How to Think about Correlation?

Good science isn't restricting the entire universe to two possible hypothesis. Think about it logically. What scientist can publish a paper that says that genes don't predispose anyone to cancer or smoking because he just observed smoking correlates with tar?

You literally say smoking must cause tar because it's the only possibility in your two hypothesis... one must occur. Sure if this was math and you defined your universe to be one of two possibilities... but no person in the real world will take that seriously.

In math you don't introduce ad-hoc axioms and rules to your formal analysis, but the scientific method exists because all observation in the real universe is ad-hoc. Anything goes and that's why the best way to make sense of it is through correlation.

Which brings me full circle back to my original method. Can you introduce any "adhoc hypothesis" to render a causation established by directly controlling the causative event invalid? You can't. Therefore the actual formal scientific way to determining causation is by placing yourself in the system.

The only way the graph method can be applied to the real world is if you map out every possible scenario and assumption into that graph. Then you can say based off of what we know and assume A must cause B. But more often then not a scientist is likely just establishing some correlations and coming to a qualitative conclusion that takes into account all the assumptions and hypothesis that we're aware about in the real world.

Absolutely. Science is about using mathematics and logic to abductively reason about and tease out underlying facts. In fact I see provaility in science and causation as orthogonal issues:

Consider a stable supplied voltage and a variable voltage divider (potentiometer). Suppose measuring the voltage once has intrinsic RMS voltage noise associated with it. So while we can never prove experimentally the exact value of the voltage at the voltage divider with zero error, we can drive the error arbitrarily low by oversampling, which means repeating the experiment over and over. In the general case we never actually reach the true value, but that doesn't mean "voltage can not be measured". Trying to measure this voltage inappropriately (say outside the reference range of an ADC, or with an ADC with too low a resolution such that the noise is systematically rounded in the same direction doesn't mean it's impossible to measure voltage, just that the measurement attempt is flawed.

Similar with causation. Of course caution is always advised with naive correlation. The problem is not propagation of such caution, but the pretense that no techniques exist to gather evidence of causation, and the omission of such techniques when sharing such a rant.

You have to mention this explicitly. Because it's also presumed not controversial that genetics can predispose someone to cancer. Additionally if you restrict your universe to two hypothesis but then suddenly I can pick up adhoc "red herrings"/assumptions like this anywhere seems inconsistent with your point.

>But the point is the universe of plausible hypotheses is not infinite, either. Scientists in the real world are not dealing with the set of logically possible statements consistent with raw sensory experience. They're building models of the world, and testing and challenging a few of their beliefs at a time, to gradually come closer to the truth.

This is factually wrong. We don't know whether plausible hypothesis are finite or infinite. This is the very reason why nothing can actually be proven. The entire reason why nothing can be proven with science is because we can't specify a range and domain for the universe because we simply don't know.

Formally in the scientific world your method is flawed. This is what I'm saying. In the mathematical world your assumptions and logic can hold but not in science. I get that your toy example was used for illustration purposes but it's failure to be applicable to the real world is also an illustration of the flaw behind the method itself.

I can bring it back to the world of axioms and theorems so . Imagine the real world as a model where there the domain and range is infinite in size and entity types. The only thing that can be assumed is probability and logic. That is what the scientific method is operating in. Therefore your methods have to account for an infinite amount of hypothesis rendering your assumptions and deductions impossible without caveating your conclusion with a limiting assumption.

In the logical model above the best you can do to establish causality it by placing yourself into the system. Any other method requires assumptions (which is commonly done).

>When they can conduct randomized controlled trials, that's great. But it's not always feasible or even possible, yet it's still often possible in those cases to test competing causal hypotheses via observation. The Eddington experiment tested the (causal) theory of General Relativity against the (causal) theory of Newtonian gravity, but there was no intervention. The discovery of iridium in a giant crater off the coast of Mexico was a key confirmation for the (causal) dinosaur meteor extinction theory, but there was no intervention there either. These were causal hypotheses tested by observation.

Of course. I'm saying we have limited tools so we tend to jump to conclusions and make assumptions based off of our intuition and understanding of probable outcomes. But this does not discount the fact that these assumptions were made.

I am saying that the ONLY way to determine causality to a degree in a universe where the domain and codomain is unknowable is by placing your self into the system and making yourself the causal factor. But this is not how we operate in practice. Clearly many scientists assume causality based only off of correlating evidence do in no part to, like you said, how hard it is to actually create these experiments.

>The point is RCTs are a special case of a more general causal logic. RCTs work because they sever the statistical link to confounders, but only provided your assumptions about the RCTs are correct.

Ok we're probably on the same page then. This is exactly what I'm saying.

All of our science is founded off of a few foundational axioms/assumptions. The recursive assumption that logic is true, the assumption probability is true and maybe a few more intuitive adhoc foundational assumptions that the random number generator is random and that influencing the causal event does not produce side effects. The scientific method is foundational because it operates within these foundational assumptions and so does determining causality by placing yourself into the system.

The "RCT" thing you introduced is not foundational because you must create new assumptions outside of the foundational ones we use for all of science. You make up stuff. Which is fine, but it's no different then using correlation to imply causality. It's the same type of "assumption." We're so against the term correlation does not imply causation but these and similar leaps of assumptions are common place, often done and allow us to come to conclusions without actually proving anything.

I don't think you've understood my example. We may have philosophical disagreements left over even once you've understood my example. But so far you still don't seem to understand it. For example, you say:

You have to mention [that the smoking → tar causation is presumed not controversial] explicitly. Because it's also presumed not controversial that genetics can predispose someone to cancer.

Right, of course it's not controversial that genetics can predispose someone to cancer. That's the whole point. That's why it's among the two hypotheses being tested. It is ruled out (under the assumptions of the causal super-graph, which includes that hypothesis and an alternative hypothesis as the world of possibilities) by the statistical data. Please try to understand this point and acknowledge that it is the case.

You may question the usefulness of this model of science. But so far you don't seem to understand what I'm even saying with this.

This is factually wrong. We don't know whether plausible hypothesis are finite or infinite.

"Plausible", as I'm using it, is a partially subjective term: plausible in the view of subject matter experts. As you're using it, it appears to be an objective term, meaning something akin to "logically possible", perhaps with some ill-defined notion of probability (but what are your priors?). I'm talking about real scientists in the real world. You seem to be talking about some idealized scientist which doesn't, and can't, exist. General Relativity was not plausible in the 17th century. It was not even conceived of then. It became plausible in the 20th century. What's the plausible (as judged by actual scientists, not in some hypothetical abstract logic) alternative hypothesis to the meteor killing the dinosaurs? There is none, so far as I know. The set of hypotheses under consideration changes over time due to the imagination and skepticism of real flesh and blood scientists, not due to some Platonic logic which no scientist can follow.

Formally in the scientific world your method is flawed. This is what I'm saying. In the mathematical world your assumptions and logic can hold but not in science. I get that your toy example was used for illustration purposes but it's failure to be applicable to the real world is also an illustration of the flaw behind the method itself.

… The only thing that can be assumed is probability and logic. That is what the scientific method is operating in.

To be quite frank, I think it is your model which fails to be applicable to the real world. You'll be hard-pressed to find a single practicing scientist who sees themself as a logic/probability machine. In the real world we need to operate, provisionally, with certain assumptions about how the world works, or else we can understand nothing. Those assumptions can always be challenged. But they are held, provisionally, so that we can make progress in our understanding.

>The point is RCTs are a special case of a more general causal logic. RCTs work because they sever the statistical link to confounders, but only provided your assumptions about the RCTs are correct.

Ok we're probably on the same page then. This is exactly what I'm saying.

No, my whole point is that RCTs are just one special case of the more general theory of causal inference, and that more general theory can be applied elsewhere too. The assumptions underlying an RCT are not exceptional, and like everything else in science, they can be challenged.

If you have a population of 1000 people come in and take pills A or B, as determined by a random number generator, and then the ones who took pill A survive and the ones who took pill B die, the RCT says B killed them. But perhaps unbeknownst to you, there was a source of gamma rays due to an experiment in the next room which was simultaneously flipping bits on your random number generator and also irradiating your subjects with lethal doses. The gamma rays would then be the confounder.

"Science" does not rule out this possibility in theory. But in practice, it may be implausible to the experts. (Remember: plausibility is partially subjective, a function of the current state of scientific understanding! It can change over time!) So they are not worried about it when interpreting the results of the RCT.

Similarly, the universe of logically possible causal explanations consists mostly of hypotheses that no scientist currently takes seriously (i.e. considers plausible). Of the ones on the table, so to speak, the predictions can be tested to leave only one causal explanation (for the time being) standing.

Let's clarify something here. I never said it's not useful. I have a very clear definition of science, a very concrete model of what is should be. Usefulness is not what I'm talking about here, I'm more talking about the true nature of science and causation rather then how useful it is to do science or perform even your methodological technique of commuting a dependency graph (which I agree can be useful at times). But continue, I will reiterate your points to prove to you that I completely understand and that I always understood from the beginning.

>"Plausible", as I'm using it, is a partially subjective term: plausible in the view of subject matter experts. As you're using it, it appears to be an objective term, meaning something akin to "logically possible",

So let me describe what I'm thinking when you say "plausible hypothesis." I am thinking your talking about a some predicate statement that can either be true or false that is reasonable to predict or state due to your intuitive sense of how likely that hypothesis is to occur. I totally get it.

>I'm talking about real scientists in the real world. You seem to be talking about some idealized scientist which doesn't, and can't, exist. General Relativity was not plausible in the 17th century.

The irony here is that your initial example about smoking isn't even real world, it's highly idealized. You assume a fixed set of rules and hypotheses in your universe. I claimed that your idealized example can't function in the real world but your counter claim is that because the real world is imperfect your commuting model of drawing a dependency graph is valid because people in the "real" world make assumptions. Do you see the inconsistency here? Fear not, despite the inconsistency, I get your point and I will address it.

>What's the plausible (as judged by actual scientists, not in some hypothetical abstract logic) alternative hypothesis to the meteor killing the dinosaurs? There is none, so far as I know.

I assume you're saying this to point out how there's only a single "plausible hypothesis" about how the dinos became extinct, correct? I assume yes.

If so then this is my response: There was a period in time where a geocentric universe was the only hypothesis available with no alternative model in existence. There was a period of time where creationism was the only possible hypothesis for the origin of life. There was a period of time where a flat earth was the only plausible hypothesis about the world as we know it. These are all hypothesis that were at one time the sole single "plausible hypothesis" that was overturned at a later date.

"Plausibility" is usually defined relative to the amount of knowledge humans have rather then the total knowledge (aka mathematical domain) of the universe. So if humans have no other evidence about the spherical nature of the globe then a flat earth becomes a plausible hypothesis. But because the universe is not fully discovered or known the more accurate definition of "total knowledge of the universe" the right definition.

Because we currently have no way of knowing all the knowledge in the universe there is know true way to accurately gauge how plausible a hypothesis is. Because like the flat earth, like geocentrism, like creationism there is an infinite amount of models and hypothesis that can be discovered at any point in time that will render your current hypothesis, implausible.

Of course let me point out that I'm not saying that this does not make all of science not useful. It's the best tool we have. We can use it and assume the world is flat until we know better. My point in this case is to acknowledge your point and point out yes it's useful but here is the flaw and the problems with it.

>To be quite frank, I think it is your model which fails to be applicable to the real world. You'll be hard-pressed to find a single practicing scientist who sees themself as a logic/probability machine.

I think you're misunderstanding here. Statistics is founded on probability, the very fact that statistics is used in all of science indicates that all scientists assume probability to be true. I think you don't really understand the nature of probability.

Probability is just an axiomatic set of statements and theorems and formulas. You plug in a probability into this function and you get another numerical representation that we call probabilities.

By sheer coincidence probability happens to work in reality. If there is a raffle with 10 billion tickets where 10% of the tickets have my name on it, then when I draw from that raffle 1 million times around 10% of those tickets will have my name. The larger the numbers you work with the more and more reality seems to represent probability. Nobody to this day knows why this things are consistently like this. Currently we all just take this as an assumption and assume the mathematics of probability apply to the real world.

The same goes for logic. We assume logic is true, I call it a recursive statement because saying something is true is part of logic itself. These are assumptions that are the foundation of how we interpret reality as we know it.

>Those assumptions can always be challenged. But they are held, provisionally, so that we can make progress in our understanding.

Your saying this because you misunderstood me! not I you. I am saying 100% that we have to make foundational assumptions about our universe like probability and logic.

But I get where you're coming from. Your talking about adhoc assumptions like the sun in the sky exists and I don't have to use formal methodology to prove it because it's ludicrous. I get it. What I am saying goes deeper than this.

I am saying that any for any formal scientific method to truly establish causality should not need to include adhoc hypothesis and assumptions. It should as much as possible only include foundational axioms like probability and logic... Why? Because every additional assumption can be questioned as to whether it's true or not true.

I can hear what you're thinking. There's just a ton of obvious stuff in the universe that we can just assume to be true. For example the sun exists in the sky and the keyboard under my fingers exists. It's stupid to not use these assumptions in our "science." I would say all the hypothesis I mentioned fall under the category of "plausible hypothesis" the same category where the flat earth theory once belonged.

That is MY point. The scientific method itself is defined requiring only two assumptions: probability and logic is real. Your method is defined with the requirement of an adhoc amount of assumptions about the universe. It is not foundational. It is merely logical conjecture in a game of axioms and theorems.

>No, my whole point is that RCTs are just one special case of the more general theory of causal inference, and that more general theory can be applied elsewhere too. The assumptions underlying an RCT are not exceptional, and like everything else in science, they can be challenged.

Special case indeed, I wouldn't even call it a special case as it's more like a logic puzzle or a game where the entities are well defined. Rarely applicable to a universe composed of an infinite amount of possibilities.

>If you have a population of 1000 people come in and take pills A or B, as determined by a random number generator, and then the ones who took pill A survive and the ones who took pill B die, the RCT says B killed them. But perhaps unbeknownst to you, there was a sou.....

>"Science" does not rule out this possibility in theory. But in practice, it may be implausi....

Sure totally agreed. However in this experiment No causality was established! only correlation. The statement that correlation doesn't imply causation still holds. However you are assuming here that correlation implies causation.

Like I said earlier for the causal experiment truly be causal we must conduct the experiment by creating a cause with no side effects. Since it's basically impossible to confirm that your causal switch has no side effects are closest possible methodology is to account for it then assume no side effect occurs. Keep in mind, I mentioned that this assumption was a "foundational assumption" like probability and logic, this assumption is not the same as the adhoc assumptions you are bringing up to make your logic work correctly.

If this was an actual experiment this foundational assumption of no side effects would be an implicit part of the conclusion of the experiment: "Assuming there were no side effects when triggering the causal event, Pill B causes the person who eats it to die."

>Similarly, the universe of logically possible causal explanations consists mostly of hypotheses that no scientist currently takes seriously (i.e. considers plausible).

Right but the scientific method itself just uses foundational assumptions just like my controlling the causal event. In practice you take these foundational techniques and you adjust when needed. My argument and point was to say that your methodology is too adhoc and requires too many assumptions to be placed side by side with the scientific method.

Hopefully that makes it more clear because you said earlier that you were describing a "special case" which is exactly what I'm saying about your methodology. It's describing a special case that in actually you have to assume causality of other factors in order to assume causality itself, it's more like inference rather then building up the axiomatic components of causality from the ground up.