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Video
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Lecture
This is an excerpt from Chapter 10 of Uncertainty.
For those who are cheating and starting at the end [of the book], or for those in want of a review, these brief [a relative term] comments on the goal of modeling. The overarching goal of all models is to understand cause. What caused the contingent proposition of interest Y to be true? Was it X? Something else? Knowledge of cause is provided by the comprehension of essence, power, and nature; it is the kind of knowledge which moves from the particular to the general. Knowledge of cause is thus always provided through some form of induction, which is why machines can never discover cause. Machines [including AI!] do not have the power to make inductive leaps, no matter how cleverly they are programmed; machines do not have intellects and cannot, however fleetingly or minutely, grasp the infinite as beings with intellects can.
Knowledge of cause is never complete and rarely full. We can understand that it is gravity that is causing apples to fall without having total comprehension that space is being warped, though if we do know that, and much more besides, we can make better predictions over a wider variety of circumstance. Knowledge of cause, even though it is apparently closed off to us in its primary mechanism and in the very small, should be restored and understood by all as the primary purpose of science.
Second to understanding of cause, and often an apt substitute, is knowledge of determination. If we know the cause of Y, we also know what determines it. But knowing only what determines Y does not imply we also understand the cause of Y. Instrumentalist accounts of science are deterministic, in the sense I am using the word, as in ascertains. We can have an equation which, fed by X, determines uniquely and precisely the status of Y, but this does not mean X is a cause of Y. Given Y is contingent and because of under-determination, a poor phrase, it could be that what we thought was a cause of Y was not. Under-determination says that, in essence, the contingent always has another explanation, where by explanation I mean determination. Y only has one cause; it’s our knowledge of that cause that is open to doubt. Once again, we meet the distinction between ontology and epistemology. Instead of under-determination, perhaps a suitable replacement is just contingency. The danger is that explanation or determination will be mistaken for cause, somebody might think Y has more than one cause. Of course, most contingent, or observable, Y do have many forces, i.e. causes, acting in concert to being Y about, but that is only because Y is not simple.
Take Y = “The atmospheric temperature is y”. This is not a simple proposition because atmospheric temperature is the product of a vast number of constituents acting in concert, both in the air and at the air-measurement-apparatus junction. There are a plethora of causes of Y. For some measurements, in precisely controlled experiments of isolated (a relative term) environments, these causes may be sought. But the finer and more isolated these experiments grow, the greater the singularity, so to speak, of Y disappears and we must speak not only of Y but of the measurement of Y simultaneously. Cause can be influenced by measurement—and not only at the very small. How does an anthropoligist measure behavior without influencing it? Answer: he cannot. For most measurements it is hopeless to identify all its causes. Instead, a deterministic approach is taken. With temperature, physicists speak of ensembles” of gases or of “statistics”; meteorologists might speak of “parcels” or convection. These are fine approaches, as long as it is remembered, which it often is not, that these terms are deterministic and not causal. They are conglomerations of cause. Knowledge or understanding of cause can be and is had by studying these conglomerations, but the danger of over-certainty is ever present.
The danger is never avoided with probability models, the third level of modeling. Almost universally, users of probability models believe they have identified causes. This, as examples below will prove, is the cause (another cause!) of pandemic over-certainty. Causal models are deductions, and so there is no uncertainty, unless outside events add causes which were unplanned for. Causal models speak ontologically. Deterministic models also have no uncertainty. They say things will happen with certainty. Deterministic models speak epistemologically, though, and not ontologically. Outside causes intrude in deterministic models, too. We have an deterministic equation which says the projectile will certainly be at y at time t. But the location is measured as y + epsilon. This model is formally, strictly falsified since something it said was impossible happened. But everybody treats all deterministic models, and most causal ones, even those models for the very small, in for example quantum mechanics, in probabilistic terms. Look at any argument showing why the results from some collider confirms a deterministic (and probably not causal) projection, even though the observations do not lie precisely exactly with-no-deviation on the curve theory predicted. The departures from the curve do not have to be probabilistically quantified, though sometimes they are. Not all probability is quantifiable. Of course, it can only be, as I said, that if the observations do not lie precisely on the theory-determined curve, outside causes must have intruded if the theory is true. These outside causes are shrugged off even though they provide strict falsification, if we are asking whether and not assuming that the theory is true. The shrugging off is acknowledgement that complete understanding of all causes in any situation will always be lacking. Understand that the probability spoken of in these shruggings off is of the measurement and not the theory, which is assumed true.
Now probability models say nothing about cause or determination, but an unfortunate and curious culture has developed that associates parameters in probability with cause. Causal hypotheses, or theories, are accepted or rejected depending on the value estimates (or functions of these estimates) of these non-observable parameters take. Even if this made any sense, which I proved earlier does not since these tests are based on the fallacy of the false dichotomy, it cannot be that the certainty one has based on parameters could be as strong as the certainty one has in actual observables. This statement applies to judgements made of “effect size” and the like, too, which also speak of unobservable parameters. It is never remembered that one can be as certain as one likes in the value of a parameter, while remaining mostly ignorant about the observable itself. The certainty (or rather uncertainty) in parameters is always taken as certainty in the observables. Some consequences of this mistake are detailed below. But note that this is only some. As I said above, an entire book can be written on the abuses generated by parameter-based statistical methods. For instance, I do not here criticize the methods of so-called structural equations, factor analysis, machine learning, so-called neural nets, and many, many others. But once the material here is absorbed, it doesn’t take much work to identify their flaws.
We’re sick of hearing about this, but these probability models do not identify cause. They may help understand cause, in a weak way, and they lend some interest to comprehending determination, but their goal in life is only to say how our uncertainty in Y is related to some premises X. And that’s it. This is all they do. Especially when the “Ys” are related to human behavior, probability models are mute on cause. Just what are all the causes of a person receiving, say, an income of y dollars a year? Can you even imagine a successful argument which defends such a list? No. Probability models examine observables like physicians used to examine patients before the advent of surgery. Only the vaguest perceptions as to what is really happening can be had. It is therefore grieve-worthy that all who use probability models speak with such monstrous confidence. It is all a bluff, though. Bluffs that ought to be called. How?
I promised at the beginning a discussion of the We-Must-Do-Something fallacy, a malignant tumor in the body of science. Probability models, just like their causal and deterministic brethren, can and must be verified. That means only one thing: making predictions of observables never before seen. I mean never as in never. Known observables, in concert with facts already known or assumed, are used to posit a model. Since any set known set of observables can be predicted with absolute certainty by the trick of arranging clever post hoc premises, we cannot trust any measure of how close a model “fits” or explains what is already known. These measures have some slight utility in judging between potential models, yes, but as guides to the future (rather, to the unknown) they are very nearly useless. Instead, models should be posited and then predictions made. If the model makes useful predictions—where usefuleness is related to the decisions one makes with a model—the model is good, else it is bad. As I said earlier, this technique is slower than the old way of hypothesis testing, parameter estimation, and the like, which are methods fecund in “results.” And it is results which are desired by the We-Must-Do-Something fallacy. This is a fallacy that applies to the generation of results as well as the decisions made by or on the results themselves. How easy it is to say “X cause Y because of this wee p-value”! How very productive is the research which flows! No. It’s past time to slw down. What was the old saying? Trust but verify. That is how all research, particularly that on human behavior should be conducted.
Given that admonition, which I wish I could make stronger because I am sure that many who are reading it who agree that the other guy often makes these mistakes, but he won’t see the plank in his own eye, there are some general comments to made about “model building” which I haven’t elsewhere spoken of. Suppose a set of premises includes the explanatory (not causal) premise sex; call this set of premises X. From that, we can form Pr(Y|X). This is one model. But then, because of internal measures of “fit”, the researcher decides to drop sex from the premises. Call this reduced set X(-s). Speaking correctly, we now have two different models. Any change to a set of premises (such that the change produces a new set where the old set cannot be deduced from the new reduced set) produces a new model. Often in science the model will retain its old name, as a convenience. But it is a new and different model just the same. This name-retention makes it difficult to know what is being assessed. Is it the old model or the new one with the same name? But that is merely bookkeeping. Since probability is not decision, and it may be unclear what decisions are to be made on the models X and X(-s), what is the researcher to do? Well, if he is not the decision maker, he simply makes two predictions, one conditional on X and one conditional on X(-s). We then sit back and wait for reality to decide (conditional on the decision to be made) which is better. Whatever we do, we must never ever never decide on a science-wide criterion of relevance. Relevance depends on decision, and that can never be anything but individualistic.
In the language used here a theory is a model is a hypothesis is a supposition. All predictions, since they are always conditional, are scenarios are forecasts. And so forth. Now there comes a time when a theory is either believed or beloved but it hasn’t yet met the test of reality, which is the comparison with theory predictions with observables, as outlined in the previous Chapter. This is natural. Theories are to their creators like the statue was to Pygmalion: beautiful objects of love, as stated before. It is thus perfectly understandable that theory creators defend their creations. But they cannot do it with probability. Here is a cautionary tale of one attempt. Though it applies to the so-called multiverse, everything said below applies to theories or models of any kind.
There is in physics whether the so-called mutliverse exists. It is thought to by some who support a certain theory from which the multiverse is deduced; about the multiverse,. One physicist, Joseph Polchinski, wanted to prove the existence of the multiverse using probability. From his notable paper:
To conclude this section, I will make a quasi-Bayesian estimate of the likelihood that there is a multiverse. To establish a prior, I note that a multiverse is easy to make: it requires quantum mechanics and general relativity, and it requires that the building blocks of spacetime can exist in many metastable states. We do not know if this last is true. It is true for the building blocks of ordinary matter, and it seems to be a natural corollary to getting physics from geometry. So I will start with a prior of 50%. I will first update this with the fact that the observed cosmological constant is small. Now, if I consider only known theories, this pushes the odds of a multiverse close to 100%. But I have to allow for the possibility that the correct theory is still undiscovered, so I will be conservative and reduce the no-multiverse probability by a factor of two, to 25%. The second update is that the vacuum energy is nonzero. By the same (conservative) logic, I reduce the no-multiverse probability to 12%. The final update is the fact that our outstanding candidate for a theory of quantum gravity, string theory, most likely predicts a multiverse. But again I will be conservative and take only a factor of two. So this is my estimate for the likelihood that the multiverse exists: 94%.
Without taking any opinion on the existence of the multiverse [which I elsewhere do], let the theory, i.e. the very complex set of premises, which include a vast array or metaphysical, physical, and mathematical propositions, from which we can deduce the multiverse be called T. T is a complex proposition, and we are interested in whether T itself is true. Why? Because we know the multiverse is true if T is: the multiverse is a deduction or theorem of T. Polchinski wants to bring in Bayesian theory to answer whether T is true. That was mistake number one.
Mistake two is this statement, “I will start with a prior of 50%.” This makes no sense. Theories do not have probabilities. And since theories are nothing but (complex) propositions, neither do propositions have probabilities. Indeed, no thing has a probability. Probabilities are measures of knowledge, therefore they have to come equipped with gauges, i.e. conditions. In other words, all probability is conditional, as earlier proved.
Many think one natural gauge is the proposition W = “T might be true”, which is logically equivalent to T is true or it is false”. Both of these are tautologies, which we know are true conditional on our knowledge of logic and understanding of English grammar. But it makes no sense to say, as Polchinski said, Pr(T | W) = 50%. Tautologies are non-informative. The best we can do, as I pointed out earlier, is to deduce T’s contingency, which gives it a unit interval probability, sans endpoints. Of course, Polchinski may not have had the tautology in mind, but some other gauge. Call this G, which relates to come complex proposition in Polchinski’s head. Then it might be true that Pr(T|G) = 50%.
But what would this G have to look like? Well, it would have to be directly probative of T itself, which means of the propositions of which T is composed. And if Polchinski really had such a G, it is more plausible these G-propositions would already be in T to give it support. Why withhold from T knowledge relevant to multiverses? It doesn’t make sense. But then G might have nothing probative to say about T except its contingency like W, in which case 0 < Pr(T|G) < 1.
According to the rules of probability, Pr(T false | G) = 1 – Pr(T| G). But what does it mean to say T is false? Just that at least one of propositions within T is false. And if we knew that, then we would never entertain T. We would instead modify T (which really means making a brand new T) to remove or transform these troublesome propositions. If G told us which part of T was wrong we would fix it. Put all this another way. If all we had in contention for the multiverse was T, then T is all we have. We can’t judge its truth or falsehood because we have nothing to compare it to. T is it. It’s T or bust.
I’m sure (though I didn’t check) Polchinski’s numerical calculations are on the money, but the end result is meaningless. T has to be compared not against some internal gut reaction, because there is no such thing as subjective probability, but against the predictions T makes or against rival theories, which provide the only natural comparators. That is, Polchinski might have some alternative theory M in mind, a rival to T such that, given M, the multiverse is not a theory of M. Polchinski’s G makes a little more sense.
There may be, and almost certainly are, overlapping elements of T and M, sub-propositions which they share. Nevertheless, T is not deducible from M, nor vice versa, else they would be the same theory. We’ve already seen that it makes little sense to have in G propositions which duplicate the multiverse-predictability propositions in T, and the same objection applies to M. That means G is something else. The simplest would be the “freshman” G, which is There are two rival theories, T or M, and only one of which can be true (and is the only thing a freshman in physics is capable of knowing). Therefore, Pr(T|G) = Pr(M|G) = 1/2, via the statistical syllogism. But that’s as far as we can go without additional evidence, such as observations of the multiverse (which won’t be had, it is claimed) or via observables deducible from T or M. Other G are possible, but it is easy enough to see, since there is no such things as subjective probability, we’re up against the unquantifiable. Gut feeling as a decision takes the place of probability. In other words, it’s better to go out and find proof of T or M.
Naturally, everything said holds for theories of any kind, as I said. Readers will also see in the criticism of Polchinski the standard argument why hypothesis testing, whether by wee p-value or Bayes’s rule, is based on the fallacy of the false dichotomy.
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