- Cosmology relies heavily on statistics and probability to analyze astronomical data and test theories of the universe. - Bayesian probability provides a rigorous way to assign probabilities to hypotheses based on prior knowledge and new data, and update beliefs. - The universe appears "fine tuned" for life with parameter values that allow complexity; Bayesian reasoning can help assess if these are surprising coincidences. - The concordance model of cosmology posits that initial fluctuations in the early universe formed a Gaussian random field, but some anomalies in cosmic microwave background data could indicate "weirdness" beyond this simple picture.

1. Statistics in Astronomy
Peter Coles
STFC Summer School, Cardiff
27th
August 2015

4. Lecture 1
Probability

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

19. August 29, 2015

21. 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.”

22. SAY “PRECISION
COSMOLOGY”
ONE MORE TIME…

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

24. Direct versus Inverse
Reasoning
Theory
(Ω, H0…)
Observations

25. 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…

26. 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)

27. 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!

28. 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

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

30. 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).

31. 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

32. 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!

33. 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…

34. Null Hypotheses
• The frequentist approach to statistical
hypothesis testing involves the idea of a
null hypothesis H0,which is the model
you are prepared to accept unless there
is evidence to the contrary.
• Under the null hypothesis one then
constructs the sampling distribution of
some statistic Q, called f(Q).
• If the measured value of Q is unlikely
on the basis of H0 then the null
hypothesis is rejected.

35. Type I and Type II Errors
• There are two ways of making an error in this
kind of test.
• Type I is to reject the null when it is actually
true. The probability of this happening is called
the significance level (or p-value or “size”),
usually called α. It is usually chosen to be 5%
or 1%.
• The other possibility is to fail to reject the null
when it is wrong. If the probability of this
happening is β then (1-β) is called the power.

36. 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 λ.
Suppose the likelihoods for a given set of
data are P(D|A) and P(D|B,λ)

37. 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

38. 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!

39. Is there anything wrong with
Frequentism?
• The laws for manipulating probabilities are
no different
• What is different is the interpretation.
• OK to imagine an ensemble, but there is
no need to assert that it is real! (mind
projection fallacy)
• The idea of a prior is worrying for many,
but is the only way to make this reasoning
consistent

40. 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…

41. 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..

42. ˙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

43. Cosmology by Numbers
c
2
2
ρ=ρ
H==
a
a
a=a=k
kca=a
≡⇒






⇒⇒
−
8ππ
3H
3
8ππG
3
8ππG
0
3
8ππG
2
2
22
22



The “Critical Density”
This applies at any time, but we usually take
the “present” time. In general,
02
0
0
3H
8ππG
Ω=,ρ=ρ,H=H,t=t 000 ⇒

45. The Cosmic Tightrope
• We know the Universe doesn’t have either
a very large  or a very small one, or we
wouldn’t be around.
• We exist and this fact is an observation
about the Universe
• The most probable value of  is therefore
very close to unity
• Still leaves the mystery of what trained
the Universe to walk the tightrope
(inflation?)

46. Theories
Observations
FrequentistBayesian

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

48. “CONCORDANCE”

54. 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!

55. August 29, 2015

56. 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….

58. A)!|P(MM)|P(A ≠
Beware the Prosecutor’s
Fallacy!

59. Is there an Elephant in the
Room?

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

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

65. Weirdness in Phases
ΔT (θ,φ )
T
=∑∑ al,m Ylm(θ,φ)
| | [ ]ml,ml,ml, ia=a φexp
For a homogeneous and isotropic Gaussian
random field (on the sphere) the phases are
independent and uniformly distributed. Non-
random phases therefore indicate weirdness..

71. Final Points