By ‘Bayesian’ philosophy of science I mean the position that (1) the objective of science is, or should be, to increase our ‘credence’ for true theories, and that (2) the credences held by a rational thinker obey the probability calculus. However, if T is an explanatory theory (e.g. ‘the sun is powered by nuclear fusion’), then its negation ~T (‘the sun is not powered by nuclear fusion’) is not an explanation at all. Therefore, suppose (implausibly, for the sake of argument) that one could quantify ‘the property that science strives to maximise’. If T had an amount q of that, then ~T would have none at all, not 1-q as the probability calculus would require if q were a probability.
Also, the conjunction (T₁ &T₂) of two mutually inconsistent explanatory theories T₁ andT₂ (such as quantum theory and relativity) is provably false, and therefore has zero probability. Yet it embodies some understanding of the world and is definitely better than nothing.
Furthermore if we expect, with Popper, that all our best theories of fundamental physics are going to be superseded eventually, and we therefore believe their negations, it is still those false theories, not their true negations, that constitute all our deepest knowledge of physics.
What science really seeks to ‘maximise’ (or rather, create) is explanatory power.
And I am now really confused and conflicted. I would love it if someone could enlighten me on how Deutsch’s definition of explanation (hard-to-vary assertions about reality) and Bayesian probability conflict with each other. I am missing something very subtle here.
For context, I am aware of Popper and falsification, but wouldn’t a theory eventually become practically falsified within Bayesian updating if there is enough evidence against it?
I read that too a some time ago and he makes a really basic error, which made me lose some respect for him (If I was able to catch that error surely he should have, and if he didn’t, then he should have heard a correction and corrected it by now).
The error is the assumption that what Bayes does is compare between H and !H, or to take his example, ‘the sun is powered by nuclear fusion’ VS ‘the sun is not powered by nuclear fusion’. What the math really says you should do, is compare all possible hypothesis, so the term !H isn’t itself an explanation/hypothesis, it’s the sum of all other explanations/hypotheses.
I think Abram Demski (which, unlike me, is actually qualified to talk about this stuff) talked about this error in Bayes’ Law is About Multiple Hypothesis Testing (though not directly referring to Deutsch).
I don’t know if Bayes and Deutsch view of explanation actually conflict. It feels to me like he kinda wants them to conflict.
Wow, this is honestly baffling. It sounds as if Deutsch doesn’t know about the generalised form of Bayes’ theorem (I’m sure he does know, which makes me feel worse).
P(Hi|E)=P(E|Hi)P(Hi)ΣjP(E|Hj)P(Hj)
You make an excellent point. Bayes’ theorem can be applied to all possible hypotheses, not just H and ¬H.
If a top physicist can be this biased, then I cannot be surprised by anything anymore.
Bayes can explain why negative, disconfirmatory evidence counts more than positive support, and so sport a version of falsificationism. But it can’t rule out positive support, so doesn’t imply the more extreme Popperian doctrine that there is no justification.
A hard-to-vary explanation is a minimal explanation, one with no redundant parts. So hardness-to-vary is a simplicity criterion, a form of Occam’s razor. Compared to the simplicity criterion favoured by Bayesians, programme length, it is rather subjective. Neither criterion answers the hard problem,the problem of why simplicity implies truth. But Deutsch is more interested in Knowledge , which is left very vaguely defined.
In theory, Bayes is is about adjusting the credences of Every Possible Hypothesis. In practice, you don’t know every possible hypothesis, so there is some truth to Deutch’s claim that not-H is a blob … you might be able to locate some hypotheses other than H, but you have no chance of specifying all infinity.
Bayesians tend to be incurious about where hypotheses come from. That’s one of Chapman’s criticisms, that Bayes isn’t a complete epistemology because it can’t generate hypotheses. Popperians , by contrast, put a lot of emphasis on hypothesis-formation as a an informal, non-mechanistic process.
Good points. There were several chapters in Rationality: A-Z dedicating to this. According to Max Tegmark’s speculations, all mathematically possible universes exist, and we happen to be in one described by a simple Standard Model. I suspect that this question about why simple explanations are so effective in this universe is unanswerable but still fun to speculate about.
Good points about the lack of emphasis on hypothesis-formation within the Bayesian paradigm. Eliezer talks about this a little in Do Scientists Already Know This Stuff?
Sir Roger Penrose—a world-class physicist—still thinks that consciousness is caused by quantum gravity. I expect that no one ever warned him against mysterious answers to mysterious questions—only told him his hypotheses needed to be falsifiable and have empirical consequences.
I long for a deeper treatment on hypothesis-formation. Any good books on that?
. I suspect that this question about why simple explanations are so effective in this universe is unanswerable but still fun to speculate about.
What does “effective” mean? If you are using a simplicity criterion to decide between theories that already known to be predictive , as in Solomonoff induction, then simplicity doesn’t buy you any extra predictiveness.
Thank you for pointing this out, by the way. This is an important nuance. I just read this: Simple refutation of the ‘Bayesian’ philosophy of science.
And I am now really confused and conflicted. I would love it if someone could enlighten me on how Deutsch’s definition of explanation (hard-to-vary assertions about reality) and Bayesian probability conflict with each other. I am missing something very subtle here.
For context, I am aware of Popper and falsification, but wouldn’t a theory eventually become practically falsified within Bayesian updating if there is enough evidence against it?
I read that too a some time ago and he makes a really basic error, which made me lose some respect for him (If I was able to catch that error surely he should have, and if he didn’t, then he should have heard a correction and corrected it by now).
The error is the assumption that what Bayes does is compare between H and !H, or to take his example, ‘the sun is powered by nuclear fusion’ VS ‘the sun is not powered by nuclear fusion’. What the math really says you should do, is compare all possible hypothesis, so the term !H isn’t itself an explanation/hypothesis, it’s the sum of all other explanations/hypotheses.
I think Abram Demski (which, unlike me, is actually qualified to talk about this stuff) talked about this error in Bayes’ Law is About Multiple Hypothesis Testing (though not directly referring to Deutsch).
I don’t know if Bayes and Deutsch view of explanation actually conflict. It feels to me like he kinda wants them to conflict.
Wow, this is honestly baffling. It sounds as if Deutsch doesn’t know about the generalised form of Bayes’ theorem (I’m sure he does know, which makes me feel worse).
P(Hi|E)=P(E|Hi)P(Hi)ΣjP(E|Hj)P(Hj)You make an excellent point. Bayes’ theorem can be applied to all possible hypotheses, not just H and ¬H.
If a top physicist can be this biased, then I cannot be surprised by anything anymore.
Thank you very much for your response Yoav Ravid.
Bayes can explain why negative, disconfirmatory evidence counts more than positive support, and so sport a version of falsificationism. But it can’t rule out positive support, so doesn’t imply the more extreme Popperian doctrine that there is no justification.
A hard-to-vary explanation is a minimal explanation, one with no redundant parts. So hardness-to-vary is a simplicity criterion, a form of Occam’s razor. Compared to the simplicity criterion favoured by Bayesians, programme length, it is rather subjective. Neither criterion answers the hard problem,the problem of why simplicity implies truth. But Deutsch is more interested in Knowledge , which is left very vaguely defined.
In theory, Bayes is is about adjusting the credences of Every Possible Hypothesis. In practice, you don’t know every possible hypothesis, so there is some truth to Deutch’s claim that not-H is a blob … you might be able to locate some hypotheses other than H, but you have no chance of specifying all infinity.
Bayesians tend to be incurious about where hypotheses come from. That’s one of Chapman’s criticisms, that Bayes isn’t a complete epistemology because it can’t generate hypotheses. Popperians , by contrast, put a lot of emphasis on hypothesis-formation as a an informal, non-mechanistic process.
Good points. There were several chapters in Rationality: A-Z dedicating to this. According to Max Tegmark’s speculations, all mathematically possible universes exist, and we happen to be in one described by a simple Standard Model. I suspect that this question about why simple explanations are so effective in this universe is unanswerable but still fun to speculate about.
Good points about the lack of emphasis on hypothesis-formation within the Bayesian paradigm. Eliezer talks about this a little in Do Scientists Already Know This Stuff?
I long for a deeper treatment on hypothesis-formation. Any good books on that?
What does “effective” mean? If you are using a simplicity criterion to decide between theories that already known to be predictive , as in Solomonoff induction, then simplicity doesn’t buy you any extra predictiveness.