September 22, 2008 | 35 Comments
(This essay will form, when re-written more intelligently, part of Chapter 15, the final Chapter, of my book. Which is coming….soon? The material below is not easy nor brief, folks. But it is very important.)
To most of you, what I’m about to say will not be in the least controversial. But to some others, the idea that not all risk and uncertainty can be quantified is somewhat heretical.
However, the first part of my thesis is easily proved; I’ll prove the second part below.
Let some evidence we have collected—never mind how—be E = “Most people enjoy Butterfingers”. We are interested in answering the truth of this statement: A = “Joe enjoys Butterfingers.” We do not know whether A is true or false, and so we will quantify our uncertainty in A using probability, that is written like this
#1 Pr( A | E )
and which reads “The probability that A is true given the evidence E”. (The vertical bar “|” means “given.”)
In English, the word most at least means more than half; it could even mean a lot more than a half, or even nearly all—there is certainly ambiguity in its definition. But since most at least means more than half, we can partially answer our question, which is written like this
#2 0.5 < Pr( A | E ) < 1
and which reads "The probability that A is true is greater than a half but not certain given the evidence E.” This answer is the best we can do with the given evidence.
This answer is a quantification of sorts, but it is not a direct quantification like, say, the answer “The probability that A is true is 0.673.”
It is because there is ambiguity in the evidence that we cannot completely quantify the uncertainty in A. That is, the inability to articulate the precise definition of “most people” is the reason we cannot exactly quantify the probability of A.
The first person to recognize this, to my knowledge, was John Maynard Keynes is his gorgeous, but now little read, A Treatise on Probability, a book which argued that all probability statements were statements of logic To Keynes—and to us—all probability is conditional; you cannot have a probability of A, but you can have a probability of A with respect to certain evidence. Change the evidence and change the probability of A. Stating a probability of A unconditional on any evidence disconnects that statement from reality, so to speak.
Other Theories of Probability
For many reasons, Keynes’s eminently sensible idea never caught on and instead, around the same time his book was published, probability theory bifurcated into two antithetical paths. The first was called frequentism: probability was defined to be that number which is the ratio of experiments in which A will be true divided by the total numbers of experiments as that number of experiments goes to infinity1. This definition makes it difficult (an academic word meaning impossible) to answer what is the probability that Joe, our Joe, likes Butterfingers. It also makes it difficult to define the probability for any event or events that are constrained to occur less than an infinite number of times (so far, this is all events that I know of).
The second branch was subjective Bayesianism. To this group, all probabilities are experiences, feelings that give rise to numbers which are the results of bets you make with yourself or against Mother Nature (nobody makes bets with God anymore). To get the probability of A you poll your inner self, first wondering how you’d feel if A were true, then how you’d feel if A were false. The sort of ratio, or cut point, where you would feel equally good or bad becomes the probability. Subjective Bayesianism, then, was a perfect philosophy of probability for the twentieth century. It spread like mad starting in the late 1970s and still holds sway today; it is even gaining ground on frequentism.
What both of these views have in common is the belief that any statement can be given a precise, quantifiable probability. Frequentism does so by assuming that there always exists a class of events—which is to say, hard data—to which you can compare the A before you. Subjective Bayesianism, as we have seen, can always pull probabilities for any A out of thin air. In every conceivable field, journal articles using these techniques multiply. It doesn’t help that the many times probability estimates are offered in learned publications, they are written in dense mathematical script. Anything that looks so complicated must be right!
The problem is not that the mathematical theories are wrong; they almost never are. But because the math is right does not imply that it is applicable to any real-world problems.
The math often is applicable, of course; usually for simple problems and in small cases the results of which would not be in much dispute even without the use of probability and statistics. Take, for example, a medical trial with two drugs, D and P, given to equal numbers of patients for an explicitly definable disease that is either absent or present. As long as no cheating took place and the two groups of patients balanced, then if more patients got better using drug D, that drug is probably better. In fact, just knowing that drug D performed better (and no cheating and balance) is evidence enough for a rational person to prefer D over P.
All that probability can do for you in cases like this is to clean up the estimates of how much better D might be than P in new groups of patients. As long as no cheating took place and the patients were balanced, the textbook methods will give you reasonable answers. But suppose the disease the drugs treat is not as simply defined. Let’s write what we just said in mathematical notation so that certain elements become obvious.
#3 Pr ( D > P | Trial Results & No Cheating & Patients Like Before) > 0.5.
This reads, the probability that somebody gets better using drug D rather than P given the raw numbers we had from the old trial (including the old patient characteristics) and that no cheating took place in that trial and the new patients who will use the drugs “look like” the patients from the previous trial, is greater than 50% (and less than certain).
Now you can see why I repeatedly emphasized that part of the evidence that usually gets no emphasis: no cheating and patients “like” before. Incidentally, it might appear that I am discussing only medical trials and have lost sight of the original thread. I have not, which will become obvious in a moment.
Suppose the outcome of applying a sophisticated probability algorithm gave us the estimate of 0.72 for equation #3. Does writing this number more precisely help if you suppose you are the doctor who has to prescribe either D or P? Assume that no cheating took place in the old trial, then drug D is better if the patient in front of you is “like” the patients from the old trial. What is the probability she is so (given the information from the old trial)?
The word like is positively loaded with ambiguity. Not to be redundant, but write out the last question mathematically.
#4 Pr ( My patient like the others | Patients characteristics from previous trial)
The reason to be verbose in writing out the probability conditions is that it puts the matter starkly. It forces you, unlike the old ways of frequentisim and subjective Bayesianism, to specify as completely as possible the circumstances that form your estimate. Since all probability is conditional, it should always be written as such so that it is always seen as such. This is necessary because it is not just the probability from equation #3 that is important, equation #4 is, too. If you are the doctor, you do not—you should not—focus solely on probability #3 because what you really want is this:
#5 Pr ( D > P & My patient like before | Trial Results & No Cheating & Patients Character)
which is just #3 x #4. I am in no way arguing that we should abandon formal statistics which produces quantifications like equation #3. But I am saying that since, as we already know, exactly quantifying #4 is nearly impossible, we will be too confident of any decisions we make if we, as is common, substitute probability #3 for #5 because, not matter what, the probability of #3 and #4 both is always less than the probability of #3.
Appropriate caveats and exceptions are usually delineated in journal articles when using the old methods, but the results are buried in the text, which causes them to be weighed more or less importantly, and which give the reader a false sense of security. Because, in the end, we are left with the suitably highlighted number from equation #3, that comforting exact quantification reached by implementing impressive mathematical methods. That final number, which we can now see is not final at all, is tangible, and is held on to doggedly. All the evidence to the right of the bar is forgotten or downplayed because it is difficult to keep in mind.
The result to equation #3 is produced, too, only from the “hard data” of the trial, the actual physical measurements from the patients. These numbers have the happy property that they can be put into spreadsheets and databases. They are real. So real that their importance is magnified far beyond their capacity to provide all the answers. They fool people into thinking that equation #3 is the final answer, which it never is. It is always equation #5 that is important to making new decisions. Sometimes, in simple physical cases, probabilities #3 and #5 are so close as to be practically equal; but when the situation is complex, as it always is when involving humans, these two probabilities are not close.
Everything That Can Happen
The situation is actually even worse than what we have discussed so far. Probability models, the kind that spit out equation #3, are fit to the “hard data” at hand. The models that are chosen are usually picked because of habit and familiarity, but responsible practitioners also choose the models so that they fit the old data well. This is certainly a rational thing to do. The problem is that, since probability models are only designed to say something about future data, the old data does not always encompass everything that can happen and so we are limited in what we can say about the future. All we can say for certain is what has happened before might happen again. But it’s anybody’s guess whether what hasn’t happened before might happen in the future.
The probability models fit the old data well, but nobody can ever know how well they will fit future data. The result is that over reliance on “hard data” means that probabilities of extreme events are underestimated and mundane events overestimated. The simple way to state this is the system is built to engender overconfidence.2
You’re still the doctor and you still have to prescribe D or P (or nothing). No matter what you prescribe something will happen to the patient. What? And when? Perhaps the malady clears up, but how soon? Perhaps the illness is merely mitigated, but by how much? You not only have to figure out what treatment is better, but what will happen if you apply that treatment. This is a very tricky business, and is why, incidentally, there is such a variance in the ability of doctors.3 Part of the problem is explicitly defining what is meant be “the patient improves.” There is ambiguity in that word improve, in what will happen with either of the drugs is administered.
There are two separate questions here: (1) defining events and estimating their probability of occurring and (2) estimating what will happen given those events occur. Going through both of the steps is called computing a risk or decision analysis. This is an enormously broad subject which we won’t do more than touch on, only to show where more uncertainty comes in.
We have already seen that there is ambiguity in computing the probability of events. The more complex these events the more imprecise the estimate. It is also often the case that part (2) of the risk analysis is the most difficult. The events themselves cannot be articulated, either completely or unambiguously. In simple physical systems they often can be, of course, but in complex ones like the climate or ecosystems they are not. Anything involving humans is automatically complex.
Take the current (!) financial crisis as an example. Many of the banks and brokerages failed to both define the events that are now happening, and they extent of the cost of those events. How much will it cost to clean it up? Nobody knows. This is the proper answer. We might be able to bound it—more than half a billion, say—and that might be the best anybody can say (except that I have been asked to pay for it).
Too Much Certainty
What the older statistical methods and the strict reliance on hard data and fancy mathematics have done is to create a system where there is too much certainty when making conclusions about complex events. We should all, always, take any result and realize that they are conditional on everything being just so. We should realize those just so conditions that obtained in the past might not in the future.
Well, you get the idea. There is already far too much information to assimilate in one reading (I’m probably just as tired of going on and on as you are of reading all this!). As always, discussion is welcome.
1Another, common, way to say infinity is the euphemism “in te long run”. Keynes has famously said that “In the long run we shall all be dead.” It’s always been surprising to me that the same people who giggle at this quip ignore its force.
2There is obviously a lot more to say on this subject, but we’ll leave it for another time.
3A whole new field of medicine has emerged to deal with this topic. It is called evidence based medicine. Sounds good, no? What could be wrong with evidence? And it’s not entirely a bad idea, but there is an over reliance on the “hard data” and a belief that only this hard data can answer questions. We have already seen that this cannot be the case.