 3 ) 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.

Sequential Causal Observable and Its Realization
For each 1, 2, , . However, since the (G 2 ) says that only one measurement is permitted, the measurements we can define the product observable is the smallest field including the family with the root .This is also characterized by the map , , : if the commutativity condition holds (i.e., if the product observable . Using (3) iteratively, we can finally obtain the observable in .The is called the realize-

Statistical Measurement Theory in (B 2 )
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 Axiom S 1 presented below is a kind of mathematical generalization of Axiom P 1.
Thus, we can propose the statistical measurement theory (B 2 ), in which Axiom 2 and Interpretation (G) are common.
Let be an observable in a algebra .Assume that we know that the measured value Thus, by a hint of Fisher's maximum likelihood method, we have the following theorem, which is the most fundamental in this paper.
. Let be a compact set.Assume that we know that the measured value , Theorem 1 was proposed in [7] where we devoted ourselves to PMT.

Our Concern in This Paper
Note that (H 1 ) for , therefore, we see that [PMT] [SMT]. However, we have the following problem: (H 2 ) 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][3][4][5][6][7][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]).

Fisher-Bayes Method in Classical  
C Ω

Notations
We shall devote ourselves to classical case (i.e., ).From here, (or, commutative ) is, for simplicity, denoted by .Thus, we put And, for any mixed state and any observable in , we put: Also, put In order to avoid the confusion between 6) and , we do not use .Also, for any , we put: in .The existence will be shown in Section 7 (Appendix).
,F  Assume that we know that the measured value    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 That is, there exists Proof.Note that we can regard that Then, Axiom S 1 says that the probability that a mea value sured   y Y  obtained by the measurement The tim 2 ).Also, note that, (K) in Theorem 2, if oncerning the wave Also, for our opinion c function collapse in quantum mechanics, see [7].

measur
Combining Theorem 1 (Fisher's method) and T em 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 ed value obtained by a measurement As mentioned in the above, note that C rollary 1 is composed of the following two procedure:

A Simple Example of Fisher-Bayes Method (Regression Analysis in a Narrow Sense)
amp ed In this section, we examine Corollary 1 in a simple ex le.Readers will find that Corollary 1 can be regard 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   h t :

wer the ). Let
In what follows, from the measurement theoretical point of view, we shall ans problem (M   0,1, 2  be a series ordered set such that the parent map , .
Then, we get the deterministic causal operators hus, (12 Thus, we have the causal relation as follows. 2 ) Thus, we get the sequential deterministic causa servable , : Then, the realized causal observable in 3) and ( 12), obtained as follows: . Recall the (10), that is, the m value easured   for sufficiently large N. Here, Fisher's method (Theorem 1) says that it suffices to solve the problem (N) Find . , , , , Thus, we see, by the statement (K), that This (i.e., e answer to the pr lem 1.Since the above example is quite easy, the validity of Bayes' theorem in (P) may not be clear.f it is so the problem (M), we shou le problem.
the Schrödinger pi is (particularly, Interpretation (G 2 )) says that the picture is th oblem (M).Prob I , instead of ld present the following simp (Q) Infer the water level at time 1.Some may calculate and conclude as follows: However, this calculation is based on cture, and thus, the justification of this calculation (18) not assured.That is because measurement theory Heisenberg 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: (R 1 ) [=(M); Inference]: when measured data (10) is obtained, infer the unknown 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 applications.superficially different in

 
C Ω

Causal Bayes Method in Classical   Ω C
Let be the root of a tree T. Let be a sequential causal observable with the realization we have the statistical measurement , where   0 t that we know that the measured value we can infer that (S) the probability Note that we can regard that Here, we used the follo erator Then, we can define the Bayes op Thus, as the generalization of Theorem 2, w ve: Theorem 3. [Causal Bayes' theorem in classical measur , , : be a sequential causal observable with the realization F .Thus we have the statistical after the statistical measurement The following example promotes the un ]. Consider a particular case such that T series ordered set, i.e.,   Note, e Formula (3), that, be the posttest state in (T), that is, .
Then, we see that .
That is because that, for any observable we see Example 2. [Continued from the above example].For each ] Then, we see, by (22), that, for any Thus, we see that (23) Further we easily see that , , : be a sequential causal observable with the realization Here, the . Thus, using (23), we see that

5.
qu chanica he mos ament e presented without the answer to the problem (A).Also, note (U)) implies that even the convenf (Fisher's) statistics and Bayesian believe that fundamental statements ics should be always asserted in the  .Also, note that Corollary 2 is the   natural generalization of Theorem 6.3 in [5].

Conclusions
In this paper, we devote ourselves to the problem (A) in the light of the antum me l word view (cf.[7,8]).And, we show that regression analysis, which is t t fund al in statistics, is formulated as Corollary 2 in SMT (i.e., statistical measurement theory).We believe that Corollary 2 is the finest formulation of regression analysis, since no clear formulation can b Corollary 2 (or, the tional classification o statistics should be reconsidered. We expect that there is a great possibility that our proposal (i.e., statistics is based on statistical measurement theory) will be generally accepted.We of course know that the conventional statistics methodology can be good applied in many fields.Hence, we hope that our methodology in the light of the quantum mechenical word view should be examined from various points of view.

A
which is equivalent to the following equality.That is, for any , it holds: n that th e weak convergence (1) in   c B H can be identified with the weak convergence in   B H , there we see, by a usual way (cf.[10,11]), that Theorem 4 holds under the commutativity condition (2).fore,


is the height of water filling the tank at the be- 16) Pu we have the following problem that is equivalent to (N): ( ppendix s mentioned in Section 3.1, we have to prove the folwing theorem.Theorem 4. [Existence theorem of product observable].
Let  [resp. ;   ] be the smallest the equality (25) holds.This completes .This imlies that the proof.n of a sequential Remark 9.The above proof is applicable to the realizatiocausal observable   T in the case of an infinite T under a similar condition such that the Ko lds (cf.[1]).Also, in quantum case (i.e., there is a reason to infer that the unknown measured value   The proof is Thus, we omit it.