新統計算法所源自的三個基本思想

關於“實事求是、一分為二、群眾路線”的補充說明

【2025-11-05】

統計學是一門通用的科學認識方法論。作者在《哲學之於統計學》一書中所著的關於連續型隨機變量的中心化位置（簡稱央位）的自加權期望算法以及關於加權期望分段回歸分析中回歸權重的算法等新計算法深刻地受到了“實事求是、一分為二、群眾路線”這三個詞語所蘊含的思想和方法的影響。可以說，作者將這些思想和方法內化在了這些新統計算法中。因此，以下是對該書所要展示的新統計思想和方法的一個重要補充和說明。這個說明被置於該書的目錄之前。

Statistics is a universal scientific methodology. The author's new calculation methods in his book "Philosophy in Statistics," including the self-weighted expectation algorithm (or Cmean algorithm) for the centralized location (or center) of continuous random variables and the algorithm of regressive weights for weightedly expected piecewise regression analysis, are profoundly influenced by the ideas and methods embodied in the three phrases "Seeking Truth from Facts, Dividing One into Two, and Massiline (coined by the author by combining mass and line)." In fact, these ideas and methods are internalized into these new statistical algorithms. Therefore, the following is an important supplement and explanation to the new statistical ideas and methods presented in this book.

一、實事求是 (Seeking Truth from Facts)

作為一個詞語，它在中國曆史上的沿襲和演變是一個從古代成語到中國共產黨核心思想的過程。它最早見於東漢時期班固（32~92）所著的《漢書·河間獻王傳》。他在文中讚揚漢景帝劉啟的兒子劉德“修學好古，實事求是”，意為“以事實為根據尋求真理”。1914年，趙天麟校長為天津北洋大學題寫了“實事求是”的校訓，這被民國早期湖南公立工業專門學校的校長賓步程所繼承，他也為1917年遷入嶽麓書院辦學的湖南工專題寫了“實事求是”的牌匾作為校訓。毛澤東在1916~1919年間曾寄居於嶽麓書院。一般認為這是以後他引用該成語的淵源。

As a phrase, its evolution in Chinese history is a process from an ancient idiom to the core ideology of the Communist Party of China. It first appeared in the Book of Han - Biography of King Xian of Hejian, written by Ban Gu (32~92).  In the text, he praised Liu De, a son of Emperor Jing of Han, Liu Qi, for his “studiousness and love of antiquity, seeking truth from facts.” meaning “seeking truth based on facts.” In 1914, President Zhao Tianlin inscribed a motto “Seeking Truth from Facts” for Tianjin Beiyang University. This was inherited by Bin Bucheng, the principal of the Hunan Public Industrial College in the early Republic of China, who also wrote a plaque with the same motto for the Hunan Industrial College, which moved to Yuelu Academy in 1917. Mao Zedong stayed at Yuelu Academy from 1916 to 1919. This is generally considered to be the origin of his later use of the idiom.

在已出版的毛澤東著作中尋找“實事求是”一詞，最早見於1938年10月14日，毛澤東在六屆六中全會上的報告中指出：“共產黨員應是實事求是的模範，又是具有遠見卓識的模範。因為隻有實事求是，才能完成確定的任務；隻有遠見卓識，才能不失前進的方向。”1940年1月，毛澤東在《新民主主義論》中指出：“科學的態度是‘實事求是’，‘自以為是’……的態度是決不能解決問題的。”

The earliest known instance of the term “seeking truth from facts” in Mao Zedong’s published works is found on October 14, 1938, in his report to the Sixth Plenary Session of the Sixth Central Committee of the Communist Party of China. Mao stated: “Communist Party members should be models of seeking truth from facts, and also models of foresight and insight. Only by seeking truth from facts can we accomplish the assigned tasks; only with foresight and insight can we maintain our direction.” In January 1940, in “On New Democracy,” Mao Zedong pointed out: “The scientific attitude is ‘seeking truth from facts,’ while the attitude of ‘subjective-assumption’... can never solve problems.”

但是，直到1941年，他才在《改造我們的學習》一文中對“實事求是”做了簡單且完整的解釋：“實事”就是客觀存在著的一切事物，“是”就是客觀事物的內部聯係，即規律性，“求”就是我們去研究。這一現代化解釋超越了其古代的字麵意思，令其擁有了“在實踐基礎上探求科學真理”的涵義。他在該文中還尖銳地批評了黨內一些領導幹部憑“想當然”解決問題的行為。

However, it wasn’t until 1941 that he provided a simple yet complete explanation of “seeking truth from facts” in his article “Reform Our Study”: “Facts” refers to all objectively existing things; “truth” refers to the internal connections of objective things, that is, their regularity; and “seeking” means that we study them. This modern interpretation transcends its ancient literal meaning, giving it the connotation of “seeking scientific truth on the basis of practice.” In the same article, he also sharply criticized the behaviors of some leading cadres within the Party who solved problems based on “wishful thinking or assuming what it is.”

從毛澤東第一次在嶽麓書院見到“實事求是”的牌匾到1941年對它做出解釋，說明他對此進行過長期的思考和踐行。他將這一成語與馬列主義相結合所做的多次闡釋以及他個人的實踐，使其成為了中國共產黨人所應有的核心哲學思想，是他們認識和改造世界的根本方法。其在實踐中的具體體現是，尊重事物本來的麵貌，遵循客觀規律，避免主觀想象和臆斷；強調實踐是檢驗真理的唯一標準，經驗和實踐是認識世界的基礎；其目地是一切從實際出發，找到解決現實問題的方法，促進國家發展和人民福祉。因此，實事求是是一種與先驗假定、主觀主義、經驗主義、教條主義和僵化思維等相對立的思維模式。

From the moment Mao Zedong first saw the plaque bearing the inscription "Seek Truth from Facts" at Yuelu Academy to his interpretation of it in 1941, it was evident that he had engaged in long-term reflection and practice regarding this principle. His numerous interpretations of this idiom in conjunction with Marxism-Leninism, along with his personal experience, made it a core philosophical thought inherent to the Chinese Communist Party and a fundamental method for understanding and transforming the world. In practice, this manifests itself in respecting the true nature of things, following objective laws, and avoiding subjective imagination and conjecture; emphasizing that practice is the sole criterion for testing truth, and that experience and practice are the foundation for understanding the world; and aiming to proceed from reality, find solutions to real-world problems, and promote national development and the well-being of the people. Therefore, seeking truth from facts is a thinking mode that stands in contrast to a priori assumptions, subjectivism, empiricism, dogmatism, and rigid thinking.

毛澤東在1939年12月撰寫的文章《中國革命和中國共產黨》中有一句話：“認清中國的國情，乃是認清一切革命問題的基本的根據。”（《毛澤東選集》第2卷，人民出版社1991年版，第633頁）。同理，在統計學中我們可以說：“認清一個樣本中所含有的基本信息，乃是認清一切統計問題的基礎。”隻有首先認清了問題，才有可能找到解決問題的正確方法，才有可能構建正確的統計算法。

In his article The Chinese Revolution and the Chinese Communist Party written in December 1939, Mao Zedong stated, "Understanding China's national conditions is the fundamental basis for understanding all revolutionary problems." (Selected Works of Mao Zedong, Vol. 2, People's Publishing House, 1991, p. 633). Similarly, in statistics, we can say: "Understanding the basic information contained in a sample is the foundation for understanding all statistical problems." Only by first understanding the problem can we find a correct method to solve it and construct a correct statistical algorithm.

本書在探索連續可變屬性（即傳統概念係統下的連續型隨機變量）的凹-凸自權重算法時，首先考察的就是一個樣本所帶有的基本信息，即全部的樣本點測量以及其中的最小和最大值，而這些樣本“事實”構成了計算凹-凸自權重的全部基礎。除此之外，無需引入任何關於總體分布形態的先驗假定和其它外源性信息。即使是其算術均數或中位數等用來估計抽樣分布的中心化位置的統計量也不被用於自權重的計算。這正是該權重被稱為“自權重”的原因。最終，一個連續可變屬性的抽樣分布形態由其樣本測量和凹-凸自權重在二維空間裏得到描繪，而分布的中心化位置（簡稱央位或凸峰）可由其凸自加權均數估計。令人驚異的是，凹-凸自權重恰似每個樣本點在其自身位置上的點密度，且凸自權重完美地量化了每個點在分布中對央位的集中趨勢，而凹自權重則完美地量化了它們對央位的離散趨勢。

In exploring concave-convex self-weighting algorithm for continuously variable attributes (i.e., continuous random variables in the traditional conceptual system), this book first examines the fundamental information contained in a sample: all sample point measurements and their minimum and maximum. These sample "facts" form the entire basis for calculating concave-convex self-weights. Beyond this, no prior assumptions about the population distribution or other exogenous information are required. Even statistics used to estimate the central location of a sampling distribution, such as the arithmetic mean or median, are not used in the calculation of the self-weights. This is precisely why the weight is called a "self-weight." Finally, the sampling distribution shape of a continuously variable attribute is depicted in a two-dimensional space by its sample measurements and concave-convex self-weights, and the central location of the distribution (referred to as the center or convex peak) can be estimated by its convex self-weighted mean. Surprisingly, the concave-convex self-weights are exactly like the point density of each sample point at its own location, and the convex self-weights perfectly quantify the central tendency of each point in the distribution to the center; while the concave self-weights perfectly quantify their dispersive tendency to the center.

二、一分為二 (Dividing One into Two)

雖然老子（約公元前571~470）在《道德經》第四十二章中有“道生一、一生二、二生三、三生萬物”之說，但作為一個成語，“一分為二”被認為出自宋代邵雍（1012 ~1077）所著的《皇極經世緒言》卷七：“是故一分為二，二分為四”。

Although Lao Tzu (about 571~470 BC) stated in Chapter 42 of the Tao Te Ching, "The Tao gives birth to One, One gives birth to Two, Two gives birth to Three, Three gives birth to all things," the idiom "Dividing One into Two" is generally attributed to Shao Yong (1012~1077) in the Song Dynasty in his Huangji Jingshi Xuyan, Volume 7: "Therefore, One divides into two, and Two divides into four."

作為一個哲學概念，它指的是事物作為矛盾的統一體包含相互對立的兩個方麵，為此應全麵看待事物的不同方麵。常用於要求客觀認識和分析問題的語境。

As a philosophical concept, it refers to the fact that things, as a unity of opposites, contain two mutually opposing aspects, thus requiring a comprehensive view of their different aspects. It is often used in contexts demanding objective understanding and analysis of problems.

對連續可變屬性的凹-凸自權重的計算涉及到點對點的差異性和相似性，其結果則是將算術均數的算法中隱含的“每個樣本點對分布央位的貢獻相同”的等權重假定中的權重1分解為了凹自權重和凸自權重兩個部分。於是，對於一個連續可變屬性X的樣本測量{x_i}(i = 1, 2, …, n)，我們總是會有r_i + c_i = 1，其中r_i是x_i的凹自權重，c_i是其凸自權重。

The calculation of concave-convex self-weights for continuously variable attributes involves point-to-point differentialities and similarities. The result is that the weight 1 in the equal-weight assumption implicit in the arithmetic mean algorithm—that "each sample point contributes equally to the center of the distribution"—is decomposed into two parts: concave self-weights and convex self-weights. Therefore, for a sample measurements {x_i}(i = 1, 2, …, n) of a continuously variable attribute X, we always have r_i + c_i = 1, where r_i is the concave self-weight of x_i and c_i is its convex self-weight.

同樣地，在基於回歸加權途徑重建分段回歸算法時，回歸權重的計算也涉及到預測值的變異和殘差的變異，而不是像傳統上基於數值型最優化的分段回歸那樣僅僅使用了殘差的變異。對預測值變異的無視意味著丟棄了一大半的樣本信息，這在基於回歸權重估計未知臨界點的加權期望時將必然導致結果的偏差。

Similarly, when reconstructing piecewise regression algorithms based on a regressive weighting approach, the calculation of regressive weights involves both the variations of predicted values ??and the variations of residuals, unlike traditional piecewise regression based on numerical optimization which only uses the variations of residuals. Ignoring the variations of predicted values ??means discarding a more than half portion of the sample information, which will inevitably lead to a biased result when estimating the weighted expectation of an unknown threshold based on the regression weights.

無論是凹-凸自加權還是回歸加權，在使用樣本信息時，其算法均應遵循“無信息損失和無信息冗餘”這兩個基本原則，這也是“一分為二”思想的體現。

Whether using concave-convex self-weighting or regression weighting, the algorithm should adhere to the two fundamental principles of "No information loss, and No information redundancy" when using sample information; this also reflects the thinking of "Dividing One into Two."

三、群眾路線 (The Massiline)

這是一個近代漢語詞匯，最早見於1922年中國共產黨第二次代表大會通過的《組織章程決議案》：“黨的一切運動都必須到廣大群眾裏麵去。” 1925年的中央擴大執委會決議案明確指出：“中國革命運動的將來命運，全看中國共產黨會不會組織群眾、引導群眾。”

The "massiline" is a modern Chinese term, first appearing in the "Resolution on the Organizational Charter" adopted at the Second National Congress of the Communist Party of China in 1922: "ll movements of the Party must go among the broad masses." The resolution of the Enlarged Executive Committee of the Central Committee in 1925 clearly stated: "The future fate of the Chinese revolutionary movement depends entirely on whether the Communist Party of China can organize and guide the masses."

1928年6月的中國共產黨第六次代表全國大會也作出了“黨的總路線是爭取群眾”的重要論斷。同年11月，中國共產黨的領導人李立三在向浙江地區革命領導人傳達中央精神的一次工作談話中，首次使用了“群眾路線”這一概念：“在總的爭取群眾路線之下，需要盡最大的努力到下層群眾中去。”

The Sixth National Congress of the Communist Party of China in June 1928 also made the important assertion that "the Party's general line is to win over the masses." In November of the same year, Li Lisan, a leader of the Chinese Commuist Party, in a work talk and conveying the spirit of the Central Committee to the revolutionary leaders in Zhejiang Province, first time used the concept of the "massiline": "Under the general line of winning over the masses, we need to make the greatest effort to go to the grassroots."

1929年9月由陳毅起草、周恩來審定的《中央給紅四軍前委的指示信》三處提到“群眾路線”，即籌款工作要“經過群眾路線”，沒收地主豪紳財產要“經過群眾路線”，解決紅軍給養問題要“漸次做到由群眾路線去找出路”。毛澤東在同年12月的古田會議決議中，第一次針對群眾路線進行相關闡述：“黨的工作要在黨的討論和決議之後，再經過群眾路線去執行”。

In September 1929, the "Instructions from the Central Committee to the Front Committee of the Fourth Red Army," drafted by Chen Yi and reviewed by Zhou Enlai, mentioned the "massiline" in three places: fundraising should be carried out "through the massiline," the confiscation of landlords' and gentry's property should be carried out "through the massiline," and solving the Red Army's supply problem should be "gradually achieved by finding a way out through the massiline."

In the resolution of the Gutian Conference in December of the same year, Mao Zedong made his first elaboration on the massiline, stating that "the work of the Party must be implemented after the Party's discussion and resolution, and then through the massiline."

中國抗日戰爭時期，群眾路線概念的內涵被毛澤東繼續深化和發展。毛澤東在1943年6月為中央起草的《關於領導方法的若幹問題》一文中指出：“在我黨的一切實際工作中，凡屬正確的領導，必須是從群眾中來，到群眾中去。這就是說，將群眾的意見（分散的無係統的意見）集中起來（經過研究，化為集中的係統的意見），又到群眾中去做宣傳解釋，化為群眾的意見，使群眾堅持下去，見之於行動，並在群眾行動中考驗這些意見是否正確。然後再從群眾中集中起來，再到群眾中堅持下去。如此無限循環，一次比一次地更正確、更生動、更豐富。這就是馬克思主義的認識論。” （《毛澤東選集》第3卷第899頁）

During the War of Resistance Against Japan, Mao Zedong continuously deepened and developed the concept of the massiline. In his article "Some Questions Concerning Methods of Leadership" drafted for the Central Committee in June 1943, Mao Zedong pointed out: "In all practical work of our Party, all correct leadership must come from the masses and go to the masses. This means that the opinions of the masses (scattered and unsystematic opinions) are gathered (and, after study, transformed into concentrated and systematic opinions), then disseminated and explained to the masses, becoming their own opinions, which are then upheld and put into action by the masses. The correctness of these opinions is then tested in the process of mass action. Then, these opinions are gathered again from the masses and upheld once more. This cycle repeats infinitely, becoming more correct, more vivid, and more comprehensive with each iteration. This is a Marxist epistemology."(Selected Works of Mao Zedong, Vol. 3, p. 899)

1945年5月2日，毛澤東在延安出版的《解放日報》上發表了《論聯合政府》，其中進一步闡述了群眾路線：“我們共產黨人區別於其他任何政黨的又一個顯著的標誌，就是和最廣大的人民群眾取得最密切的聯係。全心全意地為人民服務，一刻也不脫離群眾；一切從人民的利益出發，而不是從個人或小集團的利益出發；向人民負責和向黨的領導機關負責的一致性；這些就是我們的出發點。”

On May 2, 1945, Mao Zedong published "On Coalition Government" in the Liberation Daily, published in Yan'an, in which he further elaborated on the massiline: "Another significant mark that distinguishes us Communists from any other political party is our closest ties with the broadest masses of the people. Serving the people wholeheartedly and never being separated from them for a moment; proceeding from the interests of the people in everything we do, not from the interests of individuals or small groups; the consistency between being responsible to the people and being responsible to the leading organs of the Party; these are our starting points."

在1945年4月23日至6月11日於延安召開的中國共產黨第七次全國代表大會上，群眾路線被確立為該黨的根本政治路線和根本組織路線。

At the Seventh National Congress of the Communist Party of China, held in Yan'an from April 23 to June 11, 1945, the massiline was established as the Party's fundamental political and organizational line.

林彪1966年11月3日在天安門城樓上講話（講話稿經毛澤東審閱）說：“毛主席的路線，是讓群眾自己教育自己，自己解放自己的路線，是‘敢’字當頭的路線，是敢於相信群眾，敢於依靠群眾，敢於放手發動群眾的路線。”

In his speech on Tiananmen Square on November 3, 1966 (the speech was reviewed by Mao Zedong), Lin Biao said: "Chairman Mao's line is one that lets the masses educate themselves and liberate themselves; it is a line that puts 'daring' first; it is a line that dares to trust the masses, dares to rely on the masses, and dares to mobilize the masses."

綜上所述，對群眾路線的解釋可被簡化為：相信群眾，依靠群眾；從群眾中來，到群眾中去；將分散和無係統的群眾意見集中起來，經過研究和整合，形成行動方案，並在群眾的行動中對方案進行檢驗和修正。如此有限循環和改進。

In summary, the interpretation of the massiline can be simplified as: trusting the masses and relying on the masses; coming from the masses and going back to the masses; gathering scattered and unsystematic opinions from the masses, studying and integrating them to form action plans, and testing and revising these plans through the actions of the masses. This process involves a limited cycle of improvement.

將這個思想應用到統計學中就是，在相信樣本點，依靠樣本點的基礎上，將樣本點分散的、無係統的點滴權重計算或測量出來並集中起來，再經過算法整合得到目標可變屬性的加權期望，最後在抽樣實踐和統計計算中檢驗加權期望的正確性、穩定性和可靠性。

Applying this idea to statistics means, based on trusting and relying on sample points, calculating or measuring the drops of scattered and unsystematic weights of the sample points and gathering them, and then integrating them through algorithms to obtain the weighted expectation of the target variable attributes, and finally verifying the correctness, stability, and reliability of the weighted expectation in sampling practice and statistical calculations.

連續可變屬性的凹-凸自權重的計算就是遵循了上述基本思想。這個算法的基本思想是，相信一個抽樣分布的中心化位置是由樣本中的所有點共同決定的，每個樣本點因其自身所在的位置對央位的貢獻存在個體差異。這一思想是對算術均數算法中等權重假定的逆向突破。也正是這一突破，迫使作者尋找如何計算出一個樣本中具有差異化的個體貢獻；反之，如果沒有這一突破，便不會有凹-凸自權重算法的誕生。

The calculation of concave-convex self-weights for continuously variable attributes follows the above basic ideas. The fundamental idea of ??this algorithm is that the center of a sampling distribution is determined by all points in the sample, and each sample point’s contribution to the center varies depending on its location. This idea represents a reverse breakthrough from the equal weight assumption in the arithmetic mean algorithm. It was precisely this breakthrough that compelled the author to find a way to calculate the differentiated individual contributions within a sample; conversely, without this breakthrough, the concave-convex self-weight algorithm would have never been developed.

同樣地，加權分段回歸中回歸權重的構建也遵循了群眾路線的思維，即一個被分割屬性上的未知臨界點是由回歸空間中所有樣本點在所有可變屬性中的表現共同決定的，每個樣本點正是以自身所在的位置對未知臨界點有一份可變的貢獻。我們所要做的就是將這份貢獻計算或測量出來，由此即可得到被分割屬性上未知臨界點的加權期望估計。作者拒絕臨界點由具有最大個體貢獻的樣本點決定的所謂“最優化”觀點，因為這一觀點體現的是某種“個人英雄主義”的魯莽。不僅如此，迭代搜索臨界點的過程中，所謂的“優化算子”將輸出一個關於其自身的完整分布，而它的極值位於該分布的邊界上。同時，作為可變屬性的目標參數也在迭代過程中發生隨機變異，也會形成一個完整的分布。這兩個分布都有各自的期望。由於優化算子或回歸權重都分別與目標參數相互關聯，它們分別與目標參數構成的聯合分布也一定存在著唯一的聯合的央位，而這個央位正是唯一可期望的參數估計。

Similarly, the construction of regressive weights in weighted piecewise regression followed the massiline approach: an unknown threshold on a segmented attribute is determined by the combined performance of all sample points in a regression space across all variable attributes. Each sample point contributes a variable amount to the unknown threshold based on its position. Our task is to calculate or measure this contribution, thereby obtaining a weighted expectation estimate of the unknown threshold on the segmented attribute. The author rejected the so-called “optimal” view that the threshold is determined by the sample point with the largest individual contribution, because this view reflects a kind of reckless "individual heroism." Furthermore, during the iterative search for the threshold, the so-called "optimizer" outputs a complete distribution about itself, with its extrema located on the boundaries of this distribution. Simultaneously, the target parameter, as a variable attribute, also undergoes random variation during the iteration process, forming another complete distribution. Both distributions have their own expectations. Since the optimizer or regression weight is respectively correlated with the target parameter, the joint distribution with the target parameter must possess a unique joint center, and this center is precisely the only expected parameter estimate.

 

中國共產黨在取得全國政權後，一直通過社會實踐、教育、宣傳和媒體等各種機會和渠道向全社會傳播這些思想和方法。在毛澤東之後的不同時期，它們也一直被用於思考和解決中國所麵臨的各種問題。因此，毫無疑問，這些思想和方法對作者產生了潛移默化的影響。因此，作者願意在此建議將本文設介紹的凹-凸自權重算法和回歸權重算法等稱為“群眾路線法”。

After seizing national power, the Chinese Communist Party has consistently disseminated these ideas and methods to the entire society through various opportunities and channels, including social practice, education, propaganda, and the media. In different periods after Mao Zedong, they have also been used to think about and solve various problems facing China. Therefore, there is no doubt that these ideas and methods have had a subtle influence on the author. Thus, the author would like to suggest referring to the concave-convex self-weighting algorithm and regressive weighting algorithm introduced in this book as the "massiline method."