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Priors

TagLast edit: 2 Oct 2020 0:26 UTC by Ruby

In the context of Bayes’s Theorem, priors refer generically to the beliefs an agent holds regarding a fact, hypothesis or consequence, before being presented with evidence. Upon being presented with new evidence, the agent can multiply their prior with a likelihood distribution to calculate a new (posterior) probability for their belief.

Examples

Suppose you had a barrel containing some number of red and white balls. You start with the belief that each ball was independently assigned red color (vs. white color) at some fixed probability. Furthermore, you start out ignorant of this fixed probability (the parameter could be anywhere between 0 and 1). Each red ball you see then makes it more likely that the next ball will be red, following a Laplacian Rule of Succession. For example, seeing 6 red balls out of 10 suggests that the initial probability used for assigning the balls a red color was .6, and that there’s also a probability of .6 for the next ball being red.

On the other hand, if you start out with the prior belief that the barrel contains exactly 10 red balls and 10 white balls, then each red ball you see makes it less likely that the next ball will be red (because there are fewer red balls remaining).

Thus our prior affects how we interpret the evidence. The first prior is an inductive prior—things that happened before are predicted to happen again with greater probability. The second prior is anti-inductive—the more red balls we see, the fewer we expect to see in the future.

As a real life example, consider two leaders from different political parties. Each one has his own beliefs—priors—about social organization and the roles of people and government in society. These differences in priors can be attributed to a wide range of factors, ranging from their educational backgrounds to hereditary differences in personality. However, neither can show that his beliefs are better than those of the other, unless he can show that his priors were generated by sources which track reality better1.

Because carrying out any reasoning at all seems to require a prior of some kind, ideal Bayesians would need some sort of priors from the moment that they were born. The question of where an ideal Bayesian would get this prior from has occasionally been a matter of considerable controversy in the philosophy of probability.

Updating prior probabilities

In informal discussion, people often talk about “updating” their priors. This is technically incorrect, as one does not change their prior probability, but rather uses it to calculate a posterior probability. However, as this posterior probability then becomes the prior probability for the next inference, talking about “updating one’s priors” is often a convenient shorthand.

References

Blog posts

See also

References

  1. Robin Hanson (2006). “Uncommon Priors Require Origin Disputes”. Theory and Decision 61 (4) 319–328. http://​​hanson.gmu.edu/​​prior.pdf

Pri­ors as Math­e­mat­i­cal Objects

Eliezer Yudkowsky12 Apr 2007 3:24 UTC
46 points
19 comments4 min readLW link

Against im­proper priors

DanielLC26 Jul 2011 23:50 UTC
6 points
21 comments2 min readLW link

A Priori

Eliezer Yudkowsky8 Oct 2007 21:02 UTC
59 points
134 comments4 min readLW link

Learn­ing the prior

paulfchristiano5 Jul 2020 21:00 UTC
78 points
26 comments8 min readLW link
(ai-alignment.com)

Learn­ing the prior and generalization

evhub29 Jul 2020 22:49 UTC
34 points
16 comments4 min readLW link

Separat­ing the roles of the­ory and di­rect em­piri­cal ev­i­dence in be­lief for­ma­tion: the ex­am­ples of min­i­mum wage and an­thro­pogenic global warming

VipulNaik25 Jun 2014 21:47 UTC
38 points
66 comments4 min readLW link

The Solomonoff Prior is Malign

Mark Xu14 Oct 2020 1:33 UTC
129 points
34 comments16 min readLW link

Pri­ors and Surprise

MichaelVassar3 Mar 2010 8:27 UTC
23 points
32 comments2 min readLW link

Pri­ors Are Useless

DragonGod21 Jun 2017 11:42 UTC
2 points
22 comments1 min readLW link

Cry­on­ics priors

AnthonyC20 Jan 2013 22:08 UTC
9 points
22 comments1 min readLW link

Re­vis­ing pri­ors and an­thropic reasoning

PhilGoetz6 Feb 2011 5:42 UTC
3 points
27 comments1 min readLW link

Sim­plic­ity pri­ors with re­flec­tive oracles

Benya_Fallenstein15 Nov 2014 6:39 UTC
1 point
0 comments6 min readLW link

Against im­proper priors

DanielLC26 Jul 2011 23:50 UTC
6 points
21 comments2 min readLW link

The prior of a hy­poth­e­sis does not de­pend on its complexity

cousin_it26 Aug 2010 13:20 UTC
34 points
69 comments1 min readLW link

The uni­ver­sal prior is malign

paulfchristiano30 Nov 2016 22:31 UTC
14 points
0 comments1 min readLW link
(ordinaryideas.wordpress.com)

Chap­ter 49: Prior Information

Eliezer Yudkowsky14 Mar 2015 19:00 UTC
8 points
6 comments15 min readLW link

Believ­ing oth­ers’ priors

rk22 Nov 2018 20:44 UTC
8 points
19 comments7 min readLW link

Trapped Pri­ors As A Ba­sic Prob­lem Of Rationality

Scott Alexander12 Mar 2021 20:02 UTC
109 points
25 comments14 min readLW link

Fre­quen­tist prac­tice in­cor­po­rates prior in­for­ma­tion all the time

Maxwell Peterson7 Nov 2020 20:43 UTC
17 points
0 comments4 min readLW link

[Question] What are some low-in­for­ma­tion pri­ors that you find prac­ti­cally use­ful for think­ing about the world?

Linch7 Aug 2020 4:37 UTC
27 points
13 comments1 min readLW link
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