Bayesian Methods in Cosmology: Summary

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Bayesian Methods in Cosmology Summary

• Peter Coles
• University of Nottingham

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The Legacy of Bayes

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

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A Load of Balls

• Two urns A and B.
• A has 999 black balls and 1 white one B has 999
  white balls and one black one.
• P( black urn A) .999, etc.
• Now shuffle the two urns, and pull out a ball
  from it. Suppose it is black. What is the
  probability it came from urn A?

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Bayesian Basics

• Probability theory is the unique way to
  generalize Boolean (deductive logic) beautiful
  paper by Cox (1946).
• Probability as a state of knowledge (versus
  probability as proportion)
• ..but you can use frequencies!
• Probabilities are always conditional, i.e. P(HI)
• Requires priors - subjectivity?

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Direct versus Inverse Reasoning
Theory (?, n, H0)
Observations
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CONCORDANCE
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Prior and Prejudice

• Priors are essential. Models versus Parameters
• You usually know more than you think..
• Flat priors usually dont make much sense.
• Maximum entropy, etc, give useful insights within
  a well-defined theory objective Bayesian
• Model priors are hard to assign.
• Even if your prior is subjective then at least
  your prejudice is on the table!

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The Flatness Problem

• Prior for ? in FRW Universe (without ?)?
• All FRW Universes have ?1 at t0.
• Jaynes principle gives
• P(?)??(?-1)-1
• Diverges at ?1

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The Cosmic Tightrope

• We shouldnt be surprised (even without
  inflation) that ?1..
• ..but what trained the Universe to walk the
  tightrope?

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Anthropic Principles and the Multiverse

• The weak anthropic principle is Bayesian (Carter
  1974)
• It states that everything is conditional on our
  existence as observers
• The multiverse makes it frequentist..

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Who needs ensembles?

• Do you need a multiverse to apply anthropic
  reasoning?
• Is this ensemble realized?
• Can we be sure it is ergodic?
• Unobservable things are generally bad
• ..but it is not always necessary to observe
  things to deduce their existence

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Issues

• Priors!
• Parameters versus Models (complexity)
• Are the cosmological parameters really that
  interesting?
• How worried should we be about WMAP?
• Nested sampling, thermodynamic integration, etc.
  Benchmarking?

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The Essence of Cosmology is Statistics

• George McVittie