Invited Talk given at "Maximum Entropy and Bayesian Methods in Science & Engineering", Ghent (Belgium), July 13th 2016

1. MaxEnt 2016
13th July, Ghent
Cosmology:
A Bayesian Perspective
Peter Coles
(@telescoper)

4. Lecture 1
Probability

5. “The Essence of Cosmology is Statistics”
George McVittie

6. Direct versus Inverse
Reasoning
Theory
(, H0…)
Observations

7. 14 July, 2016
Picture: R. Trotta

8. Urn A Urn B
999 white
1 black
999 black
1 white
P(white ball | urn is A)=0.999, etc

9. Balls
• Two urns A and B.
• A has 999 white balls and 1 black one; B
has 1 white balls and 999 black ones.
• P(white| urn A) = .999, etc.
• Now shuffle the two urns, and pull out a
ball from one of them. Suppose it is white.
What is the probability it came from urn
A?
• P(Urn A| white) requires “inverse”
reasoning: Bayes’ Theorem

10. Bayes’ Theorem
• In the toy example, X is “the urn is A” and Y is
“the ball is white”.
• Everything is calculable, and the required
posterior probability is 0.999
I)|P(Y
I)X,|I)P(Y|P(X
=I)Y,|P(X

12. The Expanding Universe

29. Picture

30. 14 July,
2016

33. Fine Tuning
• In the standard model of cosmology the
free parameters are fixed by observations
• But are these values surprising?
• Even microscopic physics seems to have
“unnecessary” features that allow
complexity to arise
• Are these coincidences? Are they
significant?
• These are matters of probability…

34. What is a Probability?
• It’s a number between 0 (impossible) and 1
(certain)
• Probabilities can be manipulated using simple
rules (“sum” for OR and “product” for “AND”).
• But what do they mean?
• Standard interpretation is frequentist
(proportions in an ensemble)

35. Bayesian Probability
• Probability is a measure of the “strength
of belief” that it is reasonable to hold.
• It is the unique way to generalize
deductive logic (Boolean Algebra)
• Represents insufficiency of knowledge to
make a statement with certainty
• All probabilities are conditional on stated
assumptions or known facts, e.g. P(A|B)
• Often called “subjective”, but at least the
subjectivity is on the table!

36. Bayes’ Theorem: Inverse
reasoning
• Rev. Thomas Bayes
(1702-1761)
• Never published
any mathematical
papers during his
lifetime
• The general form
of Bayes’ theorem
was actually given
later (by Laplace).

37. Probable Theories
I)|P(D
I)H,|I)P(D|P(H
=I)D,|P(H
• Bayes’ Theorem allows us to assign probabilities
to hypotheses (H) based on (assumed)
knowledge (I), which can be updated when data
(D) become available
• P(D|H,I) – likelihood
• P(H|I) – prior probability
• P(H|D,I) – posterior probability
• The best theory is the most probable!

38. Prior and Prejudice
• Priors are essential.
• You usually know more than you
think..
• Flat priors usually don’t make much
sense.
• Maximum entropy, etc, give useful
insights within a well-defined theory:
“objective Bayesian”
• “Theory” priors are hard to assign,
especially when there isn’t a theory…

40. Why is the Universe
(nearly) flat?
• Assume the
Universe is one of
the Friedman
family
• Q: What should we
expect, given only
this assumption?
• Ω=1 is a fixed
point (so is Ω=0)..
• The Universe is
walking a
tightrope..

41. ˙a
2
=
8πGρ
3
a
2
− kc
2
The Friedman Models
The simplest relativistic cosmological models are
remarkably similar (although the more general
ones have additional options…)
¨a= −
4πGρ
3
a
Solutions of these are complicated, except
when k=0 (flat Universe). This special case is
called the Einstein de Sitter universe.
Notice that
ρ∝
1
a3
For non-relativistic
particles (“dust”)
Curvature

44. Cosmological Parameters
We do not know how to set the initial conditions for the
expanding Universe, nor do we know precisely what forms of
matter and energy fill the Universe. What we have to do is
make models and see if they fit the observations. A “model” is
a solution of the Friedmann equation and is usually written in
terms of a set of parameters
Km
a
kcG
a
a
kc
a
a
G
a







1
33
8
33
8
2
2
2
2
2
2
22





45. TG 
Gravity Stuff

46. TG ?

47. ? TG

48. Whatever it is, Dark
Energy is a terrible name
for it…
• What is important is not so much the
energy, but the pressure…
• Dark Energy has to act like something
with negative pressure (or tension)
• 𝑝 = 𝑤𝜌𝑐2
= (𝛾 − 1)𝜌𝑐2

49. 14 July, 2016

50. Other Observations
• Supernovae (Type Ia)
• Large-scale structure measurements (BAO)
• Gravitational Lensing (Weak and Strong)
• CMB lensing
• Peculiar Velocities

52. fainter

57. EUCLID

59. SAY “PRECISION
COSMOLOGY”
ONE MORE TIME…

60. Bayesian Hypothesis Testing
Two of the advantages of this is that it
doesn’t put one hypothesis in a special
position (the null), and it doesn’t
separate estimation and testing.
Suppose Dr A has a theory that makes a
direct prediction while Professor B has
one that has a free parameter, say .
Suppose the likelihoods for a given set of
data are P(D|A) and P(D|B,)

61. Occam’s Razor





λ)B,|(DB)|(λdλ
A)|(D
(B)
(A)
λ)B,|(Dλ)(B,dλ
A)|(D(A)
D)|λ(B,dλ
D)|(A
=
D)|(B
D)|(A
PrPr
Pr
Pr
Pr
PrPr
PrPr
Pr
Pr
Pr
Pr
Occam
factor

62. Why does this help?
• Rigorous Form of Ockham’s Razor: the hypothesis
with fewest free parameters becomes most
probable.
• Can be applied to one-off events (e.g. Big Bang)
• It’s mathematically consistent!
• It can even make sense of the Anthropic
Principle…

63. Bayesian estimation


aI)d,aa|xI)p(x|ap(a=K
I),aa|xI)p(x|aKp(a=I),xx|ap(a
m
mnm
mnmnm
.......
............
111
1
11111
This involves finding the posterior
distribution of the parameters given the
data and any prior information.
Evidence!

65. A)!|P(MM)|P(A 
Beware the Prosecutor’s
Fallacy!

67. The “Maximally Boring
Universe”?
• There are many unanswered theoretical
questions!
• So far the questions we’ve asked have
been the “easy” ones
• Now that this “boring” stuff is out of the
way, cosmology will start to get
interesting!
• Because we now have a better idea what
to ask!

68. “CONCORDANCE”

70. Ingredients of the Standard
Cosmology
•General Relativity
•Cold Dark Matter
•Cosmological Constant
•Cosmological Principle
•Primordial Gaussian fluctuations
•Inflation
•Baryons
•Neutrinos
•Radiation…

71. Questionable Aspects of the
Standard Cosmology
•General Relativity
•Cold Dark Matter
•Cosmological Constant
•Cosmological Principle
•Primordial Gaussian fluctuations
•Inflation
•Baryons
•Neutrinos
•Radiation…

72. Cosmology is an exercise in data compression
Cosmology is a massive
exercise in data
compression...
….but it is worth looking at
the information that has
been thrown away to check
that it makes sense!

74. “If tortured sufficiently, data
will confess to almost
anything”
Fred Menger

75. Theories
Observations
FrequentistBayesian

76. Precision Cosmology
“…as we know, there are known knowns;
there are things we know we know. We also
know there are known unknowns; that is to
say we know there are some things we do not
know. But there are also unknown unknowns
-- the ones we don't know we don't know.”

77. How Weird is the Universe?
• The (zero-th order) starting point is
FLRW.
• The concordance cosmology is a “first-
order” perturbation to this
• In it (and other “first-order” models),
the initial fluctuations were a
statistically homogeneous and isotropic
Gaussian Random Field (GRF)
• These are the “maximum entropy”
initial conditions having “random
phases” motivated by inflation.
• Anything else would be weird….

78. Beyond the Power
Spectrum
• So far what we have discovered is
largely based on second-order
statistics…
• This is fine as long as we don’t throw
away important clues…
• ..ie if the fluctuations are statistically
homogeneous and istropic, and
Gaussian..

79. Weirdness in Phases
ΔT (θ,φ)
T
= ∑ ∑ al,mYlm(θ,φ)
   ml,ml,ml, ia=a exp
For a homogeneous and isotropic Gaussian
random field (on the sphere) the phases are
independent and uniformly distributed. Non-
random phases therefore indicate weirdness..

86. The Prosecutor’s Fallacy!
P(A|M)P(M|A)

87. CMB Anomalies
•Type I – obvious problems with data
(e.g. foregrounds)
•Type II – anisotropies and alignments
(North-South, Axis of Evil..)
•Type III – localized features, e.g. “The
Cold Spot”
•Type IV – Something else (even/odd
multipoles, magnetic fields, ?)

89. Low Quadrupole?

90. Parity Violation?

91. (from Copi et al. 2005)

92. (from Hansen et al. 2004)