[aston]

I don't really like this view of hypothesis. Or maybe I just don't understand it.

Take this hypothesis: 2+2 = 4. There are an infinite number of things 2+2 could equal, but I, like Einstein, without any empirical testing of my mathematical hypothesis, believe it to be 4. If you believed the sum to be 5, again without any testing, I don't see how you or I have any different number of bits dedicated to our answers. Your bits, however, are wrong.

[jey]

There's plenty of empirical evidence for 2 + 2 = 4, it's just so pervasive that we fail to recognize it as such. Take two apples, place them on a table. Now take two more apples, and place them on the table. How many apples are on the table? 3? 4? 5? I have an astronomically high probability assigned to "2+2=4", but if I kept doing this experiment and suddenly ended up with 3 apples every time, I'd have to decrease the probability I assign to 2+2=4, and increase the probability assigned to 2+2=3.

The apples example is from "How to Convince me that 2 + 2 = 3": http://www.overcomingbias.com/2007/09/how-to-convince.html

Interesting essay on Bayesian reasoning: http://yudkowsky.net/bayes/technical.html

[Tichy]

From the article:

"But from a Bayesian perspective, you need an amount of evidence roughly equivalent to the complexity of the hypothesis"

But what is the "complexity of the hypothesis"? Without a proper definition, there is not much left to the article, or is there?

[Eliezer]

Fixed with linky.

[cellis]

By my contribution, I am guilty by association. But do you realize how pointless this discussion is getting? Ha. Only on hacker news.

2+2 = 4. Thus it is written.

[jey]

IMHO, I think Bayesianism is more interesting than a lot of the stuff discussed here. The idea that all reasoning can be done as Bayesian inference is pretty damn cool. If someone looks at the above and thinks "Whoa, Bayesianism is neat! I should go read more about it!" then I'm happy, but if everyone is just thinking "Duh, of course 2 + 2 = 4"... well, shit.

[aston]

I don't disagree, but my hypothetical assumes that no one has done the experiment before making their belief known.

[jey]

Sounds like an unreasonable assumption. Either way, it's not about "wrong" or "right", it's only about what can be derived from evidence. Without evidence you have no justification to assign a higher probability to "2 + 2 = 4" than "2 + 2 = 3".

Caveat: It's true that some knowledge does have to be put into the prior distribution. Apparently you can't just bootstrap from a zero-information ("maximum entropy") prior. So it's possible that 2 + 2 = 4 is burned into our brains at birth.

[whacked_new]

> So it's possible that 2 + 2 = 4 is burned into our brains at birth.

Yup, there is plentiful evidence... not specifically the addition, but 2 and 4 are innately perceptible quantities.

[whacked_new]

I don't even think it is a good idea to think about theoretical physics as the blogger does. Math doesn't require any probabilistic validation whatsoever. Biology needs lots of it because there are too many unknowns. Physics is much closer to math than to biology, and evaluating it like one does a biological experiment, to me, is perplexing. For 2 + 2 = 4, there doesn't seem to be any need for "bits of information." And even though the author is probably talking about hunches and intuition, the probability that a theoretical physics can go about his field thinking in hunches seems small enough... ok, maybe this statement required 27 bits of information?

Maybe I am missing something.