Inferring a mean and standard deviation

One of the most common inference problems involves assuming data following a Gaussian (also known as a Normal, Central, or Maxwellian) distribution, and inferring the mean and standard deviation of this distribution from a sample of observed independent data.

The graphical model representation for this problem is shown in Figure 4.1. The data are the n observations x1,…,xn. The mean of the Gaussian is μ and the standard deviation is σ. WinBUGS parameterizes the Gaussian distribution in terms of the mean and precision, not the mean and variance or the mean and standard deviation. These are all simply related, with the variance being σ2 and the precision being λ = 1 σ2.

Here the prior used for μ is intended to be only weakly informative. That is, it is a prior intended to convey little information about the mean, so that inference will be primarily dependent upon relevant data. It is a Gaussian centered on zero, but with very low precision (i.e., very large variance), and gives prior probability to a wide range of possible means for the data. When the goal is to estimate parameters, this sort of approach is relatively non-controversial.

Setting priors for standard deviations (or variances, or precisions) is trickier, and certainly more controversial. If there is any relevant information that helps put the data on scale, so that bounds can be set on reasonable possibilities for the standard deviation, then setting a uniform over that range is advocated by Gelman (2006).

The GCM model

The Generalized Context Model (GCM: Nosofsky, 1984, 1986) is an influential and empirically successful model of categorization. It is intended to explain how people make categorization decisions in a task where stimuli are presented, one at a time, over a sequence of trials, and must be classified into one of a small number of categories (usually two) based on corrective feedback.

The GCM assumes that stimuli are stored as exemplars, using their values along underlying stimulus dimensions, which correspond to points in a multidimensional psychological space. The GCM then assumes people make similarity comparisons between the current stimulus and the exemplars, and base their decision on the overall similarities to each category.

A key theoretical component of the GCM involves selective attention. The basic idea is that, to learn a category structure, people selectively attend to those dimensions of the stimuli that are relevant to distinguishing the categories. Nosofsky (1984) showed that selective attention could help explain previously puzzling empirical regularities in the ease with which people learn different category structures (Shepard, Hovland, & Jenkins, 1961).

We consider category learning data from the “Condensation B” condition reported by Kruschke (1993). This condition is shown in Figure 17.1, and involves eight stimuli—consisting of line drawings of boxes with different heights, with an interior line in different positions—divided into two groups of four, to make Category A and Category B stimuli. Kruschke (1993) collected data from 40 participants over 8 consecutive blocks of trials, within which each stimulus was presented once in a random order.

Throughout this book, you will use the WinBUGS (Lunn et al., 2000, 2009) software to work your way through the exercises. Although it is possible to do the exercises using the graphical user interface provided by the WinBUGS package, you can also use the Matlab or R programs to interact with WinBUGS.

In this chapter, we start by working through a concrete example using just WinBUGS. This provides an introduction to the WinBUGS interface, and the basic theoretical and practical components involved in Bayesian graphical model analysis. Completing the example will also quickly convince you that you do not want to rely on WinBUGS as your primary means for handling and analyzing data. It is not especially easy to use as a graphical user interface, and does not have all of the data management and visualization features needed for research.

Instead, we encourage you to choose either Matlab or R as your primary research computing environment, and use WinBUGS as an “add-on” that does the computational sampling part of analyses. Some WinBUGS interface capabilities will remain useful, especially in the exploratory stages of research. But either Matlab or R will be primary. Matlab and R code for every example in this book, as well as the scripts that implement the models in WinBUGS, are all available at www.bayesmodels.com.

This chapter first does a concrete example in WinBUGS, then re-works it in both Matlab and R. You should pay particular attention to the section that features your preferred research software.

The general principles of Bayesian analysis are easy to understand. First, uncertainty or “degree of belief” is quantified by probability. Second, the observed data are used to update the prior information or beliefs to become posterior information or beliefs. That's it!

To see how this works in practice, consider the following example. Assume you are given a test that consists of 10 factual questions of equal difficulty. What we want to estimate is your ability, which we define as the rate θ with which you answer questions correctly. We cannot directly observe your ability θ. All that we can observe is your score on the test.

Before we do anything else (for example, before we start to look at your data) we need to specify our prior uncertainty with respect to your ability θ. This uncertainty needs to be expressed as a probability distribution, called the prior distribution. In this case, keep in mind that θ can range from 0 to 1, and that we do not know anything about your familiarity with the topic or about the difficulty level of the questions. Then, a reasonable “prior distribution,” denoted by p (θ), is one that assigns equal probability to every value of θ. This uniform distribution is shown by the dotted horizontal line in Figure 1.1.

Now we consider your performance, and find that you answered 9 out of 10 questions correctly. After having seen these data, the updated knowledge about θ is described by the posterior distribution, denoted p (θ | D), where D indicates the observed data.

A basic challenge in understanding human cognitive development is to understand how children acquire number concepts. Since the time of Piaget (1952), the concept of number has been one of the most active areas of research in the field. This chapter focuses on one prominent current theory about the origin of integer concepts, called the “knower-level” theory (Carey, 2001; Carey & Sarnecka, 2006; Wynn, 1990, 1992).

The knower-level theory asserts that children learn the exact cardinal meanings of the first three or four number words one-by-one and in order. That is, children begin by learning the meaning of “one” first, then “two,” then “three,” and then (for some children) “four,” at which point they make an inductive leap, and infer the meanings of the rest of the words in their counting list. In the terminology of the theory, children start as PN-knowers (for “Pre-Number”), progress to one-knowers once they understand “one,” through the two-knower, three-knower, and (for some children) four-knower levels, until they eventually become CP-knowers (for “Cardinal Principle”).

There are at least two common behavioral tasks that are used to assess children's number knowledge. In the “Give-N” task, children are asked to give some number of objects, such as small toys, to the experimenter, or an experimenter substitute, such as a puppet (e.g., Frye, Braisby, Lowe, Maroudas, & Nicholls, 1989; Fuson, 1988; Schaeffer, Eggleston, & Scott, 1974; Wynn, 1990, 1992).

More than a hundred years ago, some of the world's most renowned scientists were curious whether or not people can look into the future (precognition), move objects with their mind (telekinesis), or transmit messages by thought (telepathy). These so-called extrasensory abilities were studied by members of the Society for Psychical Research, a society that included intellectual heavyweights such as William James, Carl Jung, and Alfred Wallace. Even as late as 1950, the great Alan Turing argued that his famous test for artificial intelligence should be carried out in a telepathy-proof room.

How times have changed. Scientific work on extrasensory perception, or ESP, is now conducted only by a few self-pronounced academic mavericks, on a Quixotic mission to demonstrate to the world that the phenomenon is real. In 2011, the debate on the existence of ESP was re-ignited when reputable social psychologist Dr Daryl Bem published nine ESP experiments with over 1000 subjects (Bem, 2011). On the basis of these data, Bem argued that people are able to look into the future. In Bem's first experiment, for example, subjects had to guess whether a picture was going to appear on the left or the right side of the computer screen. The location of the picture was random, and this means that, on average, subjects cannot do better than a chance rate of 50% correct—unless, of course, people can look into the future. Bem (2011) found that people guessed the upcoming location of the pictures with above-chance accuracy of 53.1%. Interestingly, this effect occurred only for erotic pictures, and was absent for neutral pictures, romantic but not erotic pictures, negative pictures, and positive pictures. It was also found that the effect was largest for extravert women.

How real is “culture”? A painful answer

Dr. Samuel Johnson is known for refuting Berkeley’s idealism by stomping his thumb against a solid stone. Whether this anecdote is historically grounded or not is of minor importance. The lesson, however, is extremely important. Some arguments and ideas are refutable or clearly confirmed when they come up against the solid rock of reality. While a naive materialist would consider stones as the hallmark of reality testing, cultural encounters may provide us with a painful experience that confirms the reality of “culture” no less than the reality of stones.

At this point you may intuitively sense my pragmatic approach: it is pain that defines our boundary in encountering reality. To say it differently, let me recall Terry Eagleton, a nonrepresentative of pragmatism, who in his book How to Read a Poem writes (2006, 8): “Reality is what turns its back upon us, resisting our infantile demand that the world should serve as our looking glass.” Well, as the following example teaches us, the world is not the looking glass of our fantasies.

From the mouse’s nose to Borges’s sculpture

The above quotation from Gulliver’s Travels is funny as it shockingly violates our cultural norms. Swift, known for his wicked sense of humor, has touched on a sensitive point. How can the smell of a beloved one be so intolerable?

For the cultural psychologist there is no greater pleasure than in showing how “natural” biological psychological processes are mediated despite their nonmediated and “natural” appearance. In this context, smell is of specific interest as it is a common belief that we have a poor sense of smell, attributed mostly to our evolutionary heritage, and that smell is such a basic perception that the role of mediation is minimal if any.