The Century of Bayes

Transcription

1 The Century of Bayes Joseph J. Retzer Ph.D., Maritz Research The Bayesian `machine together with MCMC is arguably the most powerful mechanism ever created for processing data and knowledge Berger, 2001 Introduction While most of us in marketing research have probably heard something about Bayesian Analysis, chances are we re not quite sure what it is or what, if anything, would make us want to use it. This article answers three questions about Bayesian analysis: 1. What is Bayesian analysis? After gaining an intuitive feel for Bayesian analysis, the reader will see that Bayesian analysis is surprisingly straightforward and instinctive. 2. Why should I use Bayesian analysis? This section focuses on a few of the most compelling reasons for using Bayesian analysis. th 3. Why now : given that Bayes theorem has been around since the mid 18 century, and if it in fact does lead to analyses offering numerous advantages, why has Bayesian analysis only become popular in the past 10 years? Lastly, the article briefly reviews selected applications of Bayesian analyses that have already become, or are expected to become, popular in Marketing Research. A Bit of History In 1763 the Reverend Thomas Bayes, a British mathematician and Presbyterian minister, formulated Bayes' theorem. His Essay Towards Solving a Problem in the Doctrine of Chances outlining the theorem was published posthumously by his friend and colleague, Richard Price. While his theory (also referred to as the theory of inverse probability) has been well known by both traditional and Bayesian statisticians for some time, only recently has Bayesian modeling become popular among applied researchers. As a first step toward understanding the Bayesian Revolution, we need to understand how Bayes theorem underlies Bayesian analysis in general. What is Bayesian Analysis? To begin with, Bayesian analysis is a reformulation of basic statistical theory and not a branch of statistics. This means that anything we do in traditional statistics can also be

2 done in a Bayesian framework. In fact, the flexibility underlying the Bayesian approach enables us to do many things not possible with traditional statistical analysis. The principle inherent in all of Bayesian analysis, Bayes rule, is non-controversial and follows from simple probability theory. Bayes rule is given in equation (1). PH ( D) PD ( H) PH ( ) = (1) PD ( ) Left of the equals sign reads the probability of H given D. Here we can think of H as being a distribution involving some measures ( parameters ) of interest. These might include things such as an expected value (mean), variance, regression coefficients, etc. We can think of D as the data we collect through our surveys. Bayesians quickly realized that the quantity PD ( ) in the denominator of the right hand side is unnecessary and can be ignored, since it does not involve our parameters of interest H. Hence, they often re-state the rule as, PH ( D) PD ( HPH ) ( ). (2) The symbol in this equation can be read as is proportional to. Substituting standard Bayesian terminology into equation (2), we may write: Posterior Likelihood Prior. In words, the posterior is proportional to the likelihood times the prior. Each of these pieces has fairly simple interpretation: Posterior: This is the distribution of our parameter(s) of interest. Its estimation is always our ultimate goal. Likelihood: This is where information in the data is introduced how likely is the observed data given a distribution we have in mind? Prior: This represents prior knowledge, if any, about our parameters. An intuitive description of the basic steps involved in Bayesian analysis is given as follows: 1. We begin our analysis with prior beliefs about the parameters before collecting our data. 2. Next, we collect data and represent it in terms of what is referred to as a likelihood function. 3. Finally, we update our prior beliefs using information from our newly collected data. This updated information is reflected in the posterior distribution on the parameter(s) of interest. Updating in this case implies using the Bayes mechanism or rule, in other words multiplying the prior times the likelihood.

3 Bayesian estimation and inference is ALL about updating our current beliefs with new information from the data. This describes the fundamental learning process in science as well as in everyday life. Illustrative examples include, 1. My forecast of the weather is based on prior knowledge of weather at this time of year updated by current data, e.g., cloudy vs. clear skies. 2. I base my decision on whether or not to cross the street on prior knowledge of traffic patterns updated with current data, e.g., cars on the road. Why Bayesian Analysis? How do we benefit from taking a Bayesian approach to modeling rather than using traditional statistical methodology? A few major benefits include: 1. Bayesian analysis formalizes the process of learning from data to update prior beliefs. This idea is both intuitively appealing and meaningful, but it is difficult to implement with traditional statistics. Bayes rule provides the mechanism for updating prior knowledge with new data, specifically by multiplying the prior times the likelihood. 2. It allows for the estimation of complex models which may not be possible using traditional techniques. 3. Results from Bayesian analyses are more in line with common sense interpretations. For example, consider the confidence interval provided by most standard statistical packages. Suppose we are told that a 95% confidence interval for the average overall satisfaction rating by customers, on a 10 point scale, was 5 to 7. Most would interpret that information as follows: The probability that the true average overall satisfaction lies between 5 and 7 is 95% or.95. The statement above is both a natural and intuitive way to interpret this confidence interval. However, it is also completely wrong! The correct interpretation is much more convoluted and impractical 1. Bayesian probability intervals on the other hand, may be interpreted in exactly the straightforward manner noted above. 4. The end result of any Bayesian analysis is a distribution on the parameter(s) of interest (i.e., the posterior). Why is this an advantage? Knowing the distribution of 1 The correct interpretation of this confidence interval would be to say: If we were to construct many confidence intervals (which we do not do), approximately 95% of those intervals, which are random, would contain the parameter expected value of X, average overall satisfaction.

4 a parameter provides access to all the information we may be interested in regarding that parameter. For example, with the posterior distribution we can calculate, i. Probability intervals ii. Mean, median, mode and variation iii. Distributional anomalies (e.g., outliers), iv. Linear and non-linear transformations of the estimates, etc. In other words, we don t require multiple techniques to estimate multiple parameters. For a graphical depiction of multiple characteristics pertaining to parameter estimates, based on posterior probability distributions, see Figure 1 in the section Inference on Information Theoretic Drivers: An illustration. Why now? A natural question to ask at this point is why has it only been in the past 10 years or so that we ve seen Bayesian analysis become popular? The answer is that Bayesian analysis has, until recently, been mathematically challenging. The reason Bayesian analysis can be challenging is that when estimating parameters (e.g. mean, variance etc.) using probability distributions, we often need to integrate (remember second semester calculus?). Unfortunately, many of the integrations necessary for Bayesian analysis are too complex to solve using symbolic calculus. Bayesian Simulation on the other hand, is capable of solving, through computational brute force, the complex integration problems resulting from many Bayesian models. Bayesian Simulation is based on modern sampling techniques known as Markov Chain Monte Carlo (MCMC) procedures. MCMC, made feasible through recent advancements in computational power, is capable of solving these problematic integrals which would otherwise render many Bayesian analyses intractable and therefore of little practical value. Various documents describing Bayesian Simulation in general and MCMC techniques in particular may be found on the web 2. The combination of these two phenomena, mathematical innovation (MCMC) and increased computational power, has therefore allowed Bayesian analysis to emerge as a practical solution for applied researchers. 2 An overview of Bayesian simulation (Jackman (2001)) as well as a tutorial on MCMC (Green (1999)) are freely available on the web.

5 Examples of Bayesian Analysis in Marketing Research Hierarchical Bayesian (HB) Choice-Based Conjoint (CBC) Without doubt, the biggest impact of Bayesian methods in Marketing Research has been in the area of discrete choice conjoint analysis. The methodology typically applied to these designs is referred to as Hierarchical Bayes or simply HB modeling 3. Discrete choice conjoint analysis (aka Choice-Based Conjoint (CBC)) involves asking respondents to make choices from sets of alternatives as opposed to the more traditional ratings based approach. Hierarchical Bayesian (HB) Choice-Based Conjoint (CBC): An Illustration Consider an example in which an auto manufacturer would like to discover how consumers value various automobile options such as leather interior or GPS navigation 4. A traditional ratings approach might ask respondents to rate, on a 1 to 10 scale, individual option importance. CBC would instead present respondents with a much more realistic situation in which various vehicles are described in terms of the options of interest. The respondents would then be asked to choose the alternative they most preferred. This choice task might look as follows: Please pick the automobile you would be most likely to purchase from the three described below: Alternative / Alternative / Alternative / Automobile 1 Automobile 2 Automobile 3 High power stereo Standard stereo system No Stereo Cloth interior Cloth interior Leather Interior No GPS GPS Navigation No GPS Blue tooth phone Blue tooth phone No Blue tooth phone $22,000 $23,000 $19,000 The respondent would then be presented with a number of choice tasks where the mix of options would be carefully designed to ensure sufficient comparisons of options within and across automobiles. Values (utilities) for each level of each option (e.g. how much utility is derived from having a leather interior) are then estimated allowing the auto 3 For an excellent, albeit technical, overview of Bayesian analysis in Marketing Research and Hierarchical Bayesian analysis in particular, see Rossi and Allenby (2003). A more intuitive examination of Hierarchical Bayesian analysis, as applied to discrete choice conjoint modeling in particular, is given in Johnson (2000). 4 Note, these options are typically referred to as attributes in CBC analysis.

6 manufacturer to sum utilities and determine the overall desirability of any particular configuration. Unlike traditional conjoint analysis, HB recognizes and quantifies the differences in individual s utilities. Why is knowledge of individual (disaggregate) utilities important? Consider the automobiles described above. If we are estimating the utility of high power stereo system in a new automobile, we would likely expect different values for retired seniors vs. that for teenage first time car buyers. Standard, non-hierarchical analysis (also called aggregate analysis) provides only average utility estimates across all respondents. An averaged measure likely will not accurately reflect the truth about either potential buyer. HB provides individual estimates of utility for each and every respondent. Besides its intuitive appeal, HB has been shown to produce models with superior predictive performance (see Johnson (2000)). Once we have utilities for each individual respondent, many subsequent analyses become possible. For example, we can form segments of respondents based on demographics or attitudes, and then profile the segments using attribute (options) utilities. It is also possible to segment on the utilities themselves to produce, for example, needs-based segments. Bayesian Model Updating Bayesian model updating involves repeated updating of parameter estimates, (e.g. drivers of customer satisfaction based on regression coefficients) using newly collected data. The Bayes mechanism (Bayes rule) allows us to update our knowledge of the parameters each time we collect new data while incorporating what we ve learned from past experience. This is done by simply replacing the prior distribution in the current iteration with the posterior from the previous one. Bayesian model updating therefore provides a natural framework for analyzing tracking study data and likely will become quite popular for this purpose in the future. Bayesian Model Updating: An Illustration An excellent example underscoring the usefulness of Bayesian Model Updating is based on the modeling needs of a multi-national high tech. equipment supplier. The client required ongoing measures of importance pertaining to various equipment and sales attributes in terms of how they impacted customer loyalty. These measures were required for multiple business units across various countries and regions for a number of products. In addition, client needs dictated the measures be based on accumulated evidence (i.e., not solely a function of the current snapshot of data). The measures also were to be updated quarterly with fast turn around. In order to address client needs, initial Bayesian regression beta estimates are calculated using first wave data. In subsequent quarters, the information in the new data collected is

7 combined with previous information by performing a standard Bayesian regression analysis using the previous quarter s posterior distribution as the current periods prior. This approach directly implements the concept of learning from new data by combining it with past experience. In practical terms, it also eliminates the need to merge numerous past data files together with the current file in order to re-run the analysis using the entire accumulated data set. In other words, the Bayesian Updating procedure offered a more real-time analysis than what might otherwise have been done. Inference on Information Theoretic Drivers Bayesian inference has been applied to information measures of driver importance as suggested originally by Theil and Chung (1988) and extended by Soofi et. al. (2000). Basically, the model estimates the relative information importance averaged over all orderings of the explanatory variables. This provides a much more intuitively reasonable, if somewhat unfamiliar measure of relative importance. Bayesian inference applied to information theoretic measures (see Retzer (2003)) provides answers to questions such as: Can we identify attribute importance significantly greater than 0? Can we identify significant differences in attribute importance? Can we detect changes in importance over time? Inference on Information Theoretic Drivers: An Illustration An example involving standard satisfaction survey questions is graphically depicted in Figure 1 below. This example characterizes the impact of typical drivers of Overall Satisfaction with the client company. Some drivers are more directly related to the clients product (Quality, Value for the Money) while others relate more to the characteristics of the firm and its operations (Image, Service). The box-and-whisker plot efficiently summarizes numerous characteristics of drivers of overall satisfaction. Specifically, mean, median, variation, probability intervals, outliers and relative importance are all immediately discernable from the graph. Note that, unlike confidence interval plots, the data above are reflective of the actual distribution of each measure of importance and therefore have a more meaningful interpretation. In interpreting the results depicted in Figure 1, a number of inferences can quickly be made. First, Satisfaction with Quality is clearly most important in driving overall satisfaction although it is not significantly different from Satisfaction with Value for the Money (since the boxes overlap horizontally). Also, it is clear that while both

8 Figure 1: Information Theoretic Drivers of Overall Satisfaction Posterior Density Summary = Mean = Median Information Percent Satisfaction with Quality Satisfaction with Image Satisfaction with Value for the Money Satisfaction with Service Satisfaction with Image and Satisfaction with Service play lesser roles in determining Overall Satisfaction, the importance of Satisfaction with Image is more consistently low (we re therefore surer of its impact) than that of Satisfaction with Service since its box is more compressed. Bayesian Model Averaging / Bayesian Model Selection Bayesian Model Averaging (BMA) and Bayesian Model Selection (BMS) are useful techniques applicable in a variety of settings 5. An important example is as an alternative to the often used, theoretically flawed, set of techniques known as collectively as stepwise regression. In stepwise regression, variables are added (or removed) one at a time based on significance. At each step variables are evaluated to see if they meet the criterion for staying. The criterion for retention is often based on p-values associated with individual drivers. Bayesian Model Averaging (BMA) employs multiple models when arriving at estimates of driver impact. BMA uses posterior model probabilities as weights in averaging these models. This approach directly accounts for model uncertainty ignored in the stepwise approach. In other words, rather than choosing a single regression model (i.e., with a specific set of variables) as 100% certain as in stepwise analysis, BMA may average beta parameter estimates over, for example, the top 5 most likely models. 5 See Raftery et. al. (1997) and Raftery (1995) for in depth presentations of BMA and BMS respectively.

9 The end result of BMA/BMS analysis is a reduced model (fewer drivers than what we start with) with typically superior out of sample predictive performance than what would have been produced using stepwise analysis. Summary Bayesian theory has been well known for some time however only recently has it begun to achieve popularity in numerous fields one being Marketing Research. Its recent emergence may be attributed to two factors, Advances in computational power and Development of sophisticated sampling techniques, i.e., Markov Chain Monte Carlo. Bayesian analysis is applicable to all models currently used in Marketing Research and, in addition, to some which are un-estimateable using traditional techniques. While Hierarchical Bayes analysis applied to discrete choice modeling represents the bulk of Bayesian analytics in Marketing Research today, the opportunity for taking advantage of the Bayesian approach exists elsewhere as well. Bayesian analysis adds greatly to insights in areas such as drivers analysis, model selection, model averaging, updating and prediction to name but a few. With continued increase in computational power, in conjunction with expanding interest in the Bayesian approach, it is likely that the Bayesian revolution in Market Research will continue to flourish for some time to come. Some say this will be the Century of Bayes. Well we ll see. Closing statement made by Dr. Arnold Zellner at a talk given at the University of Wisconsin-Milwaukee entitled: Bayesian Point and Turning Point Forecasting Techniques with Applications, May 3, 2002.

10 References Green, P.J. (1999), A Primer on Markov Chain Monte Carlo. ( Jackman, S. (2001), Estimation and Inference Via Bayesian Simulation: A Practical Introduction. Herbert M. Blalock Lecture Series on Advanced Topics in Social Research, Stanford University. ( Johnson, R.M. (2000), Understanding HB: An Intuitive Approach. Sawtooth Software, Inc. March. ( Raftery, A. E.(1995) Bayesian Model Selection in Social Research. Sociological Methodology. Raftery, A. E., D. Madigan and J. Hoeting (1997) Bayesian Model Averaging for Linear Regression Models. Journal of the American Statistical Association, 92: Retzer, J.J. (2003) Bayesian Inference on the Information Importance of Explanatory Variables. Presented at the IEE Conference in Honor of Arnold Zellner, Washington D.C, September ( Rossi, Peter E. and Greg M. Allenby (2003) "Bayesian Statistics and Marketing." Marketing Science, 22, Soofi, E. S., Retzer, J. J. & Yasai-Ardekani, M. (2000). A framework for measuring the importance of variables with applications to management research and decision models. Decision Sciences Journal, 31, Number 3, Theil, H., & Chung, C. (1988). Information-theoretic measures of fit for univariate and multivariate linear regressions. The American Statistician, 42,