Probabilistic dilation

When learning more makes you less certain

Question: Is it ever possible for an analyst to face a situation where there is available data which, for every possible outcome, will increase uncertainty about some key quantity in such a way that the information has negative value to the analyst? If so, doesn't this turn the analyst into a money-pump? The analyst should pay to avoid receiving the data, and would need to keep paying to avoid receiving the data.

The question is asking about a phenomenon described by Seidenfeld and Wasserman (1993) known as probabilistic dilation. It occurs when new evidence leads different Bayesian investigators into greater disagreement than they had prior to their getting the new evidence. Such evidence is not merely surprising in the sense that it contradicts one's prior conceptions; it expands everyone's uncertainty. This effect is counterintuitive because it does not depend on what the new information is actually saying.

It's hard to explain dilation with a simple example, but let me try. Suppose Lucius Malfoy tosses a fair coin twice, but the second 'toss' depends on the outcome of the first toss. It could be that Malfoy just lets the coin ride, and the second outcome is exactly the same as the first outcome. Or he could just flip the coin over so that the second outcome is the opposite of the first. He does one of these two things, but you don't know which he will do. The outcome of the first toss is either heads H1 or tails T1. Because the first toss is fair (and no spells are cast midair), you judge the probability P(H1) = ½. Whether Malfoy lets the coin ride or flips it, you judge the probability the second 'toss' ends up heads to be the same, P(H2) = ½. So what happens when you see the outcome of Malfoy's first toss? Suppose it was a head. What is your probability now that the second 'toss' will also be a head? It turns out that once you condition on the first observation, the probability of the second toss being a head dilates. It is now either zero or one, but you don't know which. It doesn't depend on chance now; it depends on Malfoy's choice, about which you have no knowledge (unless maybe you too dabble in the dark arts). Dilation occurs because the observation H1 has caused the earlier precise unconditional probability P(H2) = ½ to devolve into the vacuous interval P(H2 | H1) = [0,1].

A medical example of dilation described by de Cooman and Zaffalon (2004) can perhaps convince you of the importance of this issue. Suppose 1 out of 100 people in a population has a disease that is easy to test for. In fact, let's say the test has perfect sensitivity and perfect specificity so that, if the test result is positive, the patient surely has the disease, and if it's negative the patient surely doesn't. If we take a random person from the population, what is the probability before any tests are done that he or she has the disease? Well, we said the prevalence was 1 out of 100, so the probability would be 1 out of 100. Now suppose we are told that the person has been tested for the disease but that the test result has gone missing. What can we now say about the probability that the person has the disease? You might think it would be reasonable to revert to the earlier answer that it's just 1 out of 100, but that conclusion is wrong. It's wrong because that conclusion depends on knowing something about why the test went missing. It would only be reasonable to say that the probability is still 1 out of 100 if the reason the test result went missing had nothing to do with its value, that is, it was missing at random. Unfortunately, it is generally quite hard to be confident about why information is not available in such cases. Suppose there's a stigma associated with having the disease, and a positive result was hidden because of the stigma. In this case, the only reason a test result might go missing could be that the test was positive. If so, then the fact that it's missing reveals that the patient certainly has the disease. But, in our ignorance, it might just as well be the case that the result was unobservable because it showed a negative value, in which case the patient is surely disease-free. Intermediate cases are also possible and thus the probability of disease may, for all we know, be anywhere in the interval [0,1], and we cannot say that one value in that range is more likely than another. This is an example of dilation because our initial probability of 1 out of 100 dilates because of the information that the patient has been tested to the vacuous statement that the probability might now be either zero or one. We started out with some pretty good knowledge about the chance the patient was healthy, but some seemingly irrelevant information forces us into total ignorance about whether the patient has the disease or not.

Another example of dilation is the infamous Monty Hall problem. Suppose you are a contestant on the game show Let’s Make a Deal and the host Monty Hall shows you three doors. You get to choose a door. There's a car behind one of the doors and goats behind the other two. If you pick the door concealing the car, you get the car as a prize. If you pick one of the doors with the goats, you get nothing. You pick a door and tell Monty, but before the door you picked is opened, Monty opens one of the other two doors revealing a goat. Monty then asks whether you’d like to change your pick to the other closed door. Should you switch to the other door to improve your chances of getting the car, or stick with the one you first picked? Many people believe that switching makes no difference, and several prominent people publicly embarrassed themselves arguing the point (Crocket 2015).

The Monty Hall problem is actually pretty subtle. When you first pick a door to open, it's reasonable to say you have a ⅓ chance of winning the car with that choice, that is, before Monty opens the door showing the goat. Once the goat is revealed, there are two doors left. As explained by de Cooman and Zaffalon (2004), the probability of getting the car depends not only on whether you stick with or switch your door, but also on why Monty chose the door he opened. It could go different ways. For instance, suppose Monty has previously decided he will open door three (if you haven't picked it) whenever the car is behind door one. In that case, if you pick door one and Monty opens door two, then the car must be behind door three and you should definitely switch doors. However, if Monty decided beforehand to open door two whenever the car is behind door one, and he opens door two after you pick door one, then there are two equally likely possibilities. The car is either behind door one and you should stick with it, or the car is behind door three and you should switch. Because you can't read Monty's mind and you don't know which plan he used, once he opens a door the probability the car is behind the door you first picked is now somewhere in the interval [0, ½], and the probability it is behind the other door is now in the interval [½, 1].

The reason the probabilities in the Monty Hall problem are intervals rather than scalar values is because we don't know Monty's decision process when he has a choice of doors to open. Notice what seeing Monty open the door has wrought. The probabilities before we see the goat were evenly spread, but they were precise. After we see the goat, the probabilities are imprecise. There's more knowledge—the three doors have been whittled down to two—but there's also a sense in which there's more uncertainty, which is that imprecision about the probabilities. Thus this is another example of dilation.

Note that this dilation gives us no practical difficulty. As a game show contestant you certainly wouldn't mind your increase in uncertainty that comes from Monty opening a door to show the goat. You couldn't become a money pump in the Bayesian sense to prevent this, even theoretically, because the imprecision is not the same as probability. Such behavior would only be reasonable in this case if you believed that all uncertainty, no matter its source or nature, must be expressed as precise probability. It seems clear, however, that uncertainties come in different flavors that are not really directly interchangeable.

In practice, dilation may generally be only an esoteric theoretical concern. I don't know of any examples of dilation in practical situations that would create any problems for analysts or decision makers (but see de Cooman et al. 2010, §7). Although dilation seems highly counterintuitive to some people, others consider it a natural consequence of the interactions of partial knowledge (Walley 1991, 298f). Perhaps the phenomenon is evidence that uncertainty is a much richer idea than is usually assumed in probability theory.

One way to avoid dilation is not to use conditionalization as the updating rule for new information. Interestingly, it is possible to do this with imprecise probabilities. Grove and Halpern (1998) point out that the standard justifications for conditionalization may no longer apply when we consider sets of probabilities. And it may turn out that conditionalization may not be the most natural way to update sets of probabilities in the first place (de Cooman and Zaffalon 2004). Instead, a constraint-based updating rule may sometimes be more sensible. It's also interesting to note that dilation does not occur in interval analysis (Seidenfeld and Wasserman 1993), which is a kind of constraint analysis.

This page draws from Ferson and Siegrist (2012) and de Cooman and Zaffalon (2004) under fair use. Send us your thoughts on dilation, or maybe examples of it that you've found, and we'll try to post them on this page. Thanks to Michael Goldstein who posed the question and to Erik Quaeghebeur. This page is one of a series of sites related to uncertainty.

References

de Cooman, G., F. Hermans, A. Antonucci, M. Zaffalon (2010). Epistemic irrelevance in credal nets: the case of imprecise Markov trees. International Journal of Approximate Reasoning 51: 1029–1052. http://dx.doi.org/10.1016/j.ijar.2010.08.011

de Cooman, G., and M. Zaffalon (2004). Updating beliefs with incomplete observations. Artificial Intelligence 159(1−2): 75−125. http://www.sciencedirect.com/science/article/pii/S0004370204000827, http://arxiv.org/pdf/cs/0305044v2.pdf

Crocket, Z. 2015. The time everyone “corrected” the world’s smartest woman. Priceonomics [blog] http://priceonomics.com/the-time-everyone-corrected-the-worlds-smartest/

Ferson, S., and J. Siegrist (2012). Verified computation with probabilities. Uncertainty Quantification in Scientific Computing, edited by Andrew Dienstfrey and R.F. Boisvert, pages 95–122, Springer, New York.

Grove, A.J., and J.Y. Halpern (1998). Updating sets of probabilities. Proceedings of the Fourteenth Conference on Uncertainty in AI, pages 173−182, http://arxiv.org/abs/0906.4332

Seidenfeld, T., and L. Wasserman (1993). Dilation for sets of probabilities. The Annals of Statistics 21: 1139−1154. http://www.hss.cmu.edu/philosophy/seidenfeld/relating%20to%20Dilation/Dilation%20for%20Sets%20of%20Probabilities.pdf

Walley, P. (1991). Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, London.