Confirmation of Scientific Theories

My primary focus is in scientific confirmation. So I thought I would share several theories about confirmation of scientific theories; they are Bayesian, Likelihoodism, Popperian, and my own Differential Confirmation.

All but the Popperian theory of confirmation revolve around probability, and Karl Popper even started to concede that his view is reducible to some degree of probability. So I’ll start off with a crash course in bayesianism.

Bayesianism gets its name from Bayes’ theorem, a mathematical theorem of probability. To understand this theorem, there are 4 important things to keep track of. First is the posterior probability of a hypothesis given an observation. This is written as P(H|O). Often, in bayesianism, this is what it means to confirm a hypothesis from a given observation. To calculate this, three other values are needed, the likelihood of a given piece of evidence given a hypothesis P(E|H), the prior probability of a hypothesis P(H), and the probability of the evidence P(E). These are related by Bayes’ Theorem: P(H|E)=P(E|H)P(H)/P(E)

To show how these work, let’s have the hypothesis that a particular card drawn randomly from a deck of cards is the ace of spades. So H is “this card is the ace of spades”, and let the observation E, be that the card is black. Now to start entering values.

Prior to the observation, the chance of a card being the ace of spades is 1/52. So P(H)=1/52. The probability of a card being black randomly drawn is 1/2, since half are black and the other half are red. Finally, P(E|H)=1, this is because being an ace of spades entails that it is black, so it gets the maximum probability assignable. According to Bayes’ Theorem, the result is that the probability of this card being the ace of spades is 1/52*1 / 1/2 = 1/26. And this is in line with our intuition that there are 26 possible cards and one of them will be the ace of spades. It works! After all, it’s math, it better work. Additionally, we say that the hypothesis is confirmed because P(H|E)>P(E). Disconfirmation is when the inequality goes the other way, and we have no confirmation when they are equal to one another. In the above card case, we see that 1/26>1/52, so our hypothesis is incrementally confirmed.

So now we come to confirmation of theories. We think of scientific hypotheses as being H, observations as E, and the likelihood as being what we think of as the hypothesis’ prediction. For now, I’ll ignore the problems with filling in some of these values, as it’s very hard in some cases.

Bayesianism councils us to adopt a theory whenever its posterior is above .5. This is because .5 is often the cut off for belief versus agnosticism. Why .5? It’s because P(H) + P(not H) = 1. So in cases of where the value is .5, then P(H) = P(not H), so we cannot decide between the two of them based on probabilities (assuming of course that our degree of belief in a theory should be determined by the probabilities of the theory).

The problems with this are threefold. First is that .5 is arbitrary. There why should we believe a theory to be true just because it’s marginally above .5. Why not some really high probability like .9, so we are really sure that it’s true. Secondly, it does not capture the problems of incremental changes to theories. This is because it’s a law of mathematics that if a new theory, with some additional hypotheses about the way the world works entails an old theory, then P(new theory|E)<=P(old theory|E). So no matter how many novel predictions that the new theory makes, it cannot be more probable than the old theory. Thus, why should we ever believe in an incrementally new theory? We shouldn’t on the bayesian account. Thirdly, what if we only have 2 theories and their posteriors are low? We should still select one or the other, shouldn’t we? What if general relativity’s prior is awfully low, so it’s posterior is never very high, none the less, it makes good predictions, so shouldn’t we still believe it to be true to some extent?

This is where likelihoodism attempts to solve the problems. It’s contrastive, meaning that it only works in comparing two theories. It works by having H1 and H2, as our hypotheses and examines what predictions it makes. In terms of confirmation, all we are interested in are the likelihoods. So all we look at is whether P(E|H1)>P(E|H2). It’s satisfying in that confirmation is only based on predictions. But this suffers from the problem of conspiracies. Conspiracies make the same predictions as a standard scientific hypothesis, so there is no difference between a conspiracy’s confirmation and a real scientific confirmation. Something else needs to be added to make sure that a hypothesis is a good scientific one. And this cannot be based in mathematics.

So what makes a good scientific hypothesis? Karl Popper introduced his own theory which is sticking one’s neck out. A good hypothesis that is confirmed to a greater degree than another is one that makes lots of bold new predictions. General Relativity made for a good hypothesis because it made the bold prediction that if one takes two atomic clocks, and one is flown around the earth at a high speed, then when comparing the two, they will not match up. Additionally, it made the prediction that if I see two lightning bolts striking at the same time, you might not see them striking at the same time. These are bold claims over and above what Newtonian mechanics predicted, so it’s a good hypothesis to be tested. And since these claims came out to be true, we should prefer GR over Newtonian mechanics. However, while this gives some claim to what makes a good hypothesis, Popper only said that one can falsify hypotheses through observation, never confirm. So we never believe a theory to be true, we just say that it isn’t falsified, yet.

My view is that science is concerned with what is confirmed to a greater degree. So rather than focusing on P(H|E), we look at P(E|H)-P(H). We select theories based on which is can be confirmed to a greater degree rather than which has the higher posterior. It’s contrastive so we choose theories over other theories, like likelihoodism. But we also solve the problem of conspiracies and new theories, and arbitrary points of belief. Plus, it’s rather Popperian in nature because a theory that makes a bolder prediction will enjoy a greater degree of confirmation. It takes about another 10 pages to explain all how it solves all of this, but it does work. I’m excited about my project since it captures incremental theory modification. The only concern that I have is that priors show up.

The major problem with priors I’ll leave as a puzzle. How does one figure out the prior probability of general relativity without any evidence? No one’s figured it out, so if you can, publish it. But this is a problem that’s inherited from bayesianism, so at the very least I can push it back on them. But it still concerns me considerably.

Raven’s Paradox and Hypothetical-Deductivism

I’m on a science kick lately, if you can’t tell by the last few posts. I’ve gotten my phil science batteries recharged with my current course and I feel like sharing more of what I do with the world. So today I’m going to talk about the hypothetical-deductivism model of science and the raven’s paradox. Technical words, but not that scary. What hypothetical-deductivism (from now on HD since I’m too lazy to write it out again) is, is a model for how science confirms and disconfirms hypotheses. It’s an old model, but it’s been updated to what is now called bayesianism, which I find suffers from similar problems.

So what is HD? It’s a logical approach that has two parts, confirmation and disconfirmation. Confirmation happens when we have an hypothesis that logically implies some observation. The hypothesis that all ravens are black entails that if x is a raven, then x is black. If we find something that is a raven and is black, then this confirms that all ravens are black. It can be much more interesting than this, but it’s mainly a common sensical approach to confirmation. If an hypothesis logically entails a prediction, and that prediction is correct, then the hypothesis is confirmed. Hence the name Hypothetical-Deductivism, one deduces observations from a hypothesis and if those observations are correct, confirmation happens, yay!

Disconfirmation happens in just the opposite way, that if it entails that an observation will not happen, but it does, then the theory is disconfirmed. So when Newtonian mechanics predicted that light from stars will not bend around the gravitational field of the sun (gravitational lensing), and then we observed that it did, it disconfirmed Newtonian mechanics while confirming general relativity.

It’s a straight forward approach that came out of a time when we thought that logic could solve all of our problems. It hasn’t, and it won’t. Why is this? Two reasons: white shoes and bumblebees.

The white shoes problem, usually known as the raven’s paradox, is a problem that arises from a fact in logic. It is a logical truism that if a hypothesis implies some observation, then the negation of that observation implies the negation of the hypothesis. So in the case of ravens, the HD model of confirmation, if x is a raven then x is black, is logically equivalent to if x is not black then x is not a raven. But as before we saw that if we find something that is black and a raven, then it confirms the first implication, so a white shoe that is neither black nor a raven will confirm the second implication. So white shoes confirm that all ravens are black. This is a problem, white shoes shouldn’t confirm anything along those lines.

Second is the irrelevant conjunction problem. Again, it follows from logic. You’ll have to take my word for it, but it is a logical fact that if A implies B, then (A and C) implies B. So we can take some hypothesis about bumblebees dancing to direct others toward honey and attach it to general relativity. General relativity implies that starlight will bend, and logically so will General relativity and bumblebees dancing imply that starlight will bend. The meat of this problem is that if we observe starlight bending, then we confirm both the hypothesis of General relativity, and bumblebees dancing. Hence the name, the problem of irrelevant conjunctions. We could even take this a step further and append bumble bees to the ravens implication such that white shoes confirm that all ravens are black and that bumblebees dance to show where the honey is.

Why is this interesting? Because HD is rather much like a common sense approach to science. If an observation is derived from a hypothesis, then either it is confirmed or disconfirmed by the observation. But common sense is wrong on this point. And that’s what philosophy likes to do. It likes to start from a common sense beginning, and then take it further and see how it fares. Often, it does not do so well. which is why philosophy is interesting, it questions what we take for granted as true and often finds that it doesn’t work. That doesn’t mean it has all the answers, believe me, you don’t go into philosophy for answers, but in some sense, like science, at least we know what doesn’t work.

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For those familiar with bayesianism, the problem of irrelevant conjunctions plagues it as well. First is just that since an irrelevant hypothesis will not impact the probability of an observation, it too will be confirmed by observation. Or, in a more technical perspective, the HD model is a subset of bayesianism, HD is just when the likelihood of Pr(observation|hypothesis & irrelevant hypothesis)=1, which then means that the probability of Pr(hypothesis & irrelevant hypothesis | observation) = Pr(hypothesis & irrelevant hypothesis) / Pr (observation) under Bayes’ theorem. And this is always greater than than Pr(hypothesis & irrelevant hypothesis) because Pr(observation) < 1.

On the Mathematization of Nature

Around the time of Galileo, a major advancement occurred in science, the mathematization of phenomenon. This is not to say that mathematics had not been employed before hand, but Galileo produced a systematic approach to it. It was a major leap forward since it produced a mechanism of prediction that was robust and exact. This continued on through the work of Kepler, Newton, Einstein, and Feynman to the point that now we do science almost in a vacuum of mathematics. It was then the advent of bizarre phenomena at the level of quantum mechanics that we began to run into some interesting problems (though, it also existed before when atomic theory was developed). The problem that developed was the predictions of mathematics in describing and predicting the existence of things that were in a sense unobservable to humans. What could legitimize our inference that such entities existed?

To understand this question, we should look at competing theories, particularly one produced by Bas van Fraassen. The overview of this is simple. When one conducts scientific experiments, one observes regularities in nature at the macroscopic level. In fluid dynamics, one observes motion called brownian motion that is detectable. It was a step forward in science that the mathematics predicted the existence of atomic particles that produced such motion. The position that atomic particles exist as a result of such observations is termed realism; van Fraassen says that all we observe and thus all we can confirm is the observational level of brownian motion as a regularity in our observations, thus he remains agnostic to the existence of atomic particles. It seems peculiar to take such a stance. We’re pretty sure that atoms exist in everyday life, but van Fraassen questions just how much evidence we have for atomic particles over and above what confirms regularities at the macroscopic level.

There is a relatively good basis for this. In the contrastive empiricism developed in brief by Elliot Sober, we see a simple understanding of this in bayesian terms. The idea is that all the theories that assert the existence of atomic particles imply the existence of macroscopic theories like brownian motion. Thus, Realism implies Macroscopic Skepticism. Given that this is an implication, under rules of bayesianism, Realism cannot be more probable than Skepticism. Think of it this way, the antecedent in an implication cannot be more sure than the consequent, otherwise there will be times when the antecedent is true and the consequent is false. If rain is the only cause of humidity, then a 50% chance of rain cannot make the chance of humidity greater than 50%, since there is only one cause of humidity, rain (this is simplified, but it’s a way of understanding the rule). Thus, whatever is confirmed through observation at the macroscopic level cannot make the existence of atomic particles more likely than the macroscopic theories. So why should we adopt a theory that is less probable? The bayesian answer is that we should not.

And there is the conundrum. In terms of mathematical logic and probabilistic mathematics, one encounters the counter intuitive idea that while mathematics predicts the existence of unobservable particles, one cannot confirm them to a greater degree than macroscopic principles. By the logical choice that one should believe what is most probable, one should believe the macroscopic principles to be true, as in the mathematics of the macroscopic theories to be representative of the regularities in nature; and one should at the very least remain agnostic toward the existence of atomic particles.

Furthermore, this is all thanks to the mathematization of nature. Where one can convert microscopic theories into macroscopic regularities, thus establishing the implication that leads to a problematic conclusion. Personally, I resist this idea. My thesis is focusing on the failure of the implication in the direction that Sober asserts. Rather, I believe that it is justified that the implication runs the opposite direction, that macroscopic principles imply the existence of microscopic principles through the conversion of macroscopic principles into idealized microscopic particles, such as the Hamiltonian model that is used in quantum mechanics. Under my view, one observes macroscopic principles, idealizes them, and then develops the microscopic principles that follow from the idealization. This is just an overview of what I’m working on, but it’s a fascinating twist to the history of science that I hope to help resolve in part.

The Flow of Evidence in Atheism

One could write a book on the way evidence is related to atheism and theism, but today I’ll focus on likelihoodist relationships between observation and hypothesis.

Likelihoodism is an epistemological position that is comparative at its core as a probabilistic means of weighing evidence. But it’s quite different from it’s cousin, Bayesianism, in its way of understanding probabilities.

Kosher probabilities are values between and including 0 and 1, with 0 being no possibility, .5 being a 50-50 chance, and 1 being absolute certainty. Bayesianism is an attempt to provide objective probabilities to hypothesis through support by observation (there is also subjective Bayesianism, but that’s another story). To get an objective probability, one has a probability function: P, an hypothesis H, and observations O. The relationship between these two is given as P(H|O)=P(O|H)P(H)/P(O). Which reads as the probability of H given O is equal to the likelihood of O given H times the probability that H is true, over the probability of O occurring. This is a mathematical relationship that is very well grounded in set theory, so don’t get smart about its validity.

But Bayesianism runs into its problems. One of its famous problems is how to assign a value to P(H) without observation. Think of it this way, what is the probability that the general theory of relativity is true, sans observation? It’s tricky, likelihoodists don’t like it.

Instead, what they focus on is the objective relationship between an observation and its probability given an hypothesis; P(O|H). Thus, one looks at how probable it is that light will bend around stars, given GTR. This method also has its problems, but here is not the best place to explore epistemology at this point. If you want to learn more, check out this short paper by Elliott Sober: http://philosophy.wisc.edu/sober/BAYES%20handout%202010.pdf.

Likelihoodism also compares hypotheses differently than Bayesianism. Rather than having some absolute probability, one compares the likelihood of each observation based upon the hypothesis through a ratio: P(O|H1)/P(O|H2). If the ratio is greater than 1, then the one can say that the evidence favors H1, if lower than 1, then the evidence favors H2. This is super brief in its exposition, but hopefully you get the main thrust of it as a comparative method that weighs not which is true or false, but which one is better supported by evidence.

So then let’s look at miracles. If we look at some rough statistics from cancer.gov, we can take a fairly deadly cancer pancreatic cancer, which has a survival rate of 5%. Now let’s be generous and say that god intervenes in say 1 in a 100 times. That’s a pretty active god. We let H1 be that treatment worked, and H2 that god intervened, and finally O is that you survive. Our ratio is then 0.05/0.01 = 5. The naturalistic explanation has five times the support than the miracle based on. In other cancers, the evidential support is even greater, unless god intervenes more often in more curable cancers. In which case he’s kinda dickish.

However, what separates likelihoodism from the usual common sense epistemology is that in every survival case, H2 (the god hypothesis) is confirmed to some degree. The conceptual shift occurs not in saying that god is not confirmed, but that other theories are better confirmed. Thus, one bypasses the usual humdrum argument that atheists are simply ignoring the evidence for god’s existence. Here, the atheist can take the evidence as positive confirmation for god’s existence, but say that his non-existence is more strongly confirmed with the exact same evidence. Makes your head spin, yes, but that’s because of the failure of the uniqueness theorem (which says that a given body of evidence supports at most one proposition about that evidence). In doing so, it meets the theist half way, yes, their theories are confirmed, but that’s only half the story. With alternative hypotheses, there may be a better confirmed on in the bunch and that one is the one we should adopt. Theists aren’t necessarily making a bad inference, but they are making a bad comparison, which in this case is ignoring what the evidence can tell you.

But now we can go even further than this, we can also test if the hypothesis even makes a difference. To do this, we simply need to see that P(O|H)>P(O), that is, an hypothesis is favored if it makes the observation more likely than the observation of it happening alone. To set up something to test the efficacy of god, one could observe the survival rates of someone of another faith, set that as P(O) and then test whether a given religion H makes the survival rates increase. Or one could really find the efficacy of god in testing the survival rates of faith healing groups versus science. See, god hypotheses are testable!

A cursory look through google didn’t turn up any data on the first hypothesis, but faith healing is notoriously bad. Even intercessory prayer looked bad according to the outcomes by the STEP study. The failure of faith healing gives a strong evidence against P(O|H)>P(O). Whereas the STEP study shows that prayer doesn’t increase the observation of survival over the normal observation of survival. Look, real world testing of these hypotheses shows them to not to be false, but ill confirmed compared to alternative explanations. That is the important distinction that likelihoodism forces on us.

What makes liklihoodism an important tool of analysis is two fold. First is that certain claims about god are in fact testable, he’s not just some will o the wisp. Second, it breaks with the standard problem of evil argument in not looking at all the negative instances of god’s intervention, but allowing some positive instances to confirm it, but not as much as physicalist theories. This also makes them criticizable as irrational, not in lue of them making bad inferences (which allows us to sidestep definitions of good inference which is particularly difficult when uniqueness fails), but in ignoring rational alternatives.