h r
e p o
r p
o o
p s
o istiton
o s
n . S
s y
. S m
y b
m o
b l
o ilcal
ca ly,y
18
19
Cre
Cr d
e e
d n
e c
n e i
c n p
e i
r
n p o
r p
o os
p i
os tiio
t n
io X
n
Lik
L e
ik l
e iho
l
o
iho d
o o
d f
o ob
f
s
ob e
s r
e v
r ati
v on
ati
on D
Pr
P io
r r cr
io
e
r cr d
e e
d n
e c
n e
c
gi
g v
i e
v n o
e
bs
n o e
bs r
e v
r a
v t
a io
t n
io D
n
20
20
gi
g v
i e
v n p
e
r
n p o
r p
o os
p i
os tiio
t n
io X
n X
in p
i
r
n p o
r p
o os
p i
os tiio
t n
io X
n
21
22
Tha
Th t
a ’ts B
s a
B y
a e
y s
e ’ss The
s Th o
e r
o e
r m i
em n a n
in a u
n t
u sh
s e
h l
e l. The s
e symbol “
l “ ” m
” e
m a
e n
a s “
ns i“s p
is r
p o
r p
o o
p r
o -r‑
23
titon
o a
n l
a to
l t .o” It
” I ’ts ju
s j s
u t a r
s
e
t a r m
e i
m n
i d
n e
d r t
e h
r t a
h t w
a e s
t w h
e s o
h u
o l
u d ma
d m k
a e s
k u
e s r
u e a
r l
e a l o
l f o
l o u
f o r c
u r
r c e
r d
e e
d n
e c
n e
c s
e
24
ad
a d u
d p t
d u o 1 a
p t
t t
o 1 a h
t t e e
h n
e e d o
n f t
d o h
f t e d
h a
e d y
a .y
25
•
[
26
27
It f
I e
t f e
e les n
l a
s n t
a u
t r
u a
r l t
a o a
l t s
o a ssisgin n
g u
n n m
u e
m r
e ircal c
ca r
l c e
r d
e e
d n
e c
n e
c s i
e n c
s i e
n c r
e tra
t ian ca
i
s
n ca e
s s
e , l
s i
, l k
i e p
k o
e p k
o e
k r
e r
28
ha
h n
a d
n s
d or fl
s o i
r fl ps
p of a
s o co
f a c i
o n
i ,
n wh
, w e
h r
e e w
r
e
e w ca
e c n s
a i
n s m
i p
m l
p y co
y c u
o n
u t
n al
t a l t
l h
l t e p
h o
e p s
o ssisb
i i
b liiltites
e .s
29
We
W ’ere a
re l
a slo
s fa
o f m
a i
m liilair
a wi
r w th
t us
h u isn
i g
n g pr
p o
r b
o a
b b
a i
b liilty
it -
y t‑ a
t lak w
lk h
w e
h n r
e e
n r f
e e
f r
e r
r irn
i g
n to f
g t u
o f t
u u
t r
u e
r e
30
ev
e e
v n
e t
n st: “
s Th
: “ e
Th r
e e i
r s l
e i e
s l s
e s t
s h
s t a
h n a 1 p
a
e
n a 1 p r
e c
r e
c n
e t ch
n
a
t ch n
a c
n e t
c h
e t a
h t t
a h
t t
e o
h n
e o c
n o
c m
o i
m n
i g a
n s
g a tse
t r
e o
r i
o d
d
31
wi
w lil i
l m
l i p
m a
p c
a t t
c h
t t e E
h a
e E r
a trh a
t n
h a d ca
n
u
d ca s
u e a m
s
a
e a m s
a s e
s x
s e t
x itn
i c
n t
c iton
o .
n”
32
The B
Th a
e B y
a e
y s
e isain a
a p
n a p
p r
p o
r a
o ch i
a
s m
ch i
u
s m ch m
u
o
ch m r
o e g
r e
e g n
e e
n r
e a
r l t
a h
l t a
h n t
a h
n t i
h si, h
s o
, h w
o e
w v
e e
v r
e . I
r t
. I t
33
re
r m
e i
m n
i d
n s u
d s t
s u h
s t a
h t w
a e a
t w s
e a ssisgin p
g r
n p iror c
o r
r c e
r d
e e
d n
e c
n e
c s
e , a
s n
, a d u
n p
d u d
p a
d t
a e t
t h
e t e
h m a
e
p
m a p
p r
p o
r p
o r
p irait
a e
t ley,
y
34
to
to ev
e e
v rey f
r a
y f c
a tcu
t a
u l p
a r
l p o
r p
o o
p soistiitoin t
o h
n t a
h t m
a a
t m y o
a r m
y o a
r m y n
a o
y n t b
o e t
t b r
e t u
r e a
u b
e a o
b ut t
o h
ut t e w
h or
e wo lrd
l .
d
35S
Do
D e
o s G
e
o
s G d e
o x
d e i
x sits? C
t
a
? C n o
a
u
n o r i
u n
r i n
n e
n r c
e o
r c n
o s
n c
s ico
i u
o s e
u x
s e p
x e
p r
e irein
e c
n e
c s b
e e e
s b x
e e p
x l
p alian
i e
n d i
e n
d i
n
36N
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purely physical terms? Are there objective standards of right and wrong?
01
All of the possible answers to such questions are propositions for which
02
each of us has a prior credence (whether we admit it or not), and which we
03
update when relevant new information comes in (whether we do so cor-
04
rectly or not).
05
Bayes’s Theorem allows us to be quantitative about our degrees of belief,
06
but it also helps us keep in mind how belief works at all. Thinking about
07
credences in this way provides a number of useful lessons.
08
Prior beliefs matter. When we’re trying to understand what is true
09
about the world, everyone enters the game with some initial feeling about
10
what propositions are plausible, and what ones seem relatively unlikely. This
11
isn’t an annoying mistake that we should work to correct; it’s an absolutely
12
necessary part of reasoning in conditions of incomplete information. And
13
when it comes to understanding the fundamental architecture of reality,
14
none of us has complete information.
15
Prior credences are a starting point for further analysis, and it’s hard to
16
say that any particular priors are “correct” or “incorrect.” There are, needless
17
to say, some useful rules of thumb. Perhaps the most obvious is that simple
18
theories should be given larger priors than complicated ones. That doesn’t
19
mean that simpler theories are always correct; but if a simple theory is
20
wrong, we will learn that by collecting data. As Albert Einstein put it: “The
21
supreme goal of all theory is to make the irreducible basic elements as sim-
22
ple and as few as possible without having to surrender the adequate repre-
23
sentation of a single datum of experience.”
24
Simplicity is sometimes easy to gauge; sometimes it is less so. Consider
25
three competing theories. One says that the motion of planets and moons
26
in the solar system is governed, at least to a pretty good approximation, by
27
Isaac Newton’s theories of gravity and motion. Another says that Newto-
28
nian physics doesn’t apply at all, and that instead every celestial body has an
29
angel assigned to it, and these angels guide the planets and moons in their
30
motions through space, along paths that just coincidentally match those
31
that Newton would have predicted.
32
Most of us would probably think that the first theory is simpler than the
33
second— you get the same predictions out, without needing to invoke
34
vaguely defined angelic entities. But the third theory is that Newtonian
S35
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01
gravity is responsible for the motions of everything in the solar system ex-
02
cept for the moon, which is guided by an angel, and that angel simply
03
chooses to follow the trajectory that would have been predicted by Newton.
04
It is fairly uncontroversial to say that, whatever your opinion about the first
05
two theories, the third theory is certainly less simple than either of them. It
06
involves all of the machinery of both, without any discernible difference in
07
empirical predictions. We are therefore justified in assigning it a very low
08
prior credence. (This example seems frivolous, but analogous moves become
09
common when we start talking about the progress of biological evolution
10
or the nature of consciousness.)
11
Some people don’t like the Bayesian emphasis on priors, because they
12
seem subjective rather than objective. And that’s right— they are. It can’t
13
be helped; we have to start somewhere. On the other hand, ideally the
14
likelihoods of making certain observations can be objectively deter-
15
mined. If you have a certain theory about the world, and that theory is
16
precise and well-defined, you can say with confidence what the chances are
17
of observing various bits of data under the assumption that your theory is
18
correct. In realistic circumstances, of course, we are often stuck trying to
19
evaluate theories that aren’t so rigorously defined in the first place. (“Con-
20
sciousness transcends the physical” is a legitimate proposition, but it’s not
21
sufficiently precise to make quantitative predictions.) Nevertheless, it’s our
22
job to try to make our propositions as well-defined as possible, to the point
23
where we can use them to objectively establish the likelihoods of different
24
observations.
25
Everyone’s entitled to their own priors, but not to their own likelihoods.
26
Evidence should move us toward consensus. You might worry that
27
having subjective priors could make it hard for some people to ever reach
28
agreement. If I assign a prior credence of 0.000001 to an idea like “God
29
created the universe,” and you assign a prior credence of 0.999999 to the
30
same proposition, it would require some serious updating on the basis of
31
observations before one of us changed our view.
32
In practice, that’s a real problem. People have certain views that they’re
33
just never going to change, which in Bayesian language corresponds to pri-
34
ors set to 0 or 1. That’s too bad, and something we need to learn to deal with
35S
in the real world.
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But in principle, if we are all trying to be fair and open- minded
01
and willing to change our beliefs in the face of new information, evidence
02
will win out in the end. You can assign a very high prior credence to some
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