[SEQ RERUN] Infinite Certainty

[MinibearRex]

Today’s post, Infinite Certainty was originally published on 09 January 2008. A summary (taken from the LW wiki):

If you say you are 99.9999% confident of a proposition, you’re saying that you could make one million equally likely statements and be wrong, on average, once. Probability 1 indicates a state of infinite certainty. Furthermore, once you assign a probability 1 to a proposition, Bayes’ theorem says that it can never be changed, in response to any evidence. Probability 1 is a lot harder to get to with a human brain than you would think.

Discuss the post here (rather than in the comments to the original post).

This post is part of the Rerunning the Sequences series, where we’ll be going through Eliezer Yudkowsky’s old posts in order so that people who are interested can (re-)read and discuss them. The previous post was Absolute Authority, and you can use the sequence_reruns tag or rss feed to follow the rest of the series.

Sequence reruns are a community-driven effort. You can participate by re-reading the sequence post, discussing it here, posting the next day’s sequence reruns post, or summarizing forthcoming articles on the wiki. Go here for more details, or to have meta discussions about the Rerunning the Sequences series.

[XiXiDu]

Also see this follow-up by Yvain: Confidence levels inside and outside an argument.

[MinibearRex]

Could someone please post the following summaries?

The Fallacy of Gray

Nothing is perfectly black or white. Everything is gray. However, this does not mean that everything is the same shade of gray. It may be impossible to completely eliminate bias, but it is still worth reducing bias.

Absolute Authority

Those without the understanding of the Quantitative way will often map the process of arriving at beliefs onto the social domains of Authority. They think that if Science is not infinitely certain, or if it has ever admitted a mistake, then it is no longer a trustworthy source, and can be ignored. This cultural gap is rather difficult to cross.

Infinite Certainty

If you say you are 99.9999% confident of a proposition, you’re saying that you could make one million equally likely statements and be wrong, on average, once. Probability 1 indicates a state of infinite certainty. Furthermore, once you assign a probability 1 to a proposition, Bayes’ theorem says that it can never be changed, in response to any evidence. Probability 1 is a lot harder to get to with a human brain than you would think.

0 And 1 Are Not Probabilities

In the ordinary way of writing probabilities, 0 and 1 both seem like entirely reachable quantities. But when you transform probabilities into odds ratios, or log-odds, you realize that in order to get a proposition to probability 1 would require an infinite amount of evidence.

Beautiful Math

The joy of mathematics is inventing mathematical objects, and then noticing that the mathematical objects that you just created have all sorts of wonderful properties that you never intentionally built into them. It is like building a toaster and then realizing that your invention also, for some unexplained reason, acts as a rocket jetpack and MP3 player.

Expecting Beauty

Mathematicians expect that if you dig deep enough, a stable, or even beautiful, pattern will emerge. Some people claim that this belief is unfounded. But, we have previously found order in many of the places we’ve looked for it.

Is Reality Ugly?

There are three reasons why a world governed by math can still seem messy. First, we may not actually know the math. Secondly, even if we do know all of the math, we may not have enough computing power to do the full calculation. And finally, even if we did know all the math, and we could compute it, we still don’t know where in the mathematical system we are living.

Beautiful Probability

Bayesians expect probability theory, and rationality itself, to be math. Self consistent, neat, even beautiful. This is why Bayesians think that Cox’s theorems are so important.

Trust in Math

When you find a seeming inconsistency in the rules of math, or logic, or probability theory, you might do well to consider that math has rightfully earned a bit more credibility than that. Check the proof. It is more likely that you have made a mistake in algebra, than that you have just discovered a fatal flaw in math itself.

[arundelo]

Done!

[Vladimir_Nesov]

I think you should just register another username, the problem with the old one doesn’t appear to be being solved.

[MinibearRex]

I tried a few times. Same problem.