How is Bayesian reasoning related to the scientific method?

Question

Back in September 2006, Scott Aaronson wrote a famous blog post giving 10

• Reasons to believe that N!=NP. In March 2014, he wrote a more ambitious post about

• The scientific case for N!=NP. He claims: "This post supersedes my 2006 post on the same topic, which I hereby retire." His post seems to be strongly influenced by previous intellectual exchanges with a convinced Bayesian "climate change" critic. The central Bayesian argument from the post left me with a feeling similar to the following quote from the post (replace "probability" with "science"):

  John Oliver’s deadpan response was classic: "I’m ... not sure that’s how probability works..."

  As a reaction, some computer scientists sympathetic to Scott wrote serious posts about

• Why do we think N NE NP? They actually started by asking themselves: "Why do scientists believe any particular theory?" and listed the following actual reasons: "(1) By doing Popperian experiments- experiments that really can fail. (2) Great Explanatory power. (3) (Kuhn-light) It fits into the paradigm that scientists already have."

  Other computer scientists more explicitly raised questions "with regard to the main technical argument in a recent post by Scott" by asking

• Could we have felt evidence for SDP!=P? My impression is that the main technical argument failed to convince them, but their high esteem for Scott prevents them from being blunt about this.

My question is the following: The post from 2014 tries to construct a single Bayesian argument with a single "Bayesian probability" for P!=NP. All the other posts work with multiple independent reasons, and make no attempt at all to unify this into a single Bayesian probability argument. Is subsuming multiple independent reasons into a single Bayesian judgment really in agreement with the scientific method? What does epistemology says about this? (There are statements which are either true or false, but I'm not sure whether this implies that I should only assign a single Bayesian probability to such a statement quantifying how sure I am that it's true.)

Edit Note that my confusion is not caused by using Bayesian probabilities in general, but by the procedure to subsume many different Bayesian probabilies for different facts into a single Bayesian probability for a "stronger" fact. So I'm OK that we can postulate Bayesian probabilities for facts like that N=NP would be very surprising, or that N!=NP explains many observed facts, or that P!=NP is extremely useful. But I'm confused how it should be possible to subsume these into a single Bayesian probability for a "stronger" fact like that P!=NP is true.

Answer 1

The idea that the inductive process that happens in science is best modelled by Bayesian reasoning has been around for a long time. Edwin Jaynes' book, The Logic of Science (freely available online) is usually cited in this context. I'm not going to repeat the arguments here as they have been far better articulated by others and this answer is kind of long already.

To your specific question:

I was left with the impression that the assumption that multiple independent reasons can be subsumed under a single Bayesian argument is not in agreement with the scientific method.

Within Bayesian reasoning you certainly can combine beliefs in arguments together and get a single probability. If you accept Bayesian reasoning as a model of scientific induction then you accept that you can combine arguments to yield a single probability. However, not everyone accepts the premise that Bayesian reasoning is a good model, in fact, the discussion has been rather heated over the years.

Many people think of Bayesian reasoning in science to be a model of ideal inference. As long as you give it the reasonable prior probabilities, or evidence that overwhelms the prior, you get the correct answer. If the Bayesian model does not share ones priors and there is little evidence, then you should not listen to it: GIGO.

I'll skip over the difficulty concerning giving probabilities to mathematical statements.

The practical reality preventing combining probabilities of lots arguments...

I think the main reason people don't try the approach you mention is that assigning probability to a bunch of arguments and then concluding from those probabilities (a) doesn't convince anyone and (b) is quite hard. I'll explain this further, but first I should say that the argument from the 2014 article is a very simple one

Whatever your “naïve prior probability” was that P=NP, the above considerations, together with Bayes’ Rule, suggest revising it downward.

which could be summarised as "we weren't sure but then we tried really hard to find a counter example, not finding one should only make us more sure there isn't one". Or, as long as someone thinks N=NP being true would increase the chance of finding an example of P=NP (most people would accept this), then according to Bayesian reasoning not finding examples (when you have tried to) should reduce their belief in P=NP.

More formally, this would require a prior, P(X) (where X is the statement P=NP), and a posterior P(X|d) where d is the data from each attempt to find a counter-example. With everything I have said above, we have P(X|d) < P(X). To calculate this we need to know P(d|X) and P(not d|X).

But the combination of arguments requires a whole bunch of calculation which involves how much you believe each of the separate arguments (call them Y1, Y2, ...). The calculation in this case is of P(X|Y1,Y2,Y3,...), to work this out we'd need probabilities for all the combinatorial possibilities of Y1...YN and their negation, and these would all have to be judged individually, and might be very hard to assign these probabilities.

The argument in the 2014 article only required an the widely acceptable statement that "N=NP being true increases the chance of finding an example of P=NP", the combination of many arguments requires one to provide estimates of the probability of every argument, and every interaction between arguments - pretty much everyone would disagree about the correct probabilities to at least some small extent.

One can think of it as an argument with many premises (or it is exactly that if you believe Jaynes) - the more there are the more likely it is that someone will disagree with one of them. Moreover as the degree of belief becomes important it becomes very unlikely that any two people will agree.

Answer 2

The "ideal agent" interpretation of Bayesian reasoning in science.

I thought I would add this answer, which is a more direct response to the title of the question:

How is Bayesian reasoning related to the scientific method?

Here is the story I've heard expressed by a number of scientists, statisticians, and at least one philosopher. It is certainly not the only interpretation available, but I consider it to be quite defensible. In it, Bayesian reasoning makes no claim to determining truth on its own, only to giving an all important "standard of evidence" to scientific field.

The story goes roughly as follows:

It's how it works: Bayesian reasoning is a good description of how judgements are made in science, in particular, it can be used to summarise most (if not all) statistical inferences. It is also consistent with classical logic.

It's how it should work: The way we make inferences using statistics is pretty much correct (though we might be making tacit assumptions. The argument is then that these hidden assumptions are made clearer by using Bayesian reasoning.)

and we can do it more formally: We can use mathematics to describe a fictional agent that does Bayesian inference, and allow it prior knowledge that accords with the standing assumptions of our field. We can then give it our data and see what conclusions it draws. As we have agreed on the agents prior knowledge, and agreed that it reasons correctly, then we should be agreed on the conclusions it makes.

Basically, we make a model of how we think we should think and use it to check that we are thinking in that way. Clearly this requires one to accept the validity of the statistical model. That choice is up to you, just like it's up to you to decide if p<0.1 is actually meaningful in any application of classical statistics.

As for how the agent actually combines probabilities once it has been given them a scientific community, that's just Bayes rule.

Note: In actual practice, actually choosing a prior is generally avoided (as people can disagree), and people only report how a prior would change (likelihood ratios etc). Some do use explicit priors, but only in some rather specific domains

Answer 3

Hypotheses should not be assigned probabilities. There are a number of problems with trying to assign such probabilities. A theory is either right or wrong so it's a bit difficult to say what it means to say that a theory has a probability of 0.5, say. If it is just about some subjective feeling you have about it, why would that matter?

Epistemology is about how knowledge can be created. If you're tying to create new ideas you don't know what you're going to end up with, so what you have to do is propose guesses and then try to eliminate them by critical discussion. The role that probabilities can play in this is that if an idea predicts the wrong probabilities that is a criticism of that theory. No probabilities are assigned to theories themselves.

Another problem is that if you were to assign probabilities to theories what explains how you assign the probabilities? If it is, say, some physical theory then how do you assign probabilities to that theory? In reality, any such theory would have to be a guess and so all you really know is that the theories to which you have assigned probabilities are also guesses.

There are other problems. For example, Arrow's theorem and similar mathematical results raise problems for any process that purports to make decisions by weighing ideas rather than criticising them.

See "Realism and the Aim of Science" by Karl Popper and "The Beginning of Infinity" by David Deutsch, especially Chapter 13.