Treffer: Teaching Bayesian and Markov Methods in Business Analytics Curricula: An Integrated Approach

In the era of artificial intelligence (AI), big data (BD), and digital transformation (DT), analytics students should gain the ability to solve business problems by integrating various methods. This teaching brief illustrates how two such methods--Bayesian analysis and Markov chains--can be combined to enhance student learning using the Analytics Project Life Cycle Management (APLCM) approach and a case study involving qualitative forecasting. The theoretical frameworks for combining Bayesian and Markov methods are developed, and a forecasting solution is implemented in both MS Excel and Python. Based on an assessment of student learning, applying this pedagogical approach helps students better use these disjoint methods and appreciate the value of integrating them. Although this teaching brief is designed and most appropriate for graduate students with previous BA courses, it can also be used in upper-level courses within an undergraduate BA curriculum. Finally, this teaching brief provides the instructors wishing to use this pedagogical approach in their appropriate courses with the necessary resources (i.e., case study, in-class example, and the MS Excel and Python templates).

As Provided

AN0154759243;q1n01jan.22;2022Jan21.04:36;v2.2.500

Teaching Bayesian and Markov methods in business analytics curricula: An integrated approach 

In the era of artificial intelligence (AI), big data (BD), and digital transformation (DT), analytics students should gain the ability to solve business problems by integrating various methods. This teaching brief illustrates how two such methods—Bayesian analysis and Markov chains—can be combined to enhance student learning using the Analytics Project Life Cycle Management (APLCM) approach and a case study involving qualitative forecasting. The theoretical frameworks for combining Bayesian and Markov methods are developed, and a forecasting solution is implemented in both MS Excel and Python. Based on an assessment of student learning, applying this pedagogical approach helps students better use these disjoint methods and appreciate the value of integrating them. Although this teaching brief is designed and most appropriate for graduate students with previous BA courses, it can also be used in upper‐level courses within an undergraduate BA curriculum. Finally, this teaching brief provides the instructors wishing to use this pedagogical approach in their appropriate courses with the necessary resources (i.e., case study, in‐class example, and the MS Excel and Python templates).

Keywords: Business Analytics Curricula; Integrating Bayesian Analysis with Markov chains; and Qualitative Forecasting

INTRODUCTION

In this age of AI, BD, and DT, various quantitative methods, including Bayesian analysis and Markov chains, are utilized to model highly stochastic real‐world processes. A Markov chain is a stochastic model defining a sequence of possible events in which the probability of each event depends only on the state obtained in the previous event (Gagniuc, 2017). On the other hand, Bayesian analysis is a mathematical procedure that attempts to model unknown parameters using probability statements (Congdon, 2014). Bayesian models can incorporate prior information in the analysis and assign an actual probability to any hypothesis of interest. Both Bayesian and Markov methods are considered an inextricable part of the AI toolbox and have many applications in various business domains, including finance, healthcare, and technology (Korb & Nicholson, 2011; Rendle et al., 2010; Yu, 2010). Algorithmic trading models for portfolio optimization, text and speech models to retrieve insights from customer conversations, forecasting insurance loss payments, programming self‐driving cars, and predicting customer loyalty are just some practical, real‐life uses of these two methodologies.

Since Bayesian and Markov methods have a wide range of applications and are increasingly employed in industry, a BA curriculum should incorporate these methods and ensure that students have the skills to use them for data‐driven decision‐making. Additionally, teaching Bayesian analysis and Markov chains jointly enables students to make connections between various analytics algorithms in the AI toolbox—a vital skillset needed by future BA specialists to develop and deploy complex analytics products in industry. To this extent, this teaching brief provides the content for business analytics faculty to jointly teach Bayesian analysis with Markov chains using a hypothetical case study and a systematic approach named Analytics Project Life Cycle Management (APLCM). Teaching Bayesian and Markov techniques in an integrated manner through a case study and APLCM provides students with an opportunity to understand how analytics projects are executed and productionized in industry and helps them appreciate the importance of analytics in organizational success.

The remaining sections are organized as follows: First, the literature review section summarizes pertinent pedagogical approaches developed for teaching BA. Then, employing APCLM, the theory behind Markov chains and Bayesian analysis is illustrated. The procedures to integrate these two methods are explained within the context of the previously mentioned case study. A set of templates that can be used to teach these methods are developed in MS Excel and Python and are provided for BA faculty to use in their classes. The teaching brief ends with a discussion assessing the pedagogical effectiveness of the proposed integration of Bayesian and Markov methods with respect to student learning and then provides a summary along with some suggestions for other potential integration of BA methods.

RELATED LITERATURE

Various studies, including those authored by Chiang et al. (2012), Gorman and Klimberg (2014), Wilder and Ozgur (2015), Cegielski and Jones‐Farmer (2016), Rienzo and Chen (2018), and Johnson et al. (2020), investigate the best practices for designing and improving analytics curricula. These studies notably identify the methods that should be taught in analytics programs and emphasize the skills that analytics students should attain. The significant commonalities among these papers are that they all strongly advise BA programs to offer predictive modeling courses, employ case studies with datasets, and teach students programming languages, particularly Python and R.

In addition to these articles addressing the design strategies of analytics programs, there are research efforts providing guidance for BA faculty regarding how to teach quantitative methods in the era of AI, BD, and DT. Articles authored by Gros et al. (2011), Marjanovic (2012, 2013), Yang and Liu (2013), Bolloju (2014), Gupta and Raja (2015), and Yap and Drye (2018) focus on improving BA education as a whole. Other authors, such as Levine and Stephan (2011), Anderson and Williams (2019), Pinder, 2013a, 2013b), and Johnson and Berenson (2019), study the impact of using either software platforms or systematic data analysis frameworks. The primary suggestions offered in these studies deal with making connections among various analytics methods or integrating programming tools and software applications when teaching analytics techniques.

Despite Bayesian and Markov methods being increasingly useful in analytics, these two topics are not sufficiently discussed in the BA curriculum. The introductory BA textbooks (e.g., Black et al., 2018; Evans, 2014; Sharda et al., 2013; Bowerman et al., 2019; Jaggia et al., 2020) discuss Bayes' theorem but not Bayesian estimation techniques and Markov methods. One "beyond introductory" BA book, by Shmueli et al. (2017), provides excellent coverage of Bayesian methods as a supervised learning classification tool and also includes three chapters on forecasting; however, it does not discuss Markov methods. Similarly, Ledolter (2013) discusses Bayes' theorem in the context of Naïve Bayes classification but does not cover Markov methods.

There are several specialized books covering Markov and Bayesian methods. A more advanced management science text with the title "Bayesian Decision Problems and Markov Chains" (Martin, 1975) does not address the integration of Bayesian analysis with Markov chains described in this teaching brief. Levin et al. (2007) cover Markov chains from the mathematical and computer science perspectives and discuss finite Markov chains, Markov chain mixing, and continuous Markov chains. Iosifescu (2007) explains different types of Markov chains; however, his book mostly provides examples and applications in the psychology and genetics domains. Brooks et al. (2011) and Gagniuc (2017) cover Markov chains in the prediction and simulation frameworks. Congdon (2014) discusses applied Bayesian modeling, including Bayesian methods and estimation. This book is particularly instrumental in teaching Bayesian techniques in the context of analytics and predictive modeling. Hastie et al. (2017) also cover Bayesian methods in the context of analytics and predictive modeling. They discuss the applications of the Naïve Bayes classification algorithm and Bayesian neural networks. The primary issue with these specialized books, however, is that they are generally too technical for business students, target engineering and computer science students, and do not include many business‐related case studies.

To date, there are a limited number of studies discussing approaches for teaching Bayesian or Markov methods. Albert (1993) explains how to construct prior distributions and employ the Sampling‐Importance Resampling (SIR) method to simulate posterior distributions for inference problems through Minitab Macros and data visualization techniques. The study concludes that using SIR and data visualization in Minitab helps students understand the relationship between the prior and posterior distributions better. Sedlmeier and Gigerenzer (2001) employ a computerized program to train students to construct frequency representations (representation training) instead of inserting probabilities into Bayes' rule (rule training). Their study indicates that people learn faster and retain the learned material longer using representation training than using rule training. Rouder and Morey (2019) teach Bayes' theorem in a particular ratio form in which posterior belief relative to the prior belief is equal to the conditional probability of data relative to the marginal probability of data. Their study suggests that this specific ratio form allows students to understand the prior distribution's role and view Bayes' theorem as an updating mechanism. Bárcena et al. (2019) design a simulation application that students can use to observe how the prior impacts the posterior. The study suggests that students have a better understanding of Bayesian updating when using simulation software. Johnson (2003) employs game‐based learning to teach Markov chains to undergraduate students. The teaching brief uses Monopoly as a game and models its rules through Markov chains to identify the long‐run frequencies for visiting the various game properties. The study shows that using game‐based learning when teaching Markov chains helps students better understand Markov chain properties, notations, and terminology. However, these aforementioned studies do not integrate Bayesian analysis with Markov chains in the context of BA. In fact, no research demonstrating the pedagogical effectiveness of integrating any BA methods has been found. Thus, this teaching brief aims to fill this gap.

USING THE ANALYTICS PROJECT LIFE CYCLE MANAGEMENT (APLCM) APPROACH FOR TEACHING BUSINESS ANAL...

The use of analytics to make informed decisions is a foundational element of business that is necessary for competitive advantage. The primary goals of analytics projects are to derive actionable insights from organizational and external datasets and use these insights for operational, tactical, and strategic decision‐making. Since the application of analytics to complex datasets is challenging, it requires a standard framework with explicit steps to guide the project. One of the approaches used for obtaining insights from large datasets using analytics is called APLCM or Analytics Project Life Cycle Management. The APLCM approach used at companies in various industry clusters (e.g., telecommunication, pharmaceuticals, logistics) is an outgrowth of CRISP‐DM developed in 1996 by Teradata and Daimler AG along with other leading organizations at that time (Shearer, 2000). Adopting the APLCM approach for the BA classroom serves to motivate future BA specialists to develop skills that follow industry best practices.

As depicted in Figure 1, APLCM maps the steps for effectively obtaining insights from datasets using analytics models. In the first step of APLCM, the problem statement and critical project objectives are defined based on the outline of expectations provided by project stakeholders. Additionally, the overall scope of the work and the questions that should be answered are identified in this step. In the following step, various data analysis and modeling methods that are appropriate for the given problem are explored. Having several modeling approaches in mind can help identify data requirements as well as data collection techniques. Then, in the third step, the desired data are acquired through various methods that include conducting surveys and focus groups, querying relational and nonrelational databases, as well as scrapping datasets from the Internet. Once the collected and compiled dataset is cleaned, it is analyzed in the fourth step using modeling techniques identified in the second step. Among the models explored, an appropriate analytical model is selected based on an assessment of potential performance indicators. Afterward, in the fifth step, the selected analytical model is deployed, and its performance is monitored through sensitivity analysis and regular quality control checks.

Since BA specialists in industry follow such approaches as guides in solving complex business problems, it is pedagogically desirable for faculty to employ some of them when teaching BA methods and their applications. This allows BA students to learn how to solve problems in class like they do in industrial settings. Therefore, the APLCM approach is employed for integrating Markov and Bayesian methods in this teaching brief. Versions of this approach is used in BA programs at some academic institutions, including Northeastern University (Northeastern Graduate Programs Staff, 2020)

THE CLASSROOM INTEGRATION OF BAYESIAN AND MARKOV METHODS

An Overview

The particular case study focuses on marketing analytics because Bayesian analysis and Markov chains have been used in business to study and monitor brand loyalty and brand switching patterns to make forecasts of market share since the 1960s (e.g., Harary & Lipstein, 1962; Maffei, 1961). Additionally, business students are typically familiar with marketing concepts, making this type of case study relatable and easy to grasp. This case study is then utilized in the classroom to simplify the complex theories and mathematical computations involved in teaching both Bayesian and Markov methods and Bayesian enhancements of Markov chain forecasts. Students first learn how to pool a set of subjective judgments so that a series of Markov chain forecasts based on executive opinion can be made. Such qualitative forecasts, however, are not data dependent. Since Bayesian methods are data‐driven, and data‐based decision‐making is superior to intuition (Makridakis, 1986), the collection of sample information enables a Bayesian approach that combines the sample data with executive opinion to enhance a set of data‐driven Markov chain forecasts.

Defining the Problem—Aplcm Step 1

In order to motivate students, increase their curiosity, and encourage them to ask questions, a case study is provided with a hypothetical problem statement having clear objectives at the beginning of the module, as provided below.

"A national company called A&A Corp. has developed a new product that has been marketed by only one other company within the past year, and it does not expect competition from other companies in the next year. A&A Corp. has been test‐marketing this new product and monitoring purchases to develop an aggressive marketing strategy designed to win over customers from its only competitor. A meeting among three executives of A&A Corp. (i.e., the vice president of marketing—VP, the director of market research—DM, and the director of advertising—DA) is held to evaluate and analyze the interim results from test‐marketing this new product the past 3 months. After sixth months of testing, these executives (i.e., experts) must make a recommendation to the CEO as to whether the product should be marketed or not. Therefore, the company's goal is to monitor brand loyalty and brand switching patterns for this product and make forecasts of market share in both the intermediate‐ (i.e., sixth month) and long‐ (i.e., equilibrium stage) term that the available information is pointing toward."

Identifying Potential Analytics Models—Aplcm Step 2

In this second step, several candidate techniques that can solve a problem of interest are identified. Bayesian analysis and Markov chains are determined to be appropriate choices because A&A Corp. aims to forecast the market share of a prototype product based on expert opinions and sample data on brand loyalty and brand switching patterns of customers. Therefore, in this step, students are taught the theoretical foundations of Markov chains along with the Bayesian methods used to enhance the initial, Markov‐based forecasts. The mathematical notations used are given in Table A1 of Appendix 1. Appendix 2 then describes how g expert evaluations are combined or averaged together based on their individually assigned weights from the CEO to form the initial <math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0001" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi><mo>&#215;</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">$c \times c$</annotation></semantics></math> transition matrix <bold>Q</bold> of pooled judgments that initiates the forecasting process. Appendix 3 then provides the needed classroom documents—an introduction to Markov chains, an introduction to Bayesian analysis, and the integration of Bayesian and Markov methods to enhance the judgmental forecasts obtained via Markov chains because these are not data‐driven.

Collecting and Understanding the Data—APLCM Step 3

In the absence of actual data, a common technique for forecasting is to strive for a "consensus of 'expert' opinion." This may be accomplished by conducting panel or focus group discussions or applying brainstorming or Delphi techniques. Often employed at the conference tables of various corporations and government agencies, this qualitative (i.e., judgmental) forecasting technique takes into account views about a product, service, candidate, or political position. Pooling the resulting views leads to an overall group forecast. The major premise upon which such forecasting is based is that combined group judgment is thought to be superior to any individual forecast (O'Hagan et al., 2006).

In this case study, students are told that the three executives—the vice president of marketing (VP), the director of market research (DM), and the director of advertising (DA)—are able to make evaluations of the month‐to‐month brand loyalty and brand switching behavior of consumers using their "executive intuition" combined with their analyses of currently sparse test‐market results. This is displayed in Table 1. These evaluations or judgmental forecasts provided by the (g = 3) executives are then weighed by the CEO's assessments of their experience and familiarity with the products and given in Table 2. According to this table, the CEO assigns a weight (i.e., subjective probability) of.40 to the VP's evaluation. The VP opines that 90% of the customers who have previously purchased Product 1 will keep purchasing Product 1 while 10% of Product 1 customers will switch to Product 2.

1 TABLEEvaluations on % Repurchasing Product 1 and % Switching to Competitor's Product 2

<table><thead><tr><th>Executive</th><th>% Repurchasing 1</th><th align="left">% Switching to 2</th><th>% Repurchasing 2</th><th>% Switching to 1</th></tr></thead><tbody><tr><td>VP</td><td>90.0</td><td>10.0</td><td>70.0</td><td>30.0</td></tr><tr><td>DM</td><td>80.0</td><td>20.0</td><td>75.0</td><td>25.0</td></tr><tr><td>DA</td><td>95.0</td><td>5.0</td><td>60.0</td><td>40.0</td></tr></tbody></table>

2 TABLECEO Assessment of Executive Evaluations on Repurchasing and Brand Switching

<table><thead><tr><th align="left">Executive</th><th align="left">Prior Weights</th><th align="left">Previous Product</th><th align="left">To 1</th><th align="left">To 2</th></tr></thead><tbody><tr><td>VP</td><td>.40</td><td /><td>.90</td><td>.10</td></tr><tr><td>DM</td><td>.40</td><td>From 1</td><td>.80</td><td>.20</td></tr><tr><td>DA</td><td>.20</td><td /><td>.95</td><td>.05</td></tr><tr><td>VP</td><td>.40</td><td /><td>.30</td><td>.70</td></tr><tr><td>DM</td><td>.50</td><td>From 2</td><td>.25</td><td>.75</td></tr><tr><td>DA</td><td>.10</td><td /><td>.40</td><td>.60</td></tr></tbody></table>

After providing the students with the judgmental forecasts of the executives as well as the weights assigned by the CEO, the students are given a small random sample (n = 100) of customer repurchases of these two products in the test market that is taken at the end of the current (third month) period. Students are told that the sampled individuals are asked to name the product they purchase now and name the product they had purchased previously. The tallied results from this sample are recorded in Table 3.

3 TABLESummary of Brand Loyalty and Brand Switching Patterns for 100 Consumers

Previous Product	To 1	To 2	Previous Totals
From 1	19	1	20
From 2	17	63	80
Current totals	36	64	100

The sample dataset shows that of those 20 customers who previously bought Product 1, 19 repurchased it while 1 switched to Product 2. Moreover, of those 80 individuals who previously bought Product 2, 63 repurchased it while 17 switched to Product 1. In sum, of the 100 sampled individuals who repurchased these items, 36 currently bought Product 1 while 64 bought Product 2.

Analyzing and Modeling—APLCM Step 4

After data collection, students are instructed to follow the flow chart displayed in Figure 2 to integrate Bayesian analysis with Markov chains to enhance judgmental forecasting. First, a judgmental transition matrix <bold>Q</bold> is formed from the pooled evaluations of repurchasing and brand switching provided by g executives (i.e., experts) based on assigned weights or prior probabilities w<subs>ik</subs>, as shown in Table 2. Either judgment or recent information can be used to obtain the initial row vector <math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0002" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>&#960;</mi><mo>&#8407;</mo></mover><mn>0</mn></msub><mo>=</mo><mspace width="0.28em" /><mrow><mo>(</mo><mrow><msub><mi>&#960;</mi><msub><mn>0</mn><mn>1</mn></msub></msub><msub><mi>&#960;</mi><msub><mn>0</mn><mn>2</mn></msub></msub><mtext>...</mtext><msub><mi>&#960;</mi><msub><mn>0</mn><mi>c</mi></msub></msub></mrow><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">${\vec \pi &#95;0} = {\rm{\;}}({{\pi &#95;{{0&#95;1}}}{\pi &#95;{{0&#95;2}}} \ldots {\pi &#95;{{0&#95;c}}}})$</annotation></semantics></math> . Markov chains are then developed to make baseline judgmental forecasts for time period m and the equilibrium stage. Employing Bayesian estimation methods, the sample proportion <math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0003" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>P</mi><mrow><mo>+</mo><mn>1</mn></mrow></msub><annotation encoding="application/x-tex">${P&#95;{ + 1}}$</annotation></semantics></math> computed from a random sample of n individuals (shown in Table 3) are used to obtain the initial revised row vector <math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0004" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>&#960;</mi><mo>&#8407;</mo></mover><msub><mi>R</mi><mn>0</mn></msub></msub><mo>=</mo><mspace width="0.28em" /><mrow><mo>(</mo><mrow><msub><mi>&#960;</mi><msub><mi>R</mi><msub><mn>0</mn><mn>1</mn></msub></msub></msub><msub><mi>&#960;</mi><msub><mi>R</mi><msub><mn>0</mn><mn>2</mn></msub></msub></msub><mtext>...</mtext><msub><mi>&#960;</mi><msub><mi>R</mi><msub><mn>0</mn><mi>c</mi></msub></msub></msub></mrow><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">${\vec \pi &#95;{{R&#95;0}}} = {\rm{\;}}({{\pi &#95;{{R&#95;{{0&#95;1}}}}}{\pi &#95;{{R&#95;{{0&#95;2}}}}} \ldots {\pi &#95;{{R&#95;{{0&#95;c}}}}}})$</annotation></semantics></math> . Then, Bayes' theorem is applied to develop the revised transition matrix <bold>P</bold>. Markov chains are then utilized to make actual forecasts for time period m and the equilibrium stage, and a set of model performance metrics is used to assess the results.

Classroom emphasis stresses that Bayesian enhanced Markov chain‐based forecasts, indicating trends in patterns of loyalty and migration with respect to a specific choice as well as for competing choices, are intended to guide decisions under the Markov model assumption that no further interventions will be occurring to alter these forecasted trends as well as to plan strategies to offset any negative interferences with these forecasts, should they occur.

Stage 1: Develop the Transition Matrix Q

In this classroom activity, students are asked to combine the evaluations provided by the three executives regarding brand loyalty and brand switching so that the company can forecast its product's market share. Alternative approaches are explained in Appendix 2 along with the details for obtaining the <math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0005" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mo>&#215;</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">$2 \times 2$</annotation></semantics></math> transition matrix <bold>Q</bold> for Product 1 and Product 2 by multiplying the prior weights (i.e., subjective probabilities) assigned by the CEO to the executive evaluations of repurchasing and brand switching given in Table 2. The MS Excel formulas to compute the transition matrix <math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0006" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi><mo>=</mo><mo>(</mo>.870.130.285.715<mo>)</mo></mrow><annotation encoding="application/x-tex">$Q = ({ \def\eqcellsep{&}\begin{array}{@{}*{2}{c}@{}} {.870}&{.130}\\ {.285}&{.715} \end{array} })$</annotation></semantics></math> are displayed in Figure 3.

Stage 2: Obtain Initial Probability Vector and Compute m ‐Period and Equilibrium State Foreca...

According to the brand loyalty and switching patterns for 100 sampled individuals, Product 1 is estimated to have achieved a 20% market share (i.e., <math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0007" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>&#960;</mi><msub><mn>0</mn><mn>1</mn></msub></msub><mo>=</mo><mo>.</mo><mn>20</mn></mrow><annotation encoding="application/x-tex">${\pi &#95;{{0&#95;1}}} =.20$</annotation></semantics></math> ) in 3 months while Product 2 is predicted to have achieved an 80% market share (i.e., <math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0008" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>&#960;</mi><msub><mn>0</mn><mn>2</mn></msub></msub><mo>=</mo><mo>.</mo><mn>80</mn></mrow><annotation encoding="application/x-tex">${\pi &#95;{{0&#95;2}}} =.80$</annotation></semantics></math> ) so the probability vector is denoted as <math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0009" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>&#960;</mi><mo>&#8407;</mo></mover><mn>0</mn></msub><mo>=</mo><mrow><mo>(</mo><mrow><mo>.</mo><mn>20</mn><mrow><mspace width="0.28em" /><mspace width="0.28em" /></mrow><mo>.</mo><mn>80</mn></mrow><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">${\vec \pi &#95;0} = ({.20{\rm{\;\;}}.80})$</annotation></semantics></math> . As shown in equation (A3.3) of Appendix 3, in order to forecast the market shares for both Products 1 and 2 at the end of the 6‐month test period, this initial probability vector is multiplied by the third power of the transition matrix <bold>Q</bold> so that <math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0010" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>&#960;</mi><mo>&#8407;</mo></mover><mn>3</mn></msub><mo>=</mo><msub><mover accent="true"><mi>&#960;</mi><mo>&#8407;</mo></mover><mn>0</mn></msub><msup><mi>Q</mi><mn>3</mn></msup><mspace width="0.28em" /><mo>=</mo><mspace width="0.28em" /><mrow><mo>(</mo><mrow><mo>.</mo><mn>5893</mn><mrow><mspace width="0.28em" /><mspace width="0.28em" /></mrow><mo>.</mo><mn>4107</mn></mrow><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">${\vec \pi &#95;3} = {\vec \pi &#95;0}{Q^3}{\rm{\;}} = {\rm{\;}}({.5893{\rm{\;\;}}.4107})$</annotation></semantics></math> . Therefore, the respective 6‐month market share forecasts for the two products are 58.93% and 41.07%. Moreover, the predicted equilibrium market shares are computed as 68.67% for Product 1 and 31.33% for Product 2. These sets of results could then be used to determine whether Product 1 should be marketed. However, the students must understand that the equilibrium stage results are useful as forecasts only if the transition probabilities remain the same—a condition in reality that is highly unlikely because executives for both competing products will use such current information for tactical and strategic decisions to alter the situation.

Stage 3: Computing Revised Weights and Transition Matrix P

The students are shown that the above forecasts could be strengthened by applying Bayesian analysis to sample data collected at the end of the third‐month period and displayed in Table 3. Since there are only two competing products, the conditional probabilities (i.e., the likelihoods of acquiring the specific sample of data, given that the parameter evaluations made by the experts are true) follow the binomial probability distribution.

Working separately with each "state of the transition matrix," the g = 3 executive evaluations are one‐at‐a‐time considered as the true parameters. For the first state of the matrix, the question of interest is: given that 20 customers previously purchased Product 1, what is the likelihood that 19 of them will repurchase this item if the switching probabilities of executive k (where k = VP, DM, and DA) are the true parameters? These probabilities can be computed in MS Excel using the "BINOM.DIST" function and then are multiplied by the initially assigned weights or prior probabilities to obtain the joint probabilities. As displayed in Table 4, the revised weights (i.e., posterior probabilities) for each executive's evaluations are obtained using Bayes' theorem. Figure 4 illustrates these computations using MS Excel.

4 TABLERevising the Initial Subjective Weights Assigned to Expert Evaluations

<table><thead><tr><th>State</th><th>Executive</th><th>Joint Probabilities: (Likelihood of Sample | Parameter Evaluations) &#215; (Prior Weights)</th><th>Posterior Probabilities (Revised Weights)</th></tr></thead><tbody><tr><td /><td>VP</td><td><p><math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0011" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>C</mi>1920<mo>(</mo><mo>.</mo>90<mo>)</mo>19<mo>(</mo><mo>.</mo>10<mo>)</mo>1<mo>(</mo><mo>.</mo>40<mo>)</mo><mspace width="0.28em" /><mo>=</mo><mspace width="0.28em" /><mo>.</mo>1080<annotation encoding="application/x-tex">$C&#95;{19}^{20}{({.90})^{19}}{({.10})^1}({.40}){\rm{\;}} = {\rm{\;}}.1080$</annotation></semantics></math></p></td><td><p><math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0012" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mo>.</mo>1080<mo>/</mo><mo>.</mo>2066<mspace width="0.28em" /><mo>=</mo><mspace width="0.28em" /><mo>.</mo>523<annotation encoding="application/x-tex">$.1080/.2066{\rm{\;}} = {\rm{\;}}.523$</annotation></semantics></math></p></td></tr><tr><td>1</td><td>DM</td><td><p><math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0013" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>C</mi>1920<mo>(</mo><mo>.</mo>80<mo>)</mo>19<mo>(</mo><mo>.</mo>20<mo>)</mo>1<mo>(</mo><mo>.</mo>40<mo>)</mo><mspace width="0.28em" /><mo>=</mo><mspace width="0.28em" /><mo>.</mo>0231<annotation encoding="application/x-tex">$C&#95;{19}^{20}{({.80})^{19}}{({.20})^1}({.40}){\rm{\;}} = {\rm{\;}}.0231$</annotation></semantics></math></p></td><td><p><math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0014" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mo>.</mo>0231<mo>/</mo><mo>.</mo>2066<mspace width="0.28em" /><mo>=</mo><mspace width="0.28em" /><mo>.</mo>112<annotation encoding="application/x-tex">$.0231/.2066{\rm{\;}} = {\rm{\;}}.112$</annotation></semantics></math></p></td></tr><tr><td /><td>DA</td><td><p><math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0015" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>C</mi>1920<mo>(</mo><mo>.</mo>95<mo>)</mo>19<mo>(</mo><mo>.</mo>05<mo>)</mo>1<mo>(</mo><mo>.</mo>20<mo>)</mo><mspace width="0.28em" /><mo>=</mo><mspace width="0.28em" /><mo>.</mo>0755<annotation encoding="application/x-tex">$C&#95;{19}^{20}{({.95})^{19}}{({.05})^1}({.20}){\rm{\;}} = {\rm{\;}}.0755$</annotation></semantics></math></p></td><td><p><math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0016" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mo>.</mo>0755<mo>/</mo><mo>.</mo>2066<mspace width="0.28em" /><mo>=</mo><mspace width="0.28em" /><mo>.</mo>365<annotation encoding="application/x-tex">$.0755/.2066{\rm{\;}} = {\rm{\;}}.365$</annotation></semantics></math></p></td></tr><tr><td>Sum</td><td>.2066</td><td>1.000</td></tr><tr><td /><td>VP</td><td><p><math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0017" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>C</mi>1780<mo>(</mo><mo>.</mo>30<mo>)</mo>19<mo>(</mo><mo>.</mo>70<mo>)</mo>61<mo>(</mo><mo>.</mo>40<mo>)</mo><mspace width="0.28em" /><mo>=</mo><mspace width="0.28em" /><mo>.</mo>0091<annotation encoding="application/x-tex">$C&#95;{17}^{80}{({.30})^{19}}{({.70})^{61}}({.40}){\rm{\;}} = {\rm{\;}}.0091$</annotation></semantics></math></p></td><td><p><math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0018" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mo>.</mo>0091<mo>/</mo><mo>.</mo>0488<mspace width="0.28em" /><mo>=</mo><mspace width="0.28em" /><mo>.</mo>187<annotation encoding="application/x-tex">$.0091/.0488{\rm{\;}} = {\rm{\;}}.187$</annotation></semantics></math></p></td></tr><tr><td>2</td><td>DM</td><td><p><math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0019" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>C</mi>1780<mo>(</mo><mo>.</mo>25<mo>)</mo>19<mo>(</mo><mo>.</mo>75<mo>)</mo>61<mo>(</mo><mo>.</mo>50<mo>)</mo><mspace width="0.28em" /><mo>=</mo><mspace width="0.28em" /><mo>.</mo>0397<annotation encoding="application/x-tex">$C&#95;{17}^{80}{({.25})^{19}}{({.75})^{61}}({.50}){\rm{\;}} = {\rm{\;}}.0397$</annotation></semantics></math></p></td><td><p><math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0020" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mo>.</mo>0397<mo>/</mo><mo>.</mo>0488<mspace width="0.28em" /><mo>=</mo><mspace width="0.28em" /><mo>.</mo>813<annotation encoding="application/x-tex">$.0397/.0488{\rm{\;}} = {\rm{\;}}.813$</annotation></semantics></math></p></td></tr><tr><td /><td>DA</td><td><p><math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0021" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mi>C</mi>1780<mo>(</mo><mo>.</mo>40<mo>)</mo>19<mo>(</mo><mo>.</mo>60<mo>)</mo>61<mo>(</mo><mo>.</mo>10<mo>)</mo><mspace width="0.28em" /><mo>=</mo><mspace width="0.28em" /><mo>.</mo>0000<annotation encoding="application/x-tex">$C&#95;{17}^{80}{({.40})^{19}}{({.60})^{61}}({.10}){\rm{\;}} = {\rm{\;}}.0000$</annotation></semantics></math></p></td><td><p><math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0022" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mo>.</mo>0000<mo>/</mo><mo>.</mo>0488<mspace width="0.28em" /><mo>=</mo><mspace width="0.28em" /><mo>.</mo>000<annotation encoding="application/x-tex">$.0000/.0488{\rm{\;}} = {\rm{\;}}.000$</annotation></semantics></math></p></td></tr><tr><td>Sum</td><td>.0488</td><td>1.000</td></tr></tbody></table>

To develop the revised transition matrix <bold>P</bold> (given in equation A3.9 in Appendix 3), the revised weights for the executives' evaluations obtained in Table 4 are multiplied by these executives' initial evaluations on repurchasing and brand switching displayed in Table 1. The revised transition matrix is <math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0023" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo>=</mo><mo>(</mo>.9071.0929.2594.7406<mo>)</mo></mrow><annotation encoding="application/x-tex">$P = ({ \def\eqcellsep{&}\begin{array}{@{}*{2}{c}@{}} {.9071}&{.0929}\\ {.2594}&{.7406} \end{array} })$</annotation></semantics></math> .

Stage 4: Compute Revised Probability Vector and m ‐Period and Equilibrium State Forecasts

The revised probability vector, reflecting market share for Products 1 and 2 at the current point in time, can be obtained by Bayesian estimation. As described in Appendix 3, the initially estimated market share achieved by Product 1 (i.e., 20%) is "revised" by the results obtained from the sample using equation (A3.4)

<math display="block" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0024" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>&#960;</mi><msub><mi>R</mi><msub><mn>0</mn><mn>1</mn></msub></msub></msub><mo>=</mo><mfenced separators="" open="(" close=")"><mfrac><msub><mi>I</mi><msub><mn>0</mn><mn>1</mn></msub></msub><mrow><msub><mi>I</mi><msub><mn>0</mn><mn>1</mn></msub></msub><mo>+</mo><msub><mi>I</mi><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></mfrac></mfenced><msub><mi>&#960;</mi><msub><mn>0</mn><mn>1</mn></msub></msub><mo>+</mo><mfenced separators="" open="(" close=")"><mfrac><msub><mi>I</mi><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msub><mrow><msub><mi>I</mi><msub><mn>0</mn><mn>1</mn></msub></msub><mo>+</mo><msub><mi>I</mi><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></mfrac></mfenced><msub><mi>p</mi><mrow><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\begin{equation*}{\pi &#95;{{R&#95;{{0&#95;1}}}}} = \left({\frac{{{I&#95;{{0&#95;1}}}}}{{{I&#95;{{0&#95;1}}} + {I&#95;{p + 1}}}}} \right){\pi &#95;{{0&#95;1}}} + \left({\frac{{{I&#95;{p + 1}}}}{{{I&#95;{{0&#95;1}}} + {I&#95;{p + 1}}}}} \right){p&#95;{ + 1}}\end{equation*}</annotation></semantics></math>

where <math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0025" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>&#960;</mi><msub><mi>R</mi><msub><mn>0</mn><mn>1</mn></msub></msub></msub><annotation encoding="application/x-tex">${\pi &#95;{{R&#95;{{0&#95;1}}}}}$</annotation></semantics></math> is the posterior probability or revised estimate of market share for Product 1, <math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0026" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>&#960;</mi><msub><mn>0</mn><mn>1</mn></msub></msub><mo>=</mo><mspace width="0.28em" /><mo>.</mo><mn>20</mn></mrow><annotation encoding="application/x-tex">${\pi &#95;{{0&#95;1}}} = {\rm{\;}}.20$</annotation></semantics></math> is the prior probability or initially estimated market share for Product 1 over the past 3‐month test period, <math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0027" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mrow><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>n</mi><mrow><mo>+</mo><mn>1</mn></mrow></msub><mo>/</mo><mi>n</mi><mo>=</mo><mn>36</mn><mo>/</mo><mn>100</mn><mspace width="0.28em" /><mo>=</mo><mspace width="0.28em" /><mo>.</mo><mn>36</mn></mrow><annotation encoding="application/x-tex">${p&#95;{ + 1}} = {n&#95;{ + 1}}/n = 36/100{\rm{\;}} = {\rm{\;}}.36$</annotation></semantics></math> is the current sample proportion of purchases of Product 1, given either item is previously bought, <math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0028" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>I</mi><msub><mn>0</mn><mn>1</mn></msub></msub><mo>=</mo><mn>1</mn><mo>/</mo><msubsup><mi>&#963;</mi><msub><mn>0</mn><mn>1</mn></msub><mn>2</mn></msubsup></mrow><annotation encoding="application/x-tex">${I&#95;{{0&#95;1}}} = 1/\sigma &#95;{{0&#95;1}}^2$</annotation></semantics></math> is the amount of information contained in the prior probability distribution with respect to scatter around <math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0029" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>&#960;</mi><msub><mn>0</mn><mn>1</mn></msub></msub><annotation encoding="application/x-tex">${\pi &#95;{{0&#95;1}}}$</annotation></semantics></math> , where <math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0030" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>&#963;</mi><msub><mn>0</mn><mn>1</mn></msub><mn>2</mn></msubsup><mo>=</mo><mspace width="0.28em" /><msub><mi>&#960;</mi><msub><mn>0</mn><mn>1</mn></msub></msub><mrow><mo>(</mo><mrow><mn>1</mn><mo>&#8722;</mo><msub><mi>&#960;</mi><msub><mn>0</mn><mn>1</mn></msub></msub></mrow><mo>)</mo></mrow><mo>=</mo><mo>.</mo><mn>16</mn></mrow><annotation encoding="application/x-tex">$\sigma &#95;{{0&#95;1}}^2 = {\rm{\;}}{\pi &#95;{{0&#95;1}}}({1 - {\pi &#95;{{0&#95;1}}}}) =.16$</annotation></semantics></math> , <math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0031" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>I</mi><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mi>n</mi><mo>/</mo><mrow><mo>(</mo><msub><mi>p</mi><mrow><mo>+</mo><mn>1</mn></mrow></msub><mrow><mo>(</mo><mrow><mn>1</mn><mo>&#8722;</mo><msub><mi>p</mi><mrow><mo>+</mo><mn>1</mn></mrow></msub></mrow><mo>)</mo></mrow><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">${I&#95;{p + 1}} = n/({p&#95;{ + 1}}({1 - {p&#95;{ + 1}}}))$</annotation></semantics></math> is the amount of information contained in the sample dataset of n = 100 purchases and reflects the amount of scatter in the sample proportion where the sample variance is <math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0032" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>(</mo><msub><mi>p</mi><mrow><mo>+</mo><mn>1</mn></mrow></msub><mrow><mo>(</mo><mrow><mn>1</mn><mo>&#8722;</mo><msub><mi>p</mi><mrow><mo>+</mo><mn>1</mn></mrow></msub></mrow><mo>)</mo></mrow><mo>)</mo><mo>=</mo><mo>.</mo><mn>36</mn><mo>&#215;</mo><mo>.</mo><mn>64</mn><mo>=</mo><mo>.</mo><mn>2304</mn></mrow><annotation encoding="application/x-tex">$({p&#95;{ + 1}}({1 - {p&#95;{ + 1}}})) =.36 \times.64 =.2304$</annotation></semantics></math> .

Using these values in equation (A3.4), the revised probability distribution proportion <math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0033" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>&#960;</mi><msub><mi>R</mi><msub><mn>0</mn><mn>1</mn></msub></msub></msub><annotation encoding="application/x-tex">${\pi &#95;{{R&#95;{{0&#95;1}}}}}$</annotation></semantics></math> and the revised probability vector <math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0034" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mover accent="true"><mi>&#960;</mi><mo>&#8407;</mo></mover><msub><mi>R</mi><mrow><mn>0</mn><mrow /></mrow></msub></msub><annotation encoding="application/x-tex">${\vec \pi &#95;{{R&#95;{0{\rm{}}}}}}$</annotation></semantics></math> are computed as.3577 and (.3577.6423), respectively. The forecast for market shares at the end of the 6‐month test period is obtained from <math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0035" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>&#960;</mi><mo>&#8407;</mo></mover><msub><mi>R</mi><mrow><mn>3</mn><mspace width="0.28em" /></mrow></msub></msub><mo>=</mo><msub><mover accent="true"><mi>&#960;</mi><mo>&#8407;</mo></mover><msub><mi>R</mi><mrow><mn>0</mn><mspace width="0.28em" /></mrow></msub></msub><msup><mi>P</mi><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">${\vec \pi &#95;{{R&#95;{3{\rm{\;}}}}}} = {\vec \pi &#95;{{R&#95;{0{\rm{\;}}}}}}{P^3}$</annotation></semantics></math> . Multiplying the revised probability vector <math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0036" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>&#960;</mi><mo>&#8407;</mo></mover><msub><mi>R</mi><mrow><mn>0</mn><mspace width="0.28em" /></mrow></msub></msub><mo>=</mo><mrow><mo>(</mo><mrow><mo>.</mo><mn>3577</mn><mrow><mspace width="0.28em" /><mspace width="0.28em" /></mrow><mo>.</mo><mn>6423</mn></mrow><mo>)</mo></mrow><mspace width="0.28em" /><mrow><mi>by</mi><mspace width="0.28em" /><mi>the</mi></mrow></mrow><annotation encoding="application/x-tex">${\vec \pi &#95;{{R&#95;{0{\rm{\;}}}}}} = ({.3577{\rm{\;\;}}.6423})\;{\rm{by\;the}}$</annotation></semantics></math> third power of the above <math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0037" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mo>&#215;</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">$2 \times 2$</annotation></semantics></math> transition matrix <math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0038" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo>=</mo><mo>(</mo>.9071.0929.2594.7406<mo>)</mo></mrow><annotation encoding="application/x-tex">$P = ({ \def\eqcellsep{&}\begin{array}{@{}*{2}{c}@{}} {.9071}&{.0929}\\ {.2594}&{.7406} \end{array} })$</annotation></semantics></math> yields the probability vector (.6334.3666) of 6‐month market forecasts. The results indicate that the enhanced forecast of 6‐month market share is 63.3% for Product 1 based on the projected results from the test market, assuming other conditions remain unchanged over this period. Moreover, the predicted equilibrium or "steady state" market share is 73.6% for Product 1. These forecast results are expected to be more reliable for predicting market share since they reflect actual sample data.

To assess model efficacy, various model performance metrics need to be studied. When examining patterns in brand loyalty and switching, in addition to the well‐known churn rate (Berry & Linoff, 2000), which measures the percentage of lost customers (i.e., the percentage of brand switchers over a period of time), two other simple model performance metrics named the index of loyalty (Lipstein, 1959) and the convergence percentage (Maffei, 1960) could be employed. From the revised transition matrix <bold>P</bold>, the index of loyalty, which indicates the average length‐of‐"staying power" (in the number of time periods) for Choice i, is the reciprocal of the probability of migrating from Choice i (i.e., <math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0039" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>/</mo><mo>(</mo><mrow><mn>1</mn><mo>&#8722;</mo><msub><mi>p</mi><mrow><mi>i</mi><mi>i</mi></mrow></msub></mrow><mo>)</mo></mrow><annotation encoding="application/x-tex">$1/({1 - {p&#95;{ii}}})$</annotation></semantics></math> . The convergence percentage for Choice i is its forecasted market share in time period m compared to its forecasted market share in the equilibrium stage. A value below 100% indicates how close to the equilibrium stage the current forecast is; a value exceeding 100% indicates the percent market share expected to decline at the equilibrium stage.

From the sample of 100 purchasers in Table 4, the initial churn rate for Product 1 is 5.0% (i.e., one customer out of 20 migrated to Product 2), and the initial churn rate for Product 2 is 21.3% (i.e., 17 customers out of 80 migrated to Product 1). In addition, the initial Lipstein index of loyalty for Product 1 is 20, a high "average length‐ of‐staying power" compared to that of Product 2, which is 4.7. Moreover, from the third power of the revised matrix <bold>P</bold>, which enables the computation of the intermediate 6‐month market share forecast, the churn rate for Product 1 is predicted to be 19.2%, while the churn rate for competing Product 2 is predicted to be 53.6%. Therefore, the Lipstein indices of loyalty for the two products are predicted to be 5.2 and 1.9, respectively. Furthermore, the Maffei convergence percentage for Product 1 is 86.0%, indicating how close the intermediate 6‐month (i.e., t = 3) market share forecast of 63.3% is to the equilibrium or "steady state" market share forecast of 73.6% and yielding additional information for the ultimate decision regarding the marketing of Product 1.

Figure 5 presents a time‐series plot of the Bayesian enhanced Markov chain forecasts of percent market share for Product 1, commencing in the third month of the product's test market (i.e., time period t = 0 when the sample data is obtained) through the equilibrium period (t = 17). Similar time‐series plots could also be displayed for the other metrics—projected churn rates, brand loyalty indices, and percentage convergence to equilibrium. From Figure 5, it is observed that the "steady state" market share forecast is 73.6%. Figure 5 also depicts that the convergence to equilibrium in excess of 96% occurs by time period t = 6 (9 months after Product 1 was introduced) since its market share forecast for that time period exceeds 70.8%.

Deploying and Monitoring—APLCM Step 5

Model deployment in analytics refers to applying the final model (e.g., trained model) to a future dataset for improved decision‐making. Depending on the use and requirements, the deployment phase can be as simple as generating a report and dashboard or as complex as implementing a decision support system with a front end that runs on the cloud.

In this teaching brief, deployment refers to preparing a programming application that integrates Bayesian and Markov methods, as described previously. This application takes inputs (e.g., judgmental forecasts and a sample from product purchasers) and generates outputs such as the market share of products. Various model performance metrics are also incorporated here.

Aside from the above forecasts and performance metrics provided by a BA specialist, students are told that the A&A Corp. decision should depend on such other factors as desired market share, anticipated rate of return on investment, payback period, expected future competition, and other marketing item alternatives.

Platforms in MS Excel and Python for Classroom Use

Two platforms in MS Excel and Python are developed to automatically conduct market share forecasts from datasets using Bayesian analysis with Markov chains. The MS Excel application shown in Figures 3 and 4 can be accessed online.

In this section, the implementation of Python is described. It is important to note that another programming language named R can also be very useful for teaching Markov chains and Bayesian analysis. R contains many packages bundling together code, data, documentation, and tests. R packages allow users not to program tasks from scratch, thus enabling them to solve analytics problems in a few lines. Several R packages that can be useful in teaching Markov chains and Bayesian analysis are given below:

Using Python, several functions are created that take judgmental forecasts and sample data as their parameters and produce the transition matrix, initial forecasts, posterior probabilities, revised transition matrix, and the revised probability vector. The file that contains the source code for the Python implementation is provided online as an additional document. BA faculty can download this file and import it as a package to their Python code. Then, they can call the functions available in the given Python implementation to automatically compute <bold>Q</bold>, <bold>P</bold>, and future m‐period forecasts. This Python application is flexible and can analyze more than two product choices and also predict future forecast for any period m. The code snippet of Figure 6 illustrates the custom Python package used for implementing the integration of Bayesian and Markov methods in the case study of this teaching brief. The results of this Python application are displayed in Figure 7.

EVIDENCE OF EFFECTIVENESS OF LEARNING

What follows is a discussion of how the effectiveness of learning was assessed. At the beginning of the semester, an experiment was planned to assess the effectiveness of combining the topics of Bayesian analysis with Markov chains for forecasting purposes and also to evaluate the use of a software platform. Two sections of the Data Analytics course in a BA master's program offered by the same instructor at a research university were used—one as a "treatment" group and the other as a "control" group. It is important to note that these sections include students with similar backgrounds, pursuing a quantitative graduate degree.

In the treatment group with 35 students (n = 35), this teaching brief is utilized; in the control group with 32 students (n = 32), the topics of Bayesian analysis and Markov chains were presented disjointly and students were not given a case study to analyze using the five steps of APLCM. To measure student learning, the following four assignment questions were then given to students in both sections and a numerical rubric with scores ranging from 0 to 100 on each question was used for grading.

Hotelling's T<sups>2</sups> procedure was initially used for comparing overall student performance on the vectors of their graded responses to the four common questions, and the treatment group significantly outperformed the control group (p approximately.0000). Pooled‐variance t‐tests were then employed to compare the two sections on the graded responses to each of these four questions. The results, given in Table 5, indicate that the treatment group significantly outperformed the control group on each question, demonstrating that integrating these topics within a case study better prepares students for using Markov and Bayes' methods.

5 TABLEAssessment of Student Learning: Comparing Mean Performance between Two Groups

<table><thead><tr><th align="left">Question</th><th align="left"><p><math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0040" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msub><mi>&#956;</mi><mi>treatment</mi></msub><annotation encoding="application/x-tex">${\mu &#95;{{\rm{treatment}}}}$</annotation></semantics></math></p></th><th align="left"><p><math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0041" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msub><mi>&#956;</mi><mi>control</mi></msub><annotation encoding="application/x-tex">${\mu &#95;{{\rm{control}}}}$</annotation></semantics></math></p></th><th align="left"><p><math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0042" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><mrow><msub><mi>&#956;</mi><mrow><mi>treatment</mi><mo>&#8722;</mo></mrow></msub><mspace width="0.28em" /><msub><mi>&#956;</mi><mi>control</mi></msub></mrow><annotation encoding="application/x-tex">${\mu &#95;{{\rm{treatment}} - }}\;{\mu &#95;{{\rm{control}}}}$</annotation></semantics></math></p>&#160;</th><th align="left"><p><math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0043" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msub><mi>&#963;</mi><mi>treatment</mi></msub><annotation encoding="application/x-tex">${\sigma &#95;{{\rm{treatment}}}}$</annotation></semantics></math></p></th><th align="left"><p><math display="inline" altimg="urn:x-wiley:15404595:media:dsji12249:dsji12249-math-0044" xmlns="http://www.w3.org/1998/Math/MathML"><semantics xmlns=""><msub><mi>&#963;</mi><mi>control</mi></msub><annotation encoding="application/x-tex">${\sigma &#95;{{\rm{control}}}}$</annotation></semantics></math></p></th><th align="left"><italic>t</italic></th><th align="left"><italic>p</italic> Value</th></tr></thead><tbody><tr><td>Q1</td><td>88.51</td><td>79.24</td><td>9.27</td><td>8.13</td><td>11.71</td><td>3.79</td><td>.00</td></tr><tr><td>Q2</td><td>90.14</td><td>82.37</td><td>7.77</td><td>10.67</td><td>10.34</td><td>3.02</td><td>.00</td></tr><tr><td>Q3</td><td>94.36</td><td>80.72</td><td>13.64</td><td>9.11</td><td>14.32</td><td>4.69</td><td>.00</td></tr><tr><td>Q4</td><td>89.32</td><td>78.74</td><td>10.58</td><td>8.10</td><td>13.53</td><td>3.92</td><td>.00</td></tr></tbody></table>

It must be pointed out that the other course topics were taught in the same manner in both sections by the same instructor, and the grades given to assignments and exams were similar across the two groups. Moreover, for the treatment group, there was a fifth question requiring students to integrate Bayesian analysis with Markov chains to forecast the market share of music subscription sites (e.g., Pandora and Spotify).

The results from Question 5 indicate the students were able to effectively integrate Bayesian and Markov methods using a software platform (e.g., Python). Their mean grade was 90.34 with a standard deviation of 8.71, in line with their performance on the previous four questions.

SUMMARY AND SUGGESTIONS

Many quantitative modeling techniques, including Bayesian and Markov methods, have been employed using programming languages by analytics practitioners for data‐driven decision‐making. BA programs should incorporate these two important topics into their curricula to align industry practices with analytics education. However, there has not been much research regarding how to best present Bayesian and Markov methods in the context of analytics and teach them to BA students in the era of AI, BD, and DT. This teaching brief fills this gap by providing an analytics‐oriented pedagogical approach and by using APLCM, a cross‐industry approach for data mining that maps the steps to analyze datasets using quantitative methodologies.

Through APLCM, students are exposed to data analytics project life cycle management and have an opportunity to integrate Bayesian and Markov methods to solve a market share forecast problem. In the first step of APLCM, students are introduced to the business case and given the business terminology, ensuring that they have the required business understanding. In the second step of this approach, students are introduced to the theory of Bayesian analysis and Markov chains and taught the needed formulas. In the data collection step, students discuss various data gathering techniques and are given the dataset to use in the analysis. In the fourth step, students learn to forecast the market share of a product by integrating Bayesian analysis with Markov chains using MS Excel and Python. In the last step, which is the product deployment, students implement the entire solution using Python, a programming language of choice, and then conduct a sensitivity analysis.

To assess the effectiveness of this systematic approach, two sections of a data analytics course taught by the same instructor were selected as treatment and control groups. The treatment group used this teaching brief to learn Bayesian and Markov methods jointly through an integrating case study, while the control group was taught Bayesian analysis and Markov chains disjointly using a programming language for the specific topic applications. The results indicate that on these topics the students in the treatment group significantly outperformed the students in the control group.

In summation, this teaching brief serves as a simple illustration of how BA educators can integrate multiple analytics topics to increase teaching effectiveness and help students gain more practical skills. Future pedagogical studies aimed at integrating quantitative algorithms can use this teaching brief as a guide for developing strategies that combine the following:

GRAPH: APPENDIX 1 NOTATIONS FOR TEACHING BAYESIAN AND MARKOV METHODSTable A1. Notations Used in Bayesian and Markov TheoryAPPENDIX 2 COMBINING g EVALUATIONS BASED ON ASSIGNED WEIGHTS IN JUDGMENTAL FORECASTINGAPPENDIX 3 CLASSROOM NOTES FOR INTEGRATING BAYESIAN AND MARKOV METHODS IN THE BUSINESS ANALYTICS CURRICULATable A3.1. Two‐Way Table of Sample DataTable A3.2. Obtaining the Revised Weights for Choice 1 of the c × c Revised Transition Matrix P.

GRAPH: Table S1. Evaluations on % Repurchasing Altice Product and % Switching to Avea's ProductTable S2. CEO Assessment of Executive Evaluations on Repurchasing and Brand SwitchingTable S3. Summary of Brand Loyalty and Brand Switching Patterns for 200 Consumers

GRAPH: Table S1. Evaluations on % Repurchasing Product 1 and % Switching to Competitor's Product 2Table S2. CEO Assessment of Executive Evaluations on Repurchasing and Brand SwitchingTable S3. Summary of Brand Loyalty and Brand Switching Patterns for 100 Consumers

GRAPH: Supporting information

GRAPH: Supporting information

GRAPH: Supporting information

GRAPH: Supporting information

GRAPH: Supporting information

By Marina E. Johnson; Ram Misra and Mark Berenson

Reported by Author; Author; Author

Marina E. Johnson is an assistant professor in the Information Management and Business Analytics Department at the Feliciana School of Business at Montclair State University. Dr. Johnson completed her PhD in industrial and system engineering with a focus on data mining and machine learning from The State University of New York at Binghamton. Dr. Johnson has worked at various companies, including Hugo Boss and Comcast, and implemented optimization, simulation, and predictive modeling projects. Dr. Johnson's research interests lie in the area of applications of machine learning and optimization. Dr. Johnson continuously explores educational technology tools to increase student learning outcomes.

Ram B. Misra is a professor in the Department of Information Management and Business Analytics at Montclair State University, Montclair, NJ, USA.  Prior to joining Montclair State University, Dr. Misra was an executive director at Telcordia Technologies. Misra received his PhD in operations research from Texas A&M University, College Station, Texas. Dr. Misra has published a number of papers in scholarly journals that include IEEE Transactions, the International Journal of Management Research, the International Journal of Production Research, the Naval Logistics Review, and the Decision Sciences Journal of Innovative Education.

Mark L. Berenson is a professor emeritus in information management and business analytics at the Feliciano School of Business, Montclair State University and professor emeritus in statistics at the Zicklin School of Business, Baruch College—CUNY. Over a long academic career, Dr. Berenson received several teaching awards, authored and coauthored numerous papers, and coauthored 12 books including Basic Business Statistics: Concepts and Applications with David M. Levine, Kathryn A. Szabat and David F. Stephan (14th edition, Pearson Higher Education, 2019). He is also a former president of the New York Metropolitan Area Chapter of the American Statistical Association.