Cosmology: A Bayesian Perspective
-
Upload
peter-coles -
Category
Science
-
view
1.927 -
download
1
Transcript of Cosmology: A Bayesian Perspective
![Page 1: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/1.jpg)
MaxEnt 201613th July, Ghent
Cosmology: A Bayesian Perspective
Peter Coles (@telescoper)
![Page 2: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/2.jpg)
![Page 3: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/3.jpg)
![Page 4: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/4.jpg)
Lecture 1Probability
![Page 5: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/5.jpg)
“The Essence of Cosmology is Statistics”
George McVittie
![Page 6: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/6.jpg)
Direct versus Inverse Reasoning
Theory (, H0…)
Observations
![Page 7: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/7.jpg)
3 May 2023
Picture: R. Trotta
![Page 8: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/8.jpg)
Urn A Urn B
999 white 1 black
999 black 1 white
P(white ball | urn is A)=0.999, etc
![Page 9: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/9.jpg)
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
![Page 10: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/10.jpg)
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(YI)X,|I)P(Y|P(X=I)Y,|P(X
![Page 11: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/11.jpg)
![Page 12: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/12.jpg)
The Expanding Universe
![Page 13: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/13.jpg)
![Page 14: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/14.jpg)
![Page 15: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/15.jpg)
![Page 16: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/16.jpg)
![Page 17: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/17.jpg)
![Page 18: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/18.jpg)
![Page 19: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/19.jpg)
![Page 20: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/20.jpg)
![Page 21: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/21.jpg)
![Page 22: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/22.jpg)
![Page 23: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/23.jpg)
![Page 24: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/24.jpg)
![Page 25: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/25.jpg)
![Page 26: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/26.jpg)
![Page 27: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/27.jpg)
![Page 28: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/28.jpg)
![Page 29: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/29.jpg)
Picture: R. Trotta
![Page 30: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/30.jpg)
3 May 2023
![Page 31: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/31.jpg)
![Page 32: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/32.jpg)
![Page 33: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/33.jpg)
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…
![Page 34: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/34.jpg)
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)
![Page 35: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/35.jpg)
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!
![Page 36: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/36.jpg)
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).
![Page 37: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/37.jpg)
Probable Theories
I)|P(DI)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!
![Page 38: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/38.jpg)
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…
![Page 39: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/39.jpg)
![Page 40: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/40.jpg)
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..
![Page 41: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/41.jpg)
a2=8πGρ
3 a2−kc 2
The Friedman ModelsThe 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∝1a3
For non-relativistic particles (“dust”)
Curvature
![Page 42: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/42.jpg)
![Page 43: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/43.jpg)
![Page 44: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/44.jpg)
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
akcG
aa
kcaaGa
133
833
8
2
2
2
2
22
22
![Page 45: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/45.jpg)
TG
Gravity Stuff
![Page 46: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/46.jpg)
TG ?
![Page 47: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/47.jpg)
?TG
![Page 48: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/48.jpg)
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)
![Page 49: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/49.jpg)
3 May 2023
![Page 50: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/50.jpg)
Other Observations
• Supernovae (Type Ia)• Large-scale structure measurements (BAO)• Gravitational Lensing (Weak and Strong)• CMB lensing• Peculiar Velocities
![Page 51: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/51.jpg)
![Page 52: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/52.jpg)
fainter
![Page 53: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/53.jpg)
![Page 54: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/54.jpg)
![Page 55: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/55.jpg)
![Page 56: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/56.jpg)
![Page 57: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/57.jpg)
EUCLID
![Page 58: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/58.jpg)
![Page 59: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/59.jpg)
SAY “PRECISION COSMOLOGY” ONE MORE TIME…
![Page 60: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/60.jpg)
Bayesian Hypothesis TestingTwo 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,)
![Page 61: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/61.jpg)
Occam’s Razor
λ)B,|(DB)|(λdλA)|(D
(B)(A)
λ)B,|(Dλ)(B,dλA)|(D(A)
D)|λ(B,dλD)|(A=
D)|(BD)|(A
PrPrPr
PrPr
PrPrPrPr
PrPr
PrPr
Occam factor
![Page 62: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/62.jpg)
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…
![Page 63: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/63.jpg)
Bayesian estimation
aI)d,aa|xI)p(x|ap(a=K
I),aa|xI)p(x|aKp(a=I),xx|ap(a
mmnm
mnmnm
.......
............
1111
11111
This involves finding the posterior distribution of the parameters given the data and any prior information.
Evidence!
![Page 64: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/64.jpg)
![Page 65: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/65.jpg)
A)!|P(MM)|P(A
Beware the Prosecutor’s Fallacy!
![Page 66: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/66.jpg)
![Page 67: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/67.jpg)
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!
![Page 68: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/68.jpg)
“CONCORDANCE”
![Page 69: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/69.jpg)
![Page 70: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/70.jpg)
Ingredients of the Standard Cosmology
•General Relativity •Cold Dark Matter•Cosmological Constant•Cosmological Principle•Primordial Gaussian fluctuations•Inflation•Baryons•Neutrinos•Radiation…
![Page 71: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/71.jpg)
Questionable Aspects of the Standard Cosmology
•General Relativity •Cold Dark Matter•Cosmological Constant•Cosmological Principle•Primordial Gaussian fluctuations•Inflation•Baryons•Neutrinos•Radiation…
![Page 72: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/72.jpg)
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!
![Page 73: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/73.jpg)
![Page 74: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/74.jpg)
“If tortured sufficiently, data will confess to almost
anything”
Fred Menger
![Page 75: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/75.jpg)
Theories
Observations
FrequentistBayesian
![Page 76: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/76.jpg)
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.”
![Page 77: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/77.jpg)
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….
![Page 78: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/78.jpg)
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..
![Page 79: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/79.jpg)
Weirdness in PhasesΔT (θ,φ )
T=∑∑ a l,mY lm (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..
![Page 80: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/80.jpg)
![Page 81: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/81.jpg)
![Page 82: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/82.jpg)
![Page 83: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/83.jpg)
![Page 84: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/84.jpg)
![Page 85: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/85.jpg)
![Page 86: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/86.jpg)
The Prosecutor’s Fallacy!
P(A|M)P(M|A)
![Page 87: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/87.jpg)
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, ?)
![Page 88: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/88.jpg)
![Page 89: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/89.jpg)
Low Quadrupole?
![Page 90: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/90.jpg)
Parity Violation?
![Page 91: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/91.jpg)
(from Copi et al. 2005)
![Page 92: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/92.jpg)
(from Hansen et al. 2004)
![Page 93: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/93.jpg)
![Page 94: Cosmology: A Bayesian Perspective](https://reader035.fdocuments.in/reader035/viewer/2022062503/5871101e1a28abac6d8b58d5/html5/thumbnails/94.jpg)