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Statistical inference as Green's functions

Statistical inference from data is a foundational task in science. Recently, it has received growing attention for its central role in inference systems of primary interest in data sciences and machine learning. However, the understanding of statistical inference is not that solid while remains as a matter of subjective belief or as the routine procedures once claimed objective. We here show that there is an objective description of statistical inference for long sequence of exchangeable binary random variables, the prototypal stochasticity in theories and applications. A linear differential equation is derived from the identity known as de Finetti's representation theorem, and it turns out that statistical inference is given by the Green's functions. Our finding is an answer to the normative issue of science that pursues the objectivity based on data, and its significance will be far-reaching in most pure and applied fields.


Introduction
A prototype of statistical inference is arguing the probabilistic feature of binary outcome with previous observations. The related studies date back to the seminal works by Bayes [1] and Laplace [2], which was ahead of the times and still remains a significant issue to date.
Consider a black box that randomly assigns a binary number 0 or 1 in each trial, and suppose that 1 is observed times out of trials. In this situation, one may seek the conditional probability ( | ), where is the probability of 1 in a trial and is a short notation for the observation data. Intuitively, a probability density localized around = ≡ / could be a candidate. As the localization width is expected to decrease for increasing , it is supposed that lim →∞ ( | ) = ( − ∞ ) ··· (1) for ∞ ≡ lim →∞ , where ( ) is Dirac's delta function [3]. This limiting property of the data-based inference is a spirit of science [1,2,[4][5][6].

Two streams of statistical inference
However, the justification of Eq. (1) remains rather unsatisfactory. What Eq. (1) is was examined for the first time in Bayesianism [1,2,[7][8][9][10][11], and the attempt arrived at a claim that it must be a belief. According to Bayesianism, any ( | ) is possible in belief, and then the limiting behavior in Eq. (1) becomes a matter of subjective choice of ( | ) therein. It is unnatural to consider a scientific meaning or justification for the sentence/equation including such ( | ) that is provided as a belief.
Meanwhile, frequentism [12][13][14][15] is the other stream, positioned against the subjectivity. In frequentism, for an objective stance, the entity ( | ) is rejected because of apparently no objective way to quantify it. As a consequence, there remains no room to play a role in the situation where ( | ) is invoked like Eq. (1). Still, there are wide (mis)uses [3, [16][17][18][19][20] of frequentism in most science and engineering fields while expecting objective ( | ) or similar one in routine procedures. This customary but improper practice seems uncontrollable because the unintentional misuse of ( | ) in frequentism that was once claimed objective [21][22][23][24][25] is quite attractive. Such uses are the circumstantial evidences that the notion ( | ) is very natural and intuitive in the situation of statistical inference.

Goal of this work: discerning ( | ) from belief
As the central entity in statistical inference, ( | ) should be treated in a very reliable way. However, the apparent intractability of ( | ) is regarded as the reflection of the claimed subjectivity in one stream, and this provokes the rejection of ( | ) in the other stream to keep an objective stance of science. Inference studies by subjective quantification or those in the absence of the entity ( | ) cannot make a scientifically meaningful progress. Thus if the intractability of ( | ) is resolved, the nonconstructive situation will drastically change.
In this work, we will demonstrate that ( | ) can be specified in objective way for long sequence of exchangeable binary random variables [26], for which ( | ) was studied for the first time [1,2] and also which are still significant in modern theories and applications [26][27][28][29][30][31][32][33][34]. We find out that ( | ) is given by the Green's functions [3] of a differential equation derived from de Finetti's representation theorem [26]. The ( | ) obtained this way justifies Eq. (1) without additional argue or implementation.
In order to efficiently exploit the exchangeability, we introduce the random variable ≡ ∑ =1 / , the relative frequency of 1 in , and its probability . By the exchangeability, we have = ( ) ( ) for the binomial coefficient ( ). We shall derive a differential equation for ( ), and then use it to specify ( |

De Finetti's relation with finite-correction
Equation (2) is an inhomogeneous differential equation for ( ) with heterogeneity arising from delta-function peaks weighed by .
A foundational issue in the study of statistical inference is the relation between probability and frequency , which actually led to the birth of Bayesianism [1]. Later, de Finetti [26] showed that in the limit of infinite , ∫ ′ ( ′) 0 = lim →∞ ∑ ≤ . The finiterelation is, however, unknown yet [11]. The first achievement of this work is Eq.
(2) to obtain , which shows how de Finetti's relation is modified with the leading correction owing to finite . The statistical inference we will report below follows from Eq. (2).

Statistical inference via Green's functions
We denote a Green's function of the linear differential operator ̂ by On the other hand, it is straightforward to write ( ) = ∑ ( | ) ( ). Comparing these two equations for ( ) yields This remarkable finding implies that the intractability of ( | ) can be resolved and also that statistical inference is a task of finding the associated Green's functions instead of choosing a belief.
The prefactors ℎ and are chosen from the condition ̂, ( ) = ( − ), so that , ( ) becomes an instance of ( ). See Methods VI for more technical details of , ( ). An example is shown in Fig. 1(A). For a given , various values of are possible as shown in Fig. 1(B). There exist such ± = ± ( ) that impose bounds on , i.e., − ≤ ≤ + . Then, the general construction of ( ) reads for weights ≥ 0 with normalization condition ∑ = 1. Various configurations of { } manifest the freedom of ( ). A case of ( ) with uniform is shown in Fig.   1(C). We numerically confirm that does not exist for = 0 and 1. This exception is practically unproblematic as such with = 0 and 1 is not practical for large .

Extraneousness of homogenous solutions
Equation (2)  width is at most ≈ 2 √ (1 − )⁄ . Although a localization around is intuitive, to argue whether it is substantial or not is highly nontrivial. When referred to Bayesianism of subjective belief on intractable ( | ), any profile of ( | ) must be superficial. We here point out that, if it is admitted that the intractable ( | ) could result in such superficiality, why not the intuition for localization by enough data may imply the opposite. Hence, to invent and elaborate the language for ( | ) that can quantitatively carry the data-driven intuition should be a primary issue in the study of statistical inference.
We find out ( | ) = ( ), which indicates that the localization of ( | ) near = is substantial. The upper bound of the width of ( ) characterizes and quantifies the localization. The objective existence of ( ) shows ( | ) is not such an entity that can be subjectively assigned in a belief. As ( | ) turns out to be a scientific entity, the frustrating situation by the denial of ( | ) in frequentism can be also resolved. Accepting the objectively specifiable central entity ( | ) will make the inference study natural and, for the first time, scientific. As a corollary, Eq. (1) immediately follows in the fact that any instance of ( ) is normalized and its width shrinks to zero in the → ∞ limit. In this way, the objective tractability of ( | ), we have established for exchangeable binary random variables in this article, shall shed a new light to the scientific study of statistical inference.

Specified freedom as inference itself
The freedom of ( ) means various (restricted) possible forms of ( | ). Here, we emphasize that this freedom is far from an artifact, but rather a nature of inference. Interestingly, the freedom of ( ) does not allow to exclude the observation point = or to separate the distribution, i.e., any instance of ( ) is a distribution on one interval including = . We remark that there has been an aspect called neutrosophic statistics [35], which comprehends the importance of degree of indeterminacy in statistical inference. Similarly, other predictive probabilities for different 's form their own intervals. The notion of the intervals of predictive probabilities is a natural consequence of the freedom of ( ).

Interval of predictive probability
Therefore, the predictive-probability interval provides an objective language for prediction.

From belief to science
In Bayesianism, the inference is ( | ) ∝ ( | ) ( ) for conditional probability where no belief on ( ) is referred to. We did not accept such aspect that regards ( | ) as a belief too, which completely rules out a room to think of science.
We have discovered that the statistical inference is given by the Green's functions of the differential equation in Eq.
(2), i.e., ( | ) = ( ). As demonstrated above, no belief on It is the shift from the belief of everything to the (philosophy of) science equipped with explicit computations free from a belief. Our study is based on the exchangeable binary random variables yet. Various generalizations such as beyond binary case or to higher order correction remain as future works.

Statistical inference, a bottom line of science
Science pursues objectivity by inductive reasoning from observation data [36]. This objectivity is taken for granted while its concrete description may remain open. This normative issue of science was mathematically raised by Bayes in the late 18th century, and he arrived at the apparent subjectivity of statistical inference [1]. According to this subjectivity, science based on the inference from data should be a belief though this interpretation has been overlooked in the unprecedented success of science. To continue the success, science shall face the demand of the times, the scientific guide to the inference systems for big data expected or promoted in the name of artificial intelligence or machine learning. This basically asks what a bottom line of science, the inference from data, is. It is thus time to examine what the objectivity from data really means, and the rigorous procedure and result we have discovered in this work is an answer. Our finding will be a breakthrough in scientific statistical inference and science itself.
Beginning with the initial null state |0,0⟩, upon a repeated application of ̂ or ̂ on Therefore, ( 1 , 2 , … , ) can be interpreted as a probability measure when are binary random variables.
As a preliminary to our main task, we first show that ( 1 , 2 , … , ) can play the role of the probability of sequence ( 1 , 2 , … , ) that is composed of exchangeable binary random variables .

III. / -expansion of the Laplace transform of
We define a linear operator ̂ whose action is ̂̂=̂− 1 for integer and introduce the operator ⟨⋅| acting on any linear combination of | , ⟩ to give where , is the Kronecker delta.
We shall make use a few relations to materialize Eq. .··· (19) As the next step. we shall argue that lim →∞ | ( , )| < ∞ for all .

V. Differential equation with leading correction
That , = ,0 = 1 is trivial, and , −1 = ,1 = ( − 1)/2 is easy to know. Using these, one may rewrite Eq. (19) as elsewhere, for which {ℎ , } is additionally restricted. In the following, we demonstrate how these ℎ and are specified for the construction of , ( ). We first consider the case where 1/ ≤ ≤ (1 − 1/ ), and then the case of = 0,1. We below shall often denote with for simplicity, Firstly, 0 = 0 is required because ℛ( ) is not integrable on any interval including = 0. Next, integration of Eq. (24) over the infinitesimal interval including = 0 gives ℎ 0 ℋ(0 + ) = 0 in that there is no delta peak at = 0 (the distribution magnitude is assumed to be zero for < 0 without loss of generality). Then, with 0 = ℎ 0 = 0, we have the vanishing segment, , ( ) = 0 for 0 < < 1/ . As there is no delta peak at = / for ≤ − 1 (note the current interest is , ( ) whose inhomogeneity source is at = only), one may extend the vanishing profile up to = − in the use of ℎ = = 0 while obeying Eq. (24). See the black bold line from = 0 in Fig. 1(A).
Further extension of the vanishing profile to the slot labeled with = = , that is, to the region of < < + 1/ is forbidden because there is unit heterogeneity at This shows that a non-trivial profile is required for < < + 1/ , instead of vanishing one. The non-trivial profile appearing this way can be extended up to certain = ( / ) − for an integer (by the assumed differentiability of the solution of Eq. (24) in each slots between adjacent delta functions, is an integer). This is done with ℎ = ℎ and = for + 1 ≤ ≤ − 1. Note that such extension is compatible with no heterogeneity for > . Beyond = / , we assign the vanishing profile again till = 1. See the black bold line for > / in Fig. 1(A). This is also consistent with no heterogeneity in Finally, the fact that , ( ) must be a valid probability distribution dictates that ℎ ℋ( ) + ℛ( ) ≥ 0 ··· (27) for < < / . This inequality imposes a maximum value + = + ( ) on .
All the steps explained above are also possible in the backward direction from = 1 to = 0 while giving a non-trivial part on / < < with = − , the minimum of .