Apologies: I meant to have this run on Thursday, but a huge burst of snowy spring came through and knocked out power all yesterday morning.
First: Almost all models are correlational, i.e. conditional on premises that are not necessarily true. Fully causal models, in the full sense of formal, material, efficient, and final causes, are rare. Since models are almost always correlational, claims of cause discovered or confirmed is almost always (to put it politely) boasting.
Second: any change, any change at all, even one more or fewer observation, added as a premise, or any premise subtracted, creates a NEW model. As long as the added or subtracted premises changes the probability of some proposition of interest. This is not usual! Pay strict attention.
Uncertainty & Probability Theory: The Logic of Science
Video
Links: YouTube * Twitter – X * Rumble * Bitchute * Class Page * Jaynes Book * Uncertainty
HOMEWORK: Given below; see end of lecture.
Lecture
This is an excerpt from Chapter 7 of Uncertainty.
There is a difference, as there was for truth (necessary and conditional or local), between universal and partial or limited deterministic models. The model of the projectile was, in absence of any other information, partial; so was the red-blue-object model. Both models say propositions will be true or false given the stated conditions, but the partial model contains premises which are not (known to be) necessarily true. Deterministic models may also be over-loaded, which is when two partial deterministic models have different, not logically equivalent premises, but which make identical predictions about a set of propositions. More than one model can explain the same set of facts. But there can only be one true understanding of cause.
The goal is to discover universal deterministic models, which contain necessarily true premises and which lead to certainty and where the nature and essence of the events are understood. Given the results of quantum mechanics, it appears this goal cannot be met for efficient causes for some events. No full, universal efficient deterministic model of nature exists: if one did, it would be the prized Theory of Everything. Even though we are barred from complete knowledge, rich and useful conditional models abound.
Einstein, Podolsky, and Rosen famously said that “If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of the physical quantity, then there exists an element of reality corresponding to that quantity.” What might that mean to deterministic versus causal models?
Before you is a machine that has a dial marked 1 through 3 and a light. Moving the dial through its states and the light turns yellow, blue, white. From this you form the premise (with obvious shorthand) “If D1, yellow; if D2, blue; if D3, white.” This is a deterministic model. It says that, given certain conditions, certain other things happen with certainty. Extreme probabilities (0 and 1) are easily derived from deterministic models with the addition of a minor premise and some proposition of interest. For instance, add the minor premise “D2 (the dial is in position 2)” and propositions “The light is white” or “The light is chartreuse.” Given this model, these propositions are false. We could have also deduced, from these two premises, the proposition “The light is blue.”
Why did the model turn its various colors? I have no idea. How can the model be causal if we don’t know all the causes of some event? Because it turns out we don’t know all (as in all) the causes in any contingent event, yet we can sometimes understand essences. I don’t need to know, or even need to care about every cause of the light, either, not if all I am interested in is its color.
The model relates to propositions of the light’s colors, even though there are lots of facts about the machine and it milieu which exist but which we ignore. It is I hope obvious that some thing or things were the efficient cause of the light turning color. The dial played a role, but given our understanding of physics, we suspect it wasn’t the dial itself that made the light glow. The light glowed because certain elements were electrified as opposed to others, and the electrification was caused by the states of certain resistors, diodes, etc. And the states of those resistors etc.\ in turn were caused to be in those states by their chemical components being in a certain way. And so on down the chain, all the way to the bottom (which we saw above must exist).
That is, the best we can do is to end at uncertainty. We don’t know how the Ultimate Cause works; we don’t even know how electrons work, not entirely, and seem to be forbidden that knowledge. Quantum mechanics is, of course, a theory of uncertainty. And since uncertainty is a measure of knowledge, quantum mechanics is not a causal theory, but it is a probabilistic-deterministic one. We don’t know why this particle takes this spin rather than that spin. We can say, given a set of premises, the probability it takes this spin. But this is not to abandon causality. Some thing or things must be causing whatever happens to happen, something must be reducing the potentiality of being in the state spin up to being actually in that state. Quantum mechanics events cannot happen for no reason. It cannot be that nothing causes these events. How could it? It’s nothing. Nothing is the complete absence of anything. To say that events happen “spontaneously” in the sense of “from nothing” or “uncaused” is to fool oneself and to embrace a kind of mysticism or magic. EPR’s “element of reality” can be interpreted as the actuality, whatever it is, that actualizes the potential. Bell’s Theorem (appears to) prove that we can’t know what this is, but it cannot prove that it doesn’t exist. Nothing has no power.
Why this is the case was explored previously. The point here is that if we knew all the conditions and causes of why the light turned yellow, we would have a full or universal causal model. Since instead we have only limited knowledge (the dial positions), we have a local deterministic model. The terms universal and local apply to the completeness of the model. Since we never know everything, all scientific models are thus local. No matter how well we understand any system, we will never be able to understand why what is happening at the most foundational levels is happening. So whenever we speak of causal models, we’re always leaving something out. The degree to which we leave things out is great or small, but there is no difficulty is saying the dials “caused” the light to turn various colors, or that the “$x$” caused the “$y$” in the trajectory equation, as long as we keep in mind this is shorthand.
A local deterministic model is a collection of premises for which any proposition in relation to this collection is certainly true or false or irrelevant. Since all we ever have in science is local models, henceforth I’ll drop the “local.” The irrelevancy is necessary. Suppose we have our dial model and desire the (conditional) probability of “Mike likes tacos”. How do we know this proposition is irrelevant with respect to this model? Why aren’t we conducting an experiment to verify this? Because we discern the essence of the machine and of people like Mike, and we know via induction that the two things have nothing to do with one another. We can never escape induction.
But this isn’t quite the right flavor. Actually, in models, we usually start out with the propositions of interest and search for premises which make these propositions true or likely and others false or unlikely. The propositions of interest with our machine is “The light is yellow”, etc. Models which make any proposition true are trivially found. For instance, given “The light is yellow” it is true that “The light is yellow.” But we don’t accept this premise in our model because we are after understanding of causes. Adding the trivial premise does nothing to further our knowledge.
Accept the same model: now what if the light glowed red? Then our model is falsified because (our information insists) the dial must be in one of the three positions, and these we have said certainly lead to other colors. Suppose the dial in fact was in position 2 when the light turned red. We can then modify our model in the obvious way, allowing for the possibility of two colors in position 2. This turns the deterministic model into a probabilistic one. It also makes a new model. This point cannot be too highly stressed. Any change to a model such that the probabilities or certainties of the outcomes (propositions we put to the model) also change creates a new model, even though it might, through custom, retain its old name. We cannot say in general that adding new premises creates a new model, because we can always add necessarily true premises or even possibly irrelevant ones without changing the probabilities of propositions.
We needn’t capture everything that can happen to have a causal or deterministic model. For instance, here is an alternate model “If D1, yellow, otherwise something else.” This model works. It is even as accurate as the first model! Whether it is more or less useful, however, depends on decisions and actions taken based on the model. It is obviously less precise, but less precision can be a blessing. We might have thermometers which measure air temperature to the nearest hundredth degree, but most people will be content to knowing the value within two or three degrees (Fahrenheit). Indeed, in all problems, the measurements and models should be no finer than the decisions to be made. This point is taken up again later.
One reason why it is thought probability models can discern cause, especially in the computer sciences, is a because of an unrecognized bias. As of this writing, a hot topic is “deep learning”, which can be described as machine “learning” iterated, a suite of computational tricks to find “signals” buried in “noise.” The hope is that as datasets increase in size and complexity automated (deep “learning” or otherwise) algorithms will discover the causes of “outcomes”. The bias is exposed by thinking about who decides what goes into the databases as potential causes or proxies of causes. Consider the proposition “Bob spent \$1,124.52 on his credit card.” This “effect” might have been caused by the sock colors of the residents of Perth, say, or the number of sucker sticks longer than three inches in the town of Gaylord, Michigan, or anything. These odd possibilities are not in databases of credit card charges, because database creators cannot imagine how these oddities are in any way causative of the effect of interest. Items which “make the cut” are there because creators can imagine how these items are causes, or how they might facilitate or block other causes, and this is because the natures or essences of these items are known to some extent. The form of the blocking (see below) or the conditions of when the cause operates might not be known, but that an item plays some role in the causal process is at least suspected.
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