I believe that what I am trying to point at is indeed evidence, in the Bayesian sense of the word. For example, consider masks and COVID. Imagine that we empirically observe that they are effective 20% of the time and ineffective 80% of the time. Should we stop there and take it as our belief that there is a 20% chance that they are effective? No!
Suppose now that we know that when someone with COVID breathes, particles containing COVID remain in the air. Further suppose that our knowledge of physics would tell us that someone standing two feet away is likely to breathe in these particles at some concentration. And further suppose that our knowledge of how other diseases work tell us that when that concentration of virus is ingested, it is likely that you will get infected. When you incorporate all of this knowledge about physics and biology, it should shift your belief that masks are effective. It shouldn’t stay put at 20%. We’d want to shift it upward to something like 75% maybe.
When put like this, these “evidence” sound a lot like priors. The order should be different though:
First you deduce from the theory that masks are, say, 90% effective. These are the priors.
Then you run the experiments that show that masks are only effective 20% of the time.
Finally you update your beliefs downward and say that masks are 75% effective. These are the posteriors.
To a perfect Bayesian the order shouldn’t matter, but we are not perfect Bayesians and if we try to do it the other way around and apply the theory to update the probabilities we got from the experiments, we would be able to convince ourselves the probability is 75% no matter how much empirical evidence that says otherwise we have accumulated.
if we try to do it the other way around and apply the theory to update the probabilities we got from the experiments, we would be able to convince ourselves the probability is 75% no matter how much empirical evidence that says otherwise we have accumulated.
If this were true, I would agree with you. I am very much on board with the idea that we are flawed and that we should take steps to minimize the impact of these flaws, even if those steps wouldn’t be necessary for a perfect Bayesian.
However, it isn’t at all apparent to me that your assumption is true. My intuition is that it wouldn’t make much of a difference. But this sounds like a great idea for a psychology/behavioral economics experiment!
The difference can be quite large. If we get the results first, we can come up with Fake Explanations why the masks were only 20% effective in the experiments where in reality they are 75% effective. If we do the prediction first, we wouldn’t predict 20% effectiveness. We wouldn’t predict that our experiment will “fail”. Our theory says masks are effective so we would predict 75% to begin with, and when we get the results it’ll put a big dent in our theory. As it should.
If the order doesn’t matter then it seems a kind of “accumulation of priors” should be possible. It is not obviously evident to me how the perfectness of the bayesian would protect it from this. That is for a given posterior and constant evidence there exists a prior that would give that conclusion. Normally we think of the limit where the amount and weight of the observations dominates. but there might be atleast a calculation where we keep the observation constant and more and more reflect on it, changing or adding new priors.
Then the result that a bayesian will converge on the truth with additional evidende flips to mean that any evidence can be made to fit a sufficiently complex hypothesis ie that with enough reflection there is asymptotic freedom of belief that evidence can’t restrain.
In the face of a very old and experienced bayesian allmost all things it encounters will shift its beliefs very little. If the beliefs were of unknown origin one might be tempted to assume that it would be stubborness of stupidity to not be open to evidence. If you know that you know it seems such stubborness might be justifiable. But how do you know whether you know? And what kind of error is being committed when you are understubborn?
I think you may be underestimating the impact of falsifying evidence. A single observation that violates general relativity, assuming we can perfectly trust its accuracy and rule out any interference from unknown unknowns—would shake our understanding of physics if it comes tomorrow, but had we encountered the very same evidence a century ago our understanding of physics would have already been shaken (assuming the falsified theory wouldn’t be replaced with a better one). To a perfect Bayesian, the confidence at general relativity in both cases should be equal—and very low. Because physics are lawful—the don’t make “mistakes”—we are the ones who are mistaken at understanding them, so a single violation is enough to make a huge dent no matter how many confirming evidence we have managed to pile up.
Of course, in real life we can’t just say “assuming we can perfectly trust its accuracy and rule out any interference from unknown unknowns”. The accuracy of our observations is not perfect, and we can’t rule out unknown unknowns, so we must assign some probability to our observation being wrong. Because of that, a single violating evidence is not enough to completely destroy the theory. And because of that, newer evidence should have more weight—our instruments keep getting better so our observations today are more accurate. And if you go far enough back you can also question the credibility of the observations.
Another issue, which may not apply to physics but applies to many other fields, is that the world does change. A sociology experiment form 200 years ago is evidence on society from 200 years ago, so the results of an otherwise identical experiment from recent years should have more weight when forming a theory of modern society, because society does change—certainly much more than physics change.
But to the hypothetical perfect Bayesian the chronology itself shouldn’t matter—all they have to do is take all that into account when calculating how much they need to update their beliefs, and succeeding to do so it doesn’t matter in which order they apply the evidences.
The act of a single falsification shatter the whole theory seems like a calculation where the prior just gets tossed. However in most calculations the prior still affects things. If you start from somewhere and then either don’t see or see relativistic patterns for 100 years and then see a relativity violation a perfect bayesian would not end with the same end belief. Using the updated prior or the ignorant prior makes a difference and the outcome is geniunely a different degree of belief. Or I guess another way of saying that is that if you suddenly gain access to the middle-time evidence that you missed it still impacts a perfect reasoner. Gaining 100 years worth of relativity pattern increases credence for relativity even if it is already falsified.
Maybe “destroying the theory” was not a good choice of words—the theory will more likely be “demoted” to the stature of “very good approximation”. Like gravity. But the distinction I’m trying to make here is between super-accurate sciences like physics that give exact predictions and still-accurate-but-not-as-physics fields. If medicine says masks are 99% effective, and they were not effective for 100 out of 100 patients, the theory still assigned a probability of 10−200 that this would happen. You need to update it, but you don’t have to “throw it out”. But if physics says a photon should fire and it didn’t fire—then the theory is wrong. Your model did not assign any probability at all to the possibility of the photon not firing.
This means that the falsifying evidence, on its own, does not destroy the theory. But it can still weaken it severely. And my point (which I’ve detoured too far from) is that the perfect Bayesian should achieve the same final posterior no matter at which stage they apply it.
I’m basing this answer on a clarifying example from the comments section:
When put like this, these “evidence” sound a lot like priors. The order should be different though:
First you deduce from the theory that masks are, say, 90% effective. These are the priors.
Then you run the experiments that show that masks are only effective 20% of the time.
Finally you update your beliefs downward and say that masks are 75% effective. These are the posteriors.
To a perfect Bayesian the order shouldn’t matter, but we are not perfect Bayesians and if we try to do it the other way around and apply the theory to update the probabilities we got from the experiments, we would be able to convince ourselves the probability is 75% no matter how much empirical evidence that says otherwise we have accumulated.
If this were true, I would agree with you. I am very much on board with the idea that we are flawed and that we should take steps to minimize the impact of these flaws, even if those steps wouldn’t be necessary for a perfect Bayesian.
However, it isn’t at all apparent to me that your assumption is true. My intuition is that it wouldn’t make much of a difference. But this sounds like a great idea for a psychology/behavioral economics experiment!
The difference can be quite large. If we get the results first, we can come up with Fake Explanations why the masks were only 20% effective in the experiments where in reality they are 75% effective. If we do the prediction first, we wouldn’t predict 20% effectiveness. We wouldn’t predict that our experiment will “fail”. Our theory says masks are effective so we would predict 75% to begin with, and when we get the results it’ll put a big dent in our theory. As it should.
If the order doesn’t matter then it seems a kind of “accumulation of priors” should be possible. It is not obviously evident to me how the perfectness of the bayesian would protect it from this. That is for a given posterior and constant evidence there exists a prior that would give that conclusion. Normally we think of the limit where the amount and weight of the observations dominates. but there might be atleast a calculation where we keep the observation constant and more and more reflect on it, changing or adding new priors.
Then the result that a bayesian will converge on the truth with additional evidende flips to mean that any evidence can be made to fit a sufficiently complex hypothesis ie that with enough reflection there is asymptotic freedom of belief that evidence can’t restrain.
In the face of a very old and experienced bayesian allmost all things it encounters will shift its beliefs very little. If the beliefs were of unknown origin one might be tempted to assume that it would be stubborness of stupidity to not be open to evidence. If you know that you know it seems such stubborness might be justifiable. But how do you know whether you know? And what kind of error is being committed when you are understubborn?
I think you may be underestimating the impact of falsifying evidence. A single observation that violates general relativity, assuming we can perfectly trust its accuracy and rule out any interference from unknown unknowns—would shake our understanding of physics if it comes tomorrow, but had we encountered the very same evidence a century ago our understanding of physics would have already been shaken (assuming the falsified theory wouldn’t be replaced with a better one). To a perfect Bayesian, the confidence at general relativity in both cases should be equal—and very low. Because physics are lawful—the don’t make “mistakes”—we are the ones who are mistaken at understanding them, so a single violation is enough to make a huge dent no matter how many confirming evidence we have managed to pile up.
Of course, in real life we can’t just say “assuming we can perfectly trust its accuracy and rule out any interference from unknown unknowns”. The accuracy of our observations is not perfect, and we can’t rule out unknown unknowns, so we must assign some probability to our observation being wrong. Because of that, a single violating evidence is not enough to completely destroy the theory. And because of that, newer evidence should have more weight—our instruments keep getting better so our observations today are more accurate. And if you go far enough back you can also question the credibility of the observations.
Another issue, which may not apply to physics but applies to many other fields, is that the world does change. A sociology experiment form 200 years ago is evidence on society from 200 years ago, so the results of an otherwise identical experiment from recent years should have more weight when forming a theory of modern society, because society does change—certainly much more than physics change.
But to the hypothetical perfect Bayesian the chronology itself shouldn’t matter—all they have to do is take all that into account when calculating how much they need to update their beliefs, and succeeding to do so it doesn’t matter in which order they apply the evidences.
The act of a single falsification shatter the whole theory seems like a calculation where the prior just gets tossed. However in most calculations the prior still affects things. If you start from somewhere and then either don’t see or see relativistic patterns for 100 years and then see a relativity violation a perfect bayesian would not end with the same end belief. Using the updated prior or the ignorant prior makes a difference and the outcome is geniunely a different degree of belief. Or I guess another way of saying that is that if you suddenly gain access to the middle-time evidence that you missed it still impacts a perfect reasoner. Gaining 100 years worth of relativity pattern increases credence for relativity even if it is already falsified.
Maybe “destroying the theory” was not a good choice of words—the theory will more likely be “demoted” to the stature of “very good approximation”. Like gravity. But the distinction I’m trying to make here is between super-accurate sciences like physics that give exact predictions and still-accurate-but-not-as-physics fields. If medicine says masks are 99% effective, and they were not effective for 100 out of 100 patients, the theory still assigned a probability of 10−200 that this would happen. You need to update it, but you don’t have to “throw it out”. But if physics says a photon should fire and it didn’t fire—then the theory is wrong. Your model did not assign any probability at all to the possibility of the photon not firing.
And before anyone brings 0 And 1 Are Not Probabilities, remember that in the real world:
There is a probability photon could have fired and our instruments have missed it.
There is a probability that we unknowingly failed to set up or confirm the conditions that our theory required in order for the photon to fire.
We do not assign 100% probability to our theory being correct, and we can just throw it out to avoid Laplace throwing us to hell for our negative infinite score.
This means that the falsifying evidence, on its own, does not destroy the theory. But it can still weaken it severely. And my point (which I’ve detoured too far from) is that the perfect Bayesian should achieve the same final posterior no matter at which stage they apply it.