Proposal of an epistemological model for teaching Bayesian inference

Abstract

The teaching of Bayesian statistics at school is characterized by promoting the meaning of subjective probability as a calculation algorithm. Regarding the limited development of meanings at school, the history of mathematics is a source of data that provides epistemic elements that configure new meanings about mathematical notions. Therefore, to characterize the stochastic activity that underlies Bayesian problems and the context associated with their genesis, a documentary study is proposed through qualitative content analysis from the socio-epistemological perspective. The results show an organization of stochastic practices directed by estimating the ‘true value’ of the unknown parameter, from which an epistemological reference model is built as a proposal to reorient the didactic intervention of Bayesian inference.

Historical Genesis; Inference; Stochastic practices; Bayes Theorem; Socioepistemology

Resumen

La enseñanza escolar de la estadística bayesiana se caracteriza por promover un significado de la probabilidad subjetiva como algoritmo de cálculo. A propósito del desarrollo limitado de significados escolarmente, la historia de la matemática resulta una fuente de datos que provee elementos epistémicos que configuran nuevos significados sobre las nociones matemáticas. Por lo tanto, con el objetivo de caracterizar la actividad estocástica que subyace a problemas bayesianos y el contexto asociado en su génesis, se plantea un estudio de tipo documental a través del método de análisis cualitativo de contenido desde la perspectiva socioepistemológica. Los resultados muestran una organización de prácticas estocásticas direccionada por la estimación del ‘valor verdadero’ del parámetro desconocido, a partir del cual se construye un modelo epistemológico de referencia como propuesta para reorientar la intervención didáctica de la inferencia bayesiana.

Génesis histórica; Inferencia; Prácticas estocásticas; Teorema de Bayes; Socioepistemología

1 The historical research in mathematics education

The integration of the history of mathematics into mathematics education has become an area of ​​research that has entered a period of consolidation in the last decade due to contributions on didactic phenomena linked to the study of school mathematics (see Chorlay; Clark; Tzanakis, 2022; Clark et al., 2018; Furinghetti, 2020).

At least two paths of research on the history of mathematics stand out: as a form of mathematics knowledge and a resource for didactic intervention, in which we discerned some topics of interest: theoretical questions about the relationship between historical and didactic research; the design of teaching materials based on historical sources and their implementation results; the uses of history in the curriculum, textbooks, and teacher education; the construction of conceptual frameworks based on history; and the history and epistemology of mathematics as a tool for an interdisciplinary approach, among others (Chorlay; Clark; Tzanakis, 2022).

Under this scenario, pondering over historical elements should be considered a way to improve mathematical instruction processes, allowing reflection on mathematical activity and its nature. In this regard, Barbin, Guillemette, and Tzanakis (2020) point out three types of contributions associated with historical research: epistemological, cultural, and didactic.

For example, the history of mathematics can provide ways of doing mathematics quite different from those established in the school setting and thus transform teaching and deepen students’ conceptual understanding (Barbin; Guillemette; Tzanakis, 2020). It allows us to account for the genesis, evolution, and consolidation of a mathematical concept within the framework of sociocultural conditions and, consequently, to recognize the complexity surrounding the concepts and the aspects that influenced their constitution, and thus transcend algorithmic processes (Anacona, 2003). It also helps us build epistemological models that guide the design and analysis of teaching activities (Jankvist, 2009).

2 Bayesian statistics in statistics education

Various studies in statistical education have taken the historical-epistemological dimension as a research phase toward the search for teaching methods that expand stochastic activity and its associated meanings in the school setting, partly due to the call for a transformation of its teaching, which is repositioned in the school curriculum from an algorithmic vision to an instruction based on the analysis behind the data (Chaves Esquivel, 2016). In other words, it is essential to think about the teaching and learning of stochastics beyond procedures and techniques and see them as tools that allow the individual to understand and act on their environment.

Regarding the construction of meanings for stochastic notions, Ainley and Pratt (2001) suggest that these can emanate from at least two types of tasks: calculation-oriented tasks, in which the objective is to produce results through the application of statistical techniques to data sets, and concept-oriented tasks, in which the objective is the analysis of statistical processes and the context of the data. The first type of task promotes general meanings linked to forms of procedural learning, while the second type promotes situated meanings specific to the content of the data and the processes used to produce such results.

Statistical inference, which has been widely investigated in the last decade, is considered a fundamental idea of stochastic notions. About its teaching, this has been partial, as the classical perspective has been prioritized to the detriment of the Bayesian view (Borovcnik, 2012). An example of this has been the numerous studies developed within the framework of informal inference, which is strongly based on the frequentist perspective (Borovcnik, 2019; Dijke-Droogers; Drijvers; Bakker, 2020; Nilsson, 2020).

Along the same lines, Borovcnik (2012) and Vancsó, Borovcnik, and Fejes-Tóth (2021) highlight the lack of work and research on the relationship between probability and inference and point out that a few curricula exclude the Bayesian ideas. And when they are part of the syllabuses, authors such as Carranza (2014) report that stochastic activity on Bayesian probability is focused on procedures and techniques; that is, tasks oriented to calculations in which the link with inference is nonexistent are prioritized. For this reason, Bayesian inference will be considered here as a mathematical piece in the object of study.

Following the route of historical research with emphasis on Bayesian statistics, we identify in the literature epistemological studies on probability within which reference is made to the subjective or Bayesian perspective.

In this line, Batanero, Henry, and Parzysz (2005) analyze the nature of the different interpretations of probability throughout history. The results show the multifaceted nature of this notion and its duality. Specifically, Bayesian probability stands out for its relativistic or subjective nature. In this case, probability measures the degree of uncertainty specific to a person.

Based on the previous study, Batanero (2005) characterizes the different historical meanings of probability from a theoretical model based on the onto-semiotic approach. As a product, the meanings of the problem field, procedures, and other components are described. Concerning Bayesian ideas, the subjective nature linked to the personal assignment of probabilities is again highlighted.

Likewise, Carranza (2009) conducts an epistemological study on this concept to understand its origins and evolution. One of the author’s main conclusions is consistent with what has been reported, i.e., that duality is an intrinsic characteristic of probability; he also states that this duality persists to this day. Associated with this, he proposes a set of characteristic elements to examine the presence of one or another interpretation in exercises in textbooks or curricula. One characteristic element is the type of reasoning; regarding the Bayesian approach, reasoning is mentioned in the inductive and abductive sense.

Borovcnik and Kapadia’s (2014) research presents a 21st-century historical and philosophical perspective on probability. They contribute by describing the characteristics of the three most important philosophical positions. Regarding the Bayesian approach, for example, the categories of information it uses (prior information and empirical data) and its subjectivist nature stand out.

Finally, Paredes (2018) examines Bayesian interpretation in the 18th century. Specifically, he characterizes the form and role Bayes’ rule plays in Thomas Bayes’ mathematical text.

These studies focus on noticing, from the historical scenario, the characteristics and nature of the interpretations of probability, which has led them to determine frameworks to understand the different views on a horizontal plane, except for Paredes’ (2018) work. We believe that it may lead to not contemplating other elements specific to the Bayesian perspective.

About this panorama and Batanero and Álvarez-Arroyo’s (2023) call for the development of epistemological studies on probability, we are interested in this type of work as a path to finding epistemic elements that support new ways of introducing this notion in the school scenario. To strengthen the cited research, we focus on typifying the stochastic activity that underlies the relative problems of Bayesian statistics in the genesis of this notion. With this, we aim to configure a reference epistemological model that underpins the design of concept-oriented tasks as an instrument for future research.

Given the above, we raise the following research question: What is the stochastic activity promoted by Bayesian problems and their characteristics in the genesis of Bayesian statistics?

3 Theoretical-methodological elements

By the late 1980s, research in mathematics education experienced what Lerman called the social turn. This was characterized as “the emergence […] of theories that view meaning, thought, and reasoning as products of social activity” (Lerman, 2000, p. 23). This meant considering the social nature of knowledge, that is, understanding mathematical knowledge as inseparable from the social context in which it is produced.

Along these lines, some social perspectives, in the way they understand and explain the construction of knowledge in a situated way and its meanings –the social– have adopted certain notions and concepts [from the social sciences]. One has been the notion of practice, which has been used to characterize mathematical activity and its organization. For example, research by Dogruer and Akyuz (2020), dos Santos Verbisck and Bittar (2020), García-García et al. (2022), Polaki (2002), Torres-Corrales and Montiel-Espinosa (2021), Triantafillou and Potari (2010), and Wijayanti and Winsløw (2017), besides using different theoretical perspectives based on practices, recognize the mathematical practices associated with specific mathematical contents in different settings such as school, history, and the workplace.

3.1 Social construction of mathematical knowledge

Regarding the social paradigm, socio-epistemology is a theory that studies how the social and cultural circumstances of human activity frame, regulate, and constitute the production of mathematical knowledge.

From this perspective, mathematical practices are the unit from which mathematical activity is explained as a human activity. Mathematics is then conceived as something that is produced by those who participate in it in a specific context, so it is something that individuals do.

Thus, the construction of mathematical knowledge in socioepistemology is revealed in a pragmatic progression of practices, also known as the model of nesting practices. Cantoral (2020, p. 791) describes the levels of practices involved as: “The nesting of practices has three elements, actions, activities, and socially shared practices, and are regulated by the reference practice (a cultural dimension) and the social practice (a social dimension).”

Mathematical practice refers to the organized ways of doing and saying things people do around mathematical knowledge (cultured, popular, or technical). It is structured according to the context in which it is located and, in turn, gives meaning to what is done and said. Therefore, the practices are constituted in a dialectical relationship between the person who acts and the context that frames their mathematical activity (Lave, 1988).

Considering the above, this theory is considered a lens for problematizing Bayesian statistics in its historical genesis, that is, unraveling the nature of stochastic knowledge in terms of the stochastic activity that produces it and its context.

3.2 Historical-epistemological analysis from socioepistemology

Historical-epistemological (HE) studies have been an essential method in socioepistemological research to address the epistemic aspect of the research question (see Cruz-Márquez; Montiel-Espinosa, 2022; Espinoza Ramírez; Vergara Gómez; Valenzuela Zúñiga, 2018). For this reason, a HE analysis based on the proposal of Vargas-Zambrano and Montiel-Espinosa (2022) was decided upon for the development of the study.

The methodological scheme is made up of five stages and is based on qualitative content analysis (Figure 1). Documentary analysis is carried out through a historical-mathematical approach to the mathematics of the past and its objective is the construction of epistemological hypotheses, that is, the configuration of epistemological models on knowledge based on mathematical practices that integrate both aspects of mathematical activity and context.

Figure 1
– Methodological scheme for HE studies in socioepistemology

Historicization is the historical-critical, rather than chronological, study of situated epistemology that describes the construction and constitution of mathematical knowledge in its genesis, development, or transversality (Cantoral; Montiel; Reyes-Gasperini, 2015). Hence, the data analysis is made up of two sub-analyses: one contextual and one textual. These, in turn, are composed of specific elements that characterize the mathematical activity that produces mathematical knowledge and the context that constitutes it (see Chart 1).

Chart 1
– Historicization as a critical history of knowledge

Contextual analysis	Text analysis
The external critical history of knowledge through the characterization of the context of meaning	The internal critical history of knowledge through the pragmatic reconstruction of mathematical activity
Cultural context:	Socially Shared Practice:
Social characteristics of cultural groups that reflect ideological, political, and economic needs that shape a worldview and influence the rationality of the subject.	The organization of actions and activities carried out by the subject in an iterative and deliberate manner, related to the treatment of specific situations that characterize a mathematical task.
Situational context:	Activity:
Spatiotemporal factors or circumstances that affect the subject regarding mathematical knowledge.	Coordination of intentional actions of the subject directed toward a purpose to address a specific situation within the framework of a mathematical task.
Context of the specific situation:	Action:
Mathematical and situational elements that determine problems within which mathematical knowledge emerges.	The doing and/or saying of the subject before the object (what a mathematical task demands).

Contextual analysis lies in characterizing the context of meaning to reveal the sociocultural circumstances that constitute and give meaning to mathematical knowledge. This context is made up of three contextual dimensions –context of the specific situation, situational context, and cultural context– ranging from specific to general aspects that situate the text and the mathematics at play in time and place.

Textual analysis consists of reconstructing mathematical activity in a historical text to identify the author’s mathematical work and characterize its dynamics. Thus, the three lower levels of the socioepistemological model of nested practices are used –action, activity, and socially shared practice– which describe the dynamics of the mathematical doing. To recognize the actions, we ask the text: What did the subject do? And how did he do it? At the activity level: What did he do it for? And at the level of socially shared practice: Why did he do it?

4 Data generation

4.1. Text of analysis

To develop the HE studies, we chose An essay towards solving a problem in the doctrine of chances by Thomas Bayes, recognized as the first text to establish a version (for the binomial case) of the model currently known as Bayes’ theorem that shows a treatment of inductive inference, also called Bayesian inference. In fact, a few years later, Laplace’s Théorie Analytique des Probabilités [Analytic theory of probabilites] presents an updated version of the theorem, independently of Bayes’ work, but for more general models than the binomial one (Dale, 1999; Gómez Villegas, 2001; Hald, 2007).

This research takes the fourth section of Bayes’ 1763 text –appendix– as our object of analysis since it illustrates Bayesian inference by using Bayes’ theorem in some problems. Thus, such problems will allow us to reconstruct and characterize the stochastic activity associated with Bayesian inference in terms of practices.

4.2 Data source

Intending to understand and describe the spheres that make up the context of the significance of Bayesian inference in its genesis and, based on this, reconstruct in a situated way the stochastic activity associated with Bayesian problems, we searched for sources in which biographies, translations of the text, historical studies, and interpretations of the text under our objective were identified. Some of the consultation sources used can be seen in Chart 2.

Chart 2
– Sources of reference for the historicization of Bayesian inference

Source Type	Source
Primary	■ Bayes, T. (1763). An Essay Towards Solving a Problem in The Doctrine of Chances
Secondary	■ Bellhouse, D. R. (2004). The Reverend Thomas Bayes, FRS: A Biography to Celebrate the Tercentenary of His Birth
	■ Dale, A. (1999). A history of inverse probability from Thomas Bayes to Karl Pearson
	■ Hald, A. (1990). A history of probability and statistics and their applications before 1750
	■ Hald, A. (2007). A history of parametric statistical inference from Bernoulli to Fisher, 1713–1935
	■ Todhunter, I. (1865). A history of the mathematical theory of probability from the time of Pascal to that of Laplace
Tertiary	■ Daston, L. (1988). Classical Probability in the Enlightenment
	■ Gorroochurn, P. (2012). Classic Problems of Probability
	■ Hacking, I. (2006). The emergence of probability: A Philosophical Study of Early Ideas about Probability, Induction and Statistical Inference

4.3 Approach to the text of analysis

The pre-analysis consisted of an approach to Bayes’ (1763) text based on the sources of consultation. This phase involved a series of steps, such as a thorough reading of the text accompanied by interpretations of the document, the identification of transversal and nodal notions to understand the mathematical-stochastic content of the text, and the choice of extracts or sections of the text that would help answer the question. Consequently, we recognized the structure of the text and its purpose, the type of language used to refer to probability, and the stochastic notions involved, among others.

Important aspects for us to understand the content of the text were: 1) Huygens’ conception of hope or expectation due to the definition of probability under this philosophy; 2) interpretation and presentation of the probability measure as a ratio in the significance of wagers.

5 Historicization of Bayesian statistics in its genesis

Bayes’ analysis of the text –historicization– is permeated by one contextual and one textual analysis. The first allows us to situate the social, cultural, and mathematical factors that permeate the constitution of the text, and the second refers to the characterization of stochastic activity in terms of practices that underlie Bayesian problems.

5.1 Context of the significance of Bayesian statistics

The context of significance will be guided by the following questions: What characteristic events of the 17th-18th centuries surround Bayes’ work in 1763? What were the pioneering contributions in relation to inductive inference? What characteristics do Bayesian problems present, and what stochastic activity do they promote? Each question allows us to describe a level of context stratification – cultural, situational, specific situation.

5.1.1 Bayesian statistics in the 17th-18th century

The cultural context is framed by events or occurrences that usually form paradigms and give way to specific phenomena. In this case, we highlight some sociocultural events related to the genesis of inductive or Bayesian inference that paves the way to the problem of the relationship between probability and induction and culminates in the emergence of the Bayesian method.

The genesis of Bayesian inference can be traced back to the early Enlightenment, a period that generated a movement toward understanding the world –human knowledge– through rational thinking rather than religious beliefs. Some events to underscore from this stage are: 1) the religious struggles between members of the established church in England and the dissidents or nonconformists of which the Bayes family was a supporter (Bellhouse, 2004); 2) Newton as one of the main figures to lay the foundations of the scientific method based on experimentation and the construction of hypotheses; 3) the development of the philosophical doctrine of empiricism promoted by Locke and Hume; 3) the constitution of the so-called problem of induction as a rejection of inductive inference (Gutiérrez Cabria, 1982).

As for the above, in the 17th and 18th centuries, the influence of religious thinking continued in many scientists, such as Blaise Pascal, John Wilkins, and John Arbuthnot, who used mathematics to explain the laws of the natural world as an argument for divine order. In this sense, for example, Newton’s philosophy guided the English conception of probability, which interpreted the stability implied in limit theorems as evidence of divine design (Hacking, 2006). This relationship of probability with divinity can also be recognized in texts such as Pascal’s logic and wager.

On the other hand, finding ways to test inductive reasoning is a topic that dates back to Aristotle; however, it was the philosopher Hume who raised the problem of induction derived from the incompatibility between the principle of empiricism (all scientific theories –general knowledge– should be obtained from observation and experiment), and the invalidity of induction (the conclusions obtained from observations through induction have no logical validity) (Gutiérrez Cabria, 1982). This problem is characterized by the idea that past experience cannot be used to predict the future or, in other words, the problem of determining the cause or hypothesis based on observations (Glass; Hall, 2008; Hacking, 2006).

Although the resolution of the problem of induction took different philosophical directions, the incorporation of the concept of probability allowed us to determine the probable truth against the absence of guaranteeing the certainty of the conclusion; this process led to the emergence of inductive inference. This issue will be described in the situational context through the first approaches to the problem of induction from a probabilistic perspective.

5.1.2 The problem of binomial statistical inference

The situational context is made up of the roots of the philosophical problem of inductive probability through some probabilists of the 17th and 18th centuries: Jakob Bernoulli, Abraham de Moivre, and Thomas Bayes (Hald, 2007; Landro; González, 2012). Their focus was on the development of methods that would allow us to make inferences based on observations: in the first two, by trying to infer probabilities from frequencies, and in the third, by constructing a first method (Rivadulla, 1996).

According to Stigler (1986), in the late 17th century, probabilities were used to analyze games of chance, and a priori calculations were heavily emphasized. This refers to the resolution of problems such as: Given an urn containing r red balls and s black balls, the probability of getting a red ball is estimated to be $r/(r+s)$; however, the a posteriori issue of determining r and s by observing the game results had not been addressed.

Although Bernoulli’s work was published when the problem of induction had not yet been raised as a central problem of philosophy (Hacking, 2006), it is the first approach to dealing with the inverse scheme. In the fourth part of the Ars Conjectandi, Bernoulli argues that the relative frequency of an event determined from observations taken under the same circumstances becomes increasingly stable with increasing numbers of observations. If the model that fits the observations is a binomial distribution, we ask: Does the relative frequency of this model have the same properties as the empirical relative frequency? Bernoulli demonstrated this, which led to the law of large numbers for binomial variables (Stigler, 1986; Hald, 1990).

As an example of this law, the Ars Conjectandi mentions that for an urn containing 30 tokens of one type and 20 of another (unknown to the observer), at least 25,550 observations of the game would be needed to ensure that p(29/50 ≤ X/25550 ≤3 1/5>1000/1001 (Figure 2). In other words, if we take n(θ,ε,t) number of observations, it will be more than t times more likely that the ratio between the number of observed successes and the number of all observations $(Y_n=X_n/n)$ falls into the interval $[\theta -\epsilon ,\theta +\epsilon ]$ than outside it (Gorroochurn, 2012); with this, we can affirm that Y_n tends in probability to 𝜃 and thus determines the unknown proportion of tokens (see Bernoulli, 1713/2006).

Figure 2
– Example of Bernoulli’s law of large numbers

From this reasoning, Bernoulli tried to formulate the theorem inversely: if the relative frequency “converges to a specific value 𝜃," this value will define the law that governs said event (Landro; González, 2012). However, the approach was circular, which did not allow the construction of a rigorous mathematical apparatus for a theory of inductive inference. Another aspect that proved to be a limitation was the sample size, as the number of observations was too large at the time.

A second approach to solving the problem is found in the text Approximatio ad Summam Terminorum Binomii $(a+b{)}^n$ in Seriem expansi, whose content then appears in Miscellanea Analytica and the 1738 edition of The Doctrine of Chances. Continuing Bernoulli’s work on the law of large numbers, de Moivre obtains a new approximation to the binomial distribution that allows reducing the number of observations required to be able to affirm that the relative frequency Yn = Xn/ⁿ is contained in a given interval around the true value of 𝜃 with a certain degree of reliability.

The result of the approximation of the binomial distribution by the normal is the product of a game problem posed by Alexander Cuming years before (Figure 3) and in which de Moivre demonstrates that the value of the expected value sought is $2n\theta (1-\theta )(\frac{n}{n\theta }){\theta }^{n\theta }(1-\theta {)}^{n(1-\theta )}$ .

Figure 3
– Problem 87 of de Moivre’s text

A reinterpretation of this result in terms of mean gain per trial leads de Moivre to understand it as a measure of the dispersion of the random variable Y_n around the true value of 𝜃 ; that is, it allowed us to determine the probability that variable Y_n assumes values within a given interval (see Diaconis; Zabell, 1991). Therefore, considering this interpretation and the characteristics of the associated distribution, de Moivre argues that Y_n converges in probability to 𝜃 and thus can make statements about the value of the probability (Hald, 2007).

Although de Moivre also failed to formulate a method for determining a reliability interval for 𝜃 based on n and Y_n, one of his contributions was to achieve a more precise quantification of the increase in reliability with an increase in observations.

A third look at the topic comes from An essay towards solving a problem in the doctrine of chances, in which Bayes proposes a solution to the probability inversion problem. This problem arises from the interest in determining the probability of an unknown event based on the knowledge of the number of successes and failures in n observations of the event. This became the problem presented in Figure 4.

Figure 4
– Bayesian problem in Bayes’ text

To solve this, Bayes proposes a game mechanism that simulates a binomial model and, using geometric arguments, builds a rule to measure the certainty of the conclusions that can be reached for any decision. This was the first solution to the problem in the case of a binomial model. Later works, such as those of Laplace, Jeffreys, and de Finetti, provided a generalization of the method, considering other distributions and using it beyond games of chance.

According to Landro and González (2012), Bernoulli’s and de Moivre’s failures in the probability inversion issue are fundamentally due to the impossibility of treating 𝜃 as a random variable within the framework of Newtonian philosophy; it could 𝜃 only be conceived as a constant and the relative frequency as a random variable. Next, the nature of the Bayesian problem and its solution are explored in depth in the context of the specific situation.

5.1.3 Inverse probability in Bayes’ text

The context of the specific situation is defined by the specific components of the text being analyzed that allow us to describe the nature and characteristics of Bayesian problems.

According to Stigler (2013), Bayes’ posthumous work was intended as a defensive tool in the 18th century to combat Hume’s claims from a probability perspective. Likewise, given the philosophy that permeated the time, Bayes’ work also raised theological implications.

Specifically, An essay towards solving a problem in the doctrine of chances is the text in which the first solution to the problem of inverse probability is recognized. This consists of four parts: 1) Introduction or preliminary letter in which the statement of the problem and the purpose of the text are found; 2) First section that presents the base theory, that is, definitions and specific propositions; 3) Second section in which the solution method to the problem posed is constructed; 4) Appendix in which three applications of the method in specific situations are illustrated. Richard Price was responsible for the Essay’s publication and some parts of the text were incorporated by him.

In the preliminary letter, the problem to be solved is explicitly stated (Figure 4), and this can be reinterpreted as: if we know that in n trials an event has been observed p times, what can we say about the probability of the event? Furthermore, we must construct a method as a mathematical foundation to support our reasoning regarding past events that may plausibly occur again. Moreover, the greater the number of observations available to support a conclusion, the greater the reason to accept it.

Bayes’ solution is located in section two of the Essay. It is developed through a physical model consisting of a flat, square table on which balls are thrown without a target, and the probability measure is related to the geometric properties of the model (Figure 5). The model associated with the double postulate is an analogy in the geometric context of a Bernoulli experiment, and the assumption that the point where the first ball has to stop is uniformly distributed in square ABCD, which suggests assuming uniformity over the values that the random variable 𝜃 can take, which represents the probability of the unknown event or occurrence.

Figure 5
– Geometric construction associated with postulate I

After the postulate, a series of propositions follow in which preliminary processes for solving the problem, the solution, and a generalization are established. In Proposition 8, the joint probability of variables 𝜃 and $Y_n=\frac{X}{n}$ is determined and, in the consequent corollary, we obtain the marginal distribution of Y_n; Proposition 9 contains the rule to determine the probability sought for the problem; Proposition 10 poses a generalization of the result based on an analogy between the physical mechanism and the behavior that certain natural phenomena manifest for the observer, and proposes three rules to approximate the area under the distribution curve (Landro; González, 2012).

According to the experiment and geometric construction, Bayes is interested in estimating a certain area of the unit square ABCD (Figure 5) –called event M–, and measure the veracity of said conjectured estimate of the area in probabilistic terms. This is reflected in Proposition 9, which, in current notation, represents a ratio of two areas (see Dale, 1999):

$P(\frac{Ab}{AB}<\theta <\frac{Af}{AB}|Y_n=x)=\frac{{\int }_b^f(\frac{n}{\theta }){\theta }^x(1-\theta {)}^{n-x}d\theta }{{\int }_A^B(\frac{n}{\theta }){\theta }^x(1-\theta {)}^{n-x}d\theta }$

Regarding this interpretation, we must emphasize that the historical problems identified regarding Bayesian inference from the preliminary stages to the emergence of the first solution concern estimation. Indeed, estimation is connected to inductive inference since the problem consists of establishing a conjecture about an approximate value of a parameter –probability or unknown proportion– based on the observed results of the experiment in the sample.

In conclusion, the method built on Bayes’ text allowed “to determine ‘to what degree’ observations confirm a given conjecture and thus measure ‘the strength of analogical or inductive reasoning’” (Daston, 1988, p. 256, emphasis in the original), that is, to determine the probability of being right. Furthermore, two aspects that gave way to the transition from direct probabilities to inverse probabilities were considering parameter 𝜃 as a random variable and assuming equiprobability for all its possible values, reflected in a uniform distribution on the a priori probability; however, the method presented by Bayes was limited only to the case of the binomial distribution.

5.2 Stochastic activity associated with Bayesian statistics

The analysis of stochastic activity was carried out from the written production of the problems in Bayes’ (1763) text from which the ways of doing and/or saying –actions– of the subject were reconstructed. We decided to focus on the last section of the mathematical text, as it shows the use of Bayes’ method –Proposition 10 and a version of Bayes’ theorem– in three specific situations. As an example, we illustrate the analysis by returning to problem III.

Derived from familiarization with the discursive structure of the problems and the context that frames each one, we identified and called Case related sections or blocks of writing (Figure 6). Furthermore, depending on the role of the cases, the analysis was organized into moments. For problem III, the analysis of the stochastic activity was carried out in three moments: 1) Analysis of one case (for example, case I); 2) Analysis between cases (for example, from case I to case II); 3) Global analysis of all cases (from case I to case VI).

Figure 6
– Application III in Bayes’ text (1763, pp. 411-418)

For the textual analysis, the first step was to highlight specific text fragments by labeling them with numbers in brackets and interpreting them in footnotes as part of the reconstruction of stochastic activity; the second step was to identify stochastic practices through analytical questions. To recognize the actions, we asked: What and how did they do it? To characterize the activity, we ask: What did they do it for? Socially shared practice is postulated (Why did they do it?) as the result of pragmatic reconstruction –actions and activities– and the description of the context of meaning. Figure 7 is an example of the first part of the analysis corresponding to moment one of Problem III.

Figure 7
– Example of analysis of Moment 1 of Problem III

Regarding the identification and organization of practices, Figure 8 presents the subject’s actions as a temporal sequence arising from the written production of the problem. Thus, we defined the level practice activity of measuring the uncertainty about the unknown parameter and interpreting the measurement, which represents part of the dynamics of what the subject did—actions—within the framework of the problem.

Figure 8
– Organization of stochastic practices associated with Moment 1

In short, derived from the analysis of the three problems (see Paredes-Cancino, 2024), we identified different actions whose dynamics, depending on the circumstances of each problem, allowed us to characterize three activities (Figure 9).

Figure 9
– Practices identified in Bayesian problems

6 Results

The historicization was conducted through two phases: a contextual analysis and a textual analysis. The first analysis was carried out with a stratified description of the context of meaning associated with Bayesian inference in its genesis, and with this, we recognized social and cultural elements that permeated its emergence. The second analysis focused on characterizing stochastic activity, which allowed us to identify a pragmatic organization of stochastic practices linked to Bayesian knowledge.

6.1 Context of signification

The context of the production of Bayesian knowledge in its genesis is in the cultural context of the 17th century when religious beliefs as a criterion in constructing knowledge toward the use of reason were deteriorating. In this environment, in particular, the following stands out: Hume’s induction problem as the trigger that gives rise to a theory of inductive inference applicable to “sequences of events for which the consideration of the experience acquired from their repeated observation was transformed into forms of expectation about their future behavior” (Landro; González, 2012, p. 46).

Attempts at quantification regarding the theorization of inductive inference progressed through the nature of probability problems: analysis of symmetrical situations, then asymmetrical situations and known probability, and, finally, situations of unknown probability; most of these problems are framed in gambling, which was studied in other settings. The development of probability and the transition through different types of problems revealed the relevance of the renewal of the probabilistic analysis method, which led to the creation of a new paradigm: analysis of inverse probability; such analysis was a transition from reasoning from the causes to the effects to reasoning from effects to causes (Daston, 1988; Gorroochurn, 2012), that is, an inversion of Bernoulli’s and de Moivre’s probability analyses. The above constitutes the situational context.

Finally, the context of the specific situation is characterized by the Bayesian problems in Bayes’ text that gave rise to the method for solving the problem of inverse probability. The Bayesian problems identified are of unknown parameter estimates –probability or proportion–. Bayes’ main goal was to build a mathematical tool that would allow us to start from certain observations to examine the probability of the event, and that required conjecturing about the unknown probability or probability of the causes as a starting point. In particular, 𝜃 as a random variable and the equiprobability assumption related to the application of 𝜃 with a probability distribution known a priori, i.e., a uniform distribution, were part of the elements that allowed the success of Bayes’ approach for the case of the binomial distribution.

Derived from the contextual analysis, we postulate the following hypothesis: tasks in the context of estimating an unknown parameter can help students develop and use intuitive notions of Bayesian inference.

6.2 Stochastic activity

About the context of signification that frames Bayesian inference, in particular, the context of the specific situation –parameter estimation–, the stochastic activity focuses on determining to what degree the observation confirms a given conjecture about an unknown proportion or probability (parameter), that is: What is the odds that our conjecture is true? And with that, determine what is most likely regarding the true value of 𝜃.

The above is evident in the dynamics of stochastic practices involving Bayesian problems in binomial situations (Figure 9), from postulating a uniform distribution as a model a priori on the unknown parameter, until establishing a probabilistic generalization on the event of interest, through the analysis of samples of different sizes and the review of the probability model a posteriori to recognize a tendency.

Therefore, we propose to name this dynamic of actions and activities typical of tasks in the context of estimation in binomial situations of the socially shared practice of performing Bayesian inferences about the unknown parameter, particularly on a proportion.

In summary, from the contextual analysis and in addition to the textual analysis, we expand our hypothesis: Tasks in the context of estimating an unknown parameter can help students develop and use intuitive notions of Bayesian inference. Furthermore, these tasks require students to go through three stages to estimate the “true value” of the unknown proportion: 1) Generate an a posteriori probability model for the event using two sources of information (personal model on the uncertainty of an event and statistical model on observations of the event); 2) Update the a posteriori probability model based on consideration of new evidence, for example, through new data samples; 3) Look for a pattern in the probability measures of the unknown event in the a posteriori probability model to establish a generalization.

From the theoretical perspective, this epistemological hypothesis on Bayesian inference is translated into the epistemological model in Chart 3, which shows the dynamics and evolution of stochastic activity.

Chart 3
– Epistemological model on Bayesian problems in the context of estimation

Action	Activity	Socially shared practice:
1. Establish a probability distribution a priori about the unknown parameter 𝜃	Measure the uncertainty about the unknown parameter and interpret that measurement	Estimate the “true value” of the unknown parameter 𝜃
2. Build a binomial statistical model that fits the sampling data
3. Generate the probability distribution model a posteriori
4. Determine the area under the model curve a posteriori in a specific interval (α < θ < b)
1. Collect different data samples	Update the measure of uncertainty about the parameter and interpret that measure.
2. Measure the probability P(α < θ < b| p,q) for a specific interval in different samples
1. Compare two probability measures P(α < θ < b| p,q)	Identify a pattern or trend in the behavior of conditional probability measures
2. Analyze the series of probability measures P(α < θ < b| p,q)

7 Conclusions

This work seeks to contribute from the history and epistemology of mathematics to the didactics of statistical education, particularly Bayesian statistics. Regarding the research question, we identified that the stochastic activity mobilizing the problems in the genesis of Bayesian inference is framed in a context of estimation of unknown parameters, explicitly a proportion in binomial situations. Furthermore, one of the key aspects of these problems is the consideration of parameter 𝜃 as a random variable and the postulation of an a priori probability distribution on it, which permeates the individual’s stochastic ways of doing things.

The analysis and characterization of the stochastic activity of Bayesian problems resulted in an epistemological model based on stochastic practices (Chart 3), whose social emergent is the socially shared practice of making inferences about an unknown proportion or estimating the true value of the parameter. This involves practices of the level action such as establishing conjectures about the parameter, collecting data samples, and building a data model, among others, which are organized into three activities: measuring the uncertainty about the parameter, updating the probability measure of the conjecture relative to the sample, and identifying a pattern in the behavior of the probability measures.

It is worth noting that the stochastic practice of postulating an initial distribution model on the parameter is one of the characteristic epistemic elements of Bayesian inference, differentiating it from the classical perspective. In other words, we recognize that this practice is mentioned in the historical studies of Batanero (2005), Borovcnik and Kapadia (2014), and Carranza (2009). Unlike these investigations, this study explains how this practice and others identified are part of the dynamics of doing and its organization –action, activity, and socially shared practice– within the framework of the stochastic activity of Bayesian estimation problems in binomial situations. Also, we consider that this paper points out the nature of the context of the problems in the genesis of Bayesian inference; in this way, the estimate can be a good starting point for teaching this notion in schools.

In accordance with the paradigm of informal statistical inference and the characteristics that support the type of inferential reasoning (Makar; Rubin, 2009; Zieffler et al., 2008), from the Bayesian perspective, we propose that reasoning entails that the characteristic of using data as evidence is complemented with a new characteristic, i.e., the postulation of an a priori model of probability distribution on the parameter that represents the subject’s state of knowledge of the situation.

The practice-based model is a reference to enrich the teaching of inference, taking into account ideas supported by epistemology, and specifies the actions to promote in individuals within the framework of stochastic activity; consequently, the proposed model helps to develop a way of thinking and acting beyond the data from a Bayesian perspective. Similarly, this model aligns with the proposal of Eichler and Vogel (2014) and could, therefore, be considered from a probability modeling perspective.

Finally, given the nature of the practices and stochastic activity, we consider that we have the ad hoc characteristics for constructing concept-oriented tasks that promote a meaning of Bayesian probability as a degree of belief (Rivadulla, 1995). This will expand the tasks and meanings predominating in the school setting (see Carranza, 2014; Paredes-Cancino & Montiel-Espinosa, under review).

Given the above and as a perspective for the study, assessing the epistemological model of reference from the construction of tasks and their implementation to strengthen it based on empirical evidence is imperative.

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Publication Dates

• Publication in this collection
  07 Apr 2025
• Date of issue
  2025

History

• Received
  17 Dec 2023
• Accepted
  27 Feb 2024