I would however suggest that a delusion about a major branch of academia could potentially have serious results unless the belief is very carefully compartmentalized from impacting other beliefs.

That is essentially what I was getting at in paragraph 4.

This supports my position. While delusion is low-cost for most people (as I explain in paragraph 3), it is not low-cost for everyone (as I explain in paragraph 4). When delusion is high-cost, then a good strategy is to avoid commitment, to admit ignorance, when the assigned probability is below a high threshold. Paragraph 5 says that this is usually true of facts critical to the success of everyday actions. For example, crossing the street: it is a good idea to look carefully both ways before crossing a street. It's not enough to be 90% sure that there are no cars coming close enough to run over you. That is insufficiently high, because you'll be run over within days if you cross the street with such a low level of certainty. You need to be well north of 99.9% certain that there are no cars coming before you act on the assumption that there are no cars (i.e. by crossing the street). That's the only way you can cross the street day after day for eighty years without coming to harm.

It seems far from what most people would say (I don't think that term when generally used takes the pay-off into account).

People don't consciously consider it, but the brain is a machine that furthers the interest of the animal, and so the brain can I think be relied upon to take costs and benefits into account in decisions, and therefore in beliefs. For example, what does it take for a person to be convinced that there are no cars coming? If people were willing to cross the street with less than 99.9% probability that there are no cars coming, we would be seeing vastly more accidents than we do. It seems clear then to me that people don't act as if they're convinced unless the probability is extremely high. We can tell from the infrequency of accidents, that people aren't satisfied that there are no cars coming unless they've assigned an extremely high probability to it. This must be the case whatever they admit consciously.

In the meantime this does not extend to other matters. People are easily satisfied of claims about society, the economy, the government, celebrities, where the assigned probability has to be well below 99.9%.

I'm curious, given this situation, what evidence would you consider sufficient to convince you that Andrew is right? What evidence would convince you that Andrew is wrong?

That's a very difficult question to answer. I think it's hard to know ahead of time, hard to model the hypothetical situation before it happens. But I can try to reason from analogous claims. Humans are complex, and so is their biology. So, let's ask how much evidence it takes to convince the FDA that a drug works, that it does more good than harm. As you know, it's quite expensive to conduct a study that would be convincing to the FDA. Now, it could be that the FDA is far too careful. So let's suppose that the FDA is far too careful by a factor of 100. So, whatever it typically costs to prove to the FDA that a drug work, divide that by 100 to get a rough estimate of what it should take to establish whether what Andrew says is true (or false).

The first article I found says:

Estimates about the cost of developing a new drug vary widely, from a low of $800 million to nearly $2 billion per drug.

And since we're talking clinical trials, we're talking p-value of 5. That means that, if the drug doesn't work at all, there's a 1 in 20 chance that the trial will spuriously demonstrate that it works. While it depends on the particular case, my guess is that a Bayesian watching the experiment will not assign a probability all that high to the value of the drug. Add to this that even many drugs that work on average don't work at all on an alarming fraction of patients, and the fact that the drug works is a statistical fact, not a fact about each application. So we're not getting a high probability about the success of individual application from these expensive trials.

Dividing by 100, that's $8 million to $20 million.

Okay, let's divide by 100 again. That's $80 thousand to $200 thousand.

So, now I've divided by ten thousand, and the cost of establishing the truth to a sufficiently high standard comes to around a hundred thousand dollars - about a year's pay for a bright, well-educated, hard-working individual.

That doesn't seem that unreasonable to me, because the notion of a person taking a year out of his life to check something seems not at all unusual. But what about crossing the street? It doesn't cost a hundred thousand dollars to tell whether there are cars coming. Indeed not - but it's a concrete fact about a specific time and place, something we can easily and inexpensively check. There are different kinds of facts, some harder than others to check. So the question is, what kind of fact is Andrew's claim? My sense of it is that it belongs to the category of difficult-to-check.

But it might not. That really depends on what method a person comes up with to check the claim. Emily Rosa's experiment on therapeutic touch is praised because it was so inexpensive and yet so conclusive. So maybe there is an inexpensive and conclusive demonstration either pro or con Andrew's claim.

You have only pointed out an incompleteness in my account that I already pointed out. I pointed out that below 50%, the account I gave of being convinced no longer seems to hold.

The perfect is the enemy of the good. That an account does not cover all cases does not mean the account is not on the right track. A strong attack on the account would be to offer a better account. JoshuaZ already offered an alternative account by implication, which (as I understand it) that belief is simply a constant cutoff, for example, a probability assignment above 80% is belief, or maybe 50%, or maybe 90%.

But here's the thing: if you believe something, aren't you willing to act on it? We regularly explain our actions in terms of beliefs. For example, suppose you walk out of the house taking your wife's car keys. You get to your car, notice that you can't start the engine, and at that point discover that you are holding your wife's car keys. Suppose she asks you, "why did you take my keys"? The answer seems obvious: "I took these keys because I believed they were my car keys." Isn't that obvious? Of course that's why you took them.

To restate, you did something that would have been successful had those keys been your keys. To restate, you acted in a way that would have been successful had your belief been true.

And I think this is generally a principle by which we explain our actions, particularly our mistaken actions. The explanation is that we acted in a way that would have worked out had our beliefs been correct. And so, your actions reveal your beliefs. By taking your wife's car keys, you reveal your belief that they are your car keys.

So your actions reveal your beliefs. But here's the problem: your actions are a product of a combination of your probability assignments and your value assignments, the costs and benefits. That's why you are more ready to take risky action when the downside is low and the upside is high, and less ready to take risky action when the downside is high and the upside is low. So your actions are a product of a combination of probability assignments and value assignments.

But your actions meanwhile are in accordance with your beliefs.

Conclusion follows: your beliefs are a product of a combination of probability assignments and value assignments.

Now, as I said, this picture is incomplete. But it seems to hold within certain limits.