"Extraordinary claims demand extraordinary evidence."
As I've written before [at length, poorly, and desperately in need of an editor], I find the attitude of most people who use this phrase counterproductive. But instead of pointlessly arguing back and forth like they do in certain disciplines, we'll dig into the numbers involved and see what we can learn.
I know a super-genius, an Einstein-grade mind, who for decades believed that "this is the year that the Red Sox will come back and start a long series of victories," a belief unfounded in reality.
Yes, very smart people can have strong beliefs that appear nonsensical to others.
Let's say that the claim is about some proposition ("God exists," "Red Sox are a great team") which we'll call $G$. The prior belief in $G$ we'll denote $q \doteq \Pr[G]$; so a person may be a strong believer if $q = .99$ or a moderate believer if $q = .80$.
Let's call the evidence against $G$, $E$, which is a binary observable ("no Rapture","loss against the Chicago Cubs"), with the probability of observing the evidence given that $G$ is false denoted by $p \doteq \Pr[E| \neg G]$. We'll consider evidence that has symmetric error probabilities, $\Pr[E|G] = 1 - \Pr[E|\neg G]$ so the probability that we get a false positive is equal to that of a false negative, $1-p$.
For example, if $p=0.90$ there's a 10% chance of no Rapture even if God exists; if $p = 0.99$, then there's only a 1% probability of the faithful burning up in Hellfire on Earth with the rest of us sinners, when there is a God. Note that with symmetric errors, $p=0.90$ has the interesting characteristic that there's a 10% change of Rapture with no God at all, which probably would say something about the design of the experiment (psilocybin would be my guess).
Now this is the question we want to ask: for any given prior belief $q$ in $G$, how good would the evidence against it have to be (meaning how big would $p$ have to be) to convince the believer to flip her beliefs, i.e. to believe against $G$ with the probability $q$, or formally, to have $\Pr[G|E] = 1-q$. The reason to go for a flip of beliefs is empirical: no zealot like a convert.
(Really trying to goose up page views here. Was it Stephen Hawking who said a book's potential audience is halved by each formula in it? This blog must be down to individual quarks...)
For example, if JohnDCL believes in the greatness of the Red Sox with $q= 0.99$, how strong a piece of evidence of Sox suckage would be necessary for JohnDCL to think that the probability of the Red Sox being great is only 1%?
The result is $p \approx 0.9999$, in other words, JohnDCL would have to believe that the evidence only gives a false positive (it's evidence against $G$, remember) once every 10,000 tries.
Let's say the evidence is losing against the Chicago Cubs. For JohnDCL to flip his beliefs based on observing such a defeat, he'd have to believe that, were the Sox a great team, they could play the Cubs 10,000 times and lose only once. (Recall that we're assuming symmetric errors, for simplicity.)
Here are a few other values for $q$ and corresponding $p$:
$q = 0.999 \Rightarrow p \approx 0.999 999$ one false positive in a million tries;
$q = 0.9999 \Rightarrow p \approx 0.999 999 99$ one false positive in one hundred million tries;
$q = 0.99999 \Rightarrow p \approx 0.999 999 999 9$ one false positive in ten billion tries.
$q = 0.999999 \Rightarrow p \approx 0.999 999 999 999$ one false positive in one trillion tries.
(How strong is faith in the Red Sox? In God? In Quantitative Easing Forever And Ever?)
In other words, it's true that to reverse a strong belief you need extraordinary evidence. What is equally true is that the beliefs and the evidence aren't perceived equally by all participants in a conversation. People who proselytize for a cause will not be able to convince anyone else until they see the probabilities from the other person's point of view.
Of course, those who say "extraordinary claims demand extraordinary evidence" typically see the world from their own point of view only.