I recently finished reading Suzanne Buffam’s, A Pillow Book. This is a book of non-fiction poetry about thoughts and musings that may enter the mind as one drifts off to sleep, ranging from the historical consideration of pillows to comprehensive lists of sleeping aids.

I’ve spent more than a few nights drifting off to sleep considering the following question: How can I test fundamental properties of decision making? I want to proceed by making as few assumptions about human behavior as possible—with the end goal being to test only the property of interest—no more, no less. Admittedly, the reader may be thinking that this doesn’t sound very Bayesian. Doesn’t Bayesian analysis require even more assumptions than a classical approach? These are clearly the thoughts of a sleep-deprived individual.

Suppose we wanted to know whether an individual selecting among sleep aids at the local pharmacy acted as if she were evaluating their relative pros (sleep!) and cons (groggy the next day). Could her choices be described by a mathematical function that reflects an optimized balance of these evaluated pros and cons?

One way to proceed would be to run a choice experiment and apply the analysis from Falmagne (1978). In that seminal paper, Falmagne derived a collection of linear inequalities on choice probabilities that are both necessary and sufficient for choices to be described by optimization. Applied to our example, one of the inequalities would be:

(the probability of selecting melatonin over doxylamine) + (the probability of selecting doxylamine over bourbon) – (the probability of selecting melatonin over bourbon) < 1.

The beauty of this result is that the inequalities are general—they can be applied to any set of choice alternatives (not just sleep aids) and require very few assumptions about human behavior. If a person’s choices “satisfy” the inequalities, then she can be described as optimizing her choices, if not, then she cannot be choosing in this way. The hard part is reconciling the observed choice data with the inequalities.

The inequalities are just algebra and say nothing about the variability of the data. To carry out a statistical test to determine if the person’s choices truly conform to the inequalities, we need a statistical model, preferably one that forms the most direct bridge between the choice data and the inequalities of interest.

Why would Bayesian statistics make sense here? While a Bayesian approach requires the specification of a prior, this prior can, perhaps counter-intuitively, make the model simpler and more direct. As described by Lee and Vanpaemel in this special issue, and as already noted on this blog by Simon Farrell yesterday, a prior can be used to many useful ends, and specifying it to be “non-informative” may not always be the best choice.

In the above case, we could start with a very simple statistical model of the choice data and use a prior to encode the inequalities—thereby embedding the theory directly into the statistical model (see McCausland and Marley, 2014, for a nice application of this very idea). This is an example of using theory to determine the prior advocated by Lee and Vanpaemel. While we are making some assumptions, we would be using the prior to bring the theory closer to the data, not further from it. Even better, the prior would be adding constraints to the model, not making it more complex.

We could go even further with this idea. Suppose we wanted to detect individuals who, in their sleep deprived state, simply chose among sleep aids at random? We could go to the trouble of specifying a separate model for those individuals, complete with another likelihood function, or we could make the prior do the work for us. Following Rouder, Haaf, and Vandekerckhove, also in this special issue, we could use a “spike and slab” approach where we place additional prior weight on parameter values that correspond to guessing. Such a modification would be useful in detecting whether an individual is optimizing or guessing. In this case, the prior is handling guessing without requiring more complexity in the likelihood function itself. As discussed by Rouder and colleagues, this “spike and slab” approach is general and could be applied to whatever research question may be on your mind.

The key takeaway is that the specification of a prior in a Bayesian model can be based on far more than a simple “principle of indifference” argument. Using a theory-informed prior, it is possible to seamlessly integrate behavioural constraints into your statistical models. This has the benefit of making analysis both more direct and more interpretable.

To close, I encourage you to embrace your night-time thought wanderings. While you do so, please keep a broad perspective on how Bayesian model specification can be used to dig deep and evaluate precisely the research questions you want to answer. Below is a Buffam-esqe poem of my very own:

P1. Other musings may include whether peoples’ choices become more or less rational (Bayesian) when they are sleep deprived. I’ll save you the sleepless nights; they become less (Dickinson, Dummond, and Dyche, 2016).

References

Dickinson, D. L., Drummond, S. P., & Dyche, J. (2016). Voluntary sleep choice and its effects on Bayesian decisions. Behavioral Sleep Medicine, 14, 501-513.

Falmagne, J.-C. (1978). A representation theorem for finite random scale systems. Journal of Mathematical Psychology, 18, 52-72.

McCausland, W. J., & Marley, A. A. J. (2014). Bayesian inference and model comparison for random choice structures. Journal of Mathematical Psychology, 62, 33-46.