This document discusses Bayesian methods for model choice and calculating Bayes factors. It explains that the Bayes factor is used to compare two models given data, and is equal to the ratio of the marginal likelihoods of the two models. Analytical and Monte Carlo methods are described for approximating the marginal likelihoods, including importance sampling. Interpreting the log Bayes factor using Jeffrey's scale is also covered.
The computation of automorphic forms for a group Gamma is
a major problem in number theory. The only known way to approach the higher rank cases is by computing the action of Hecke operators on the cohomology.
Henceforth, we consider the explicit computation of the cohomology by using cellular complexes. We then explain how the rational elements can be made to act on the complex when it originate from perfect forms. We illustrate the results obtained for the symplectic Sp4(Z) group.
The computation of automorphic forms for a group Gamma is
a major problem in number theory. The only known way to approach the higher rank cases is by computing the action of Hecke operators on the cohomology.
Henceforth, we consider the explicit computation of the cohomology by using cellular complexes. We then explain how the rational elements can be made to act on the complex when it originate from perfect forms. We illustrate the results obtained for the symplectic Sp4(Z) group.
Dual Gravitons in AdS4/CFT3 and the Holographic Cotton TensorSebastian De Haro
Talk given at the workshop "Gravity in Three Dimensions" at the Erwin Schrödinger Institute, Vienna, April 14-24, 2009. I argue that gravity theories in AdS4 are holographically dual to either of two three-dimensional CFT's: the usual Dirichlet CFT1 where the fixed graviton acts as a source for the stress-energy tensor, or a dual CFT2 with a fixed dual graviton which acts as a source for a dual stress-energy tensor. The dual stress-energy tensor is shown to be the Cotton tensor of the Dirichlet CFT. The two CFT's are related by a Legendre transformation generated by a gravitational Chern-Simons coupling. This duality is a gravitational version of electric-magnetic duality valid at any radius r, where the renormalized stress-energy tensor is the electric field and the Cotton tensor is the magnetic field. Generic Robin boundary conditions lead to CFT's coupled to Cotton gravity or topologically massive gravity. Interaction terms with CFT1 lead to a non-zero vev of the stress-energy tensor in CFT2 coupled to gravity even after the source is removed.
1. Bayesian Case Studies, week 2
Robin J. Ryder
14 January 2013
Robin J. Ryder Bayesian Case Studies, week 2
2. Reminder: Poisson model, Conjugate Gamma prior
For the Poisson model Yi ∼ Poisson(λ) with a Γ(a, b) prior on λ,
the posterior is
n
π(λ|Y ) ∼ Γ(a + yi , b + n)
1
Robin J. Ryder Bayesian Case Studies, week 2
3. Model choice
We have an extra binary variable Zi . We would like to check
whether Yi depends on Zi , and therefore need to choose between
two models:
M1 M2
Yi |Zi = k ∼i.i.d P(λk )
Yi ∼i.i.d P(λ)
λ1 ∼ Γ(a, b)
λ ∼ Γ(a, b)
λ2 ∼ Γ(a, b)
Robin J. Ryder Bayesian Case Studies, week 2
4. The model index is a parameter
We now consider an extra parameter M ∈ {1, 2} which indicates
the model index. We can put a prior on M, for example a uniform
prior: P[M = k] = 1/2. Inside model k, we note the parameters
θk and the prior on θk is noted πk .
We are then interested in the posterior distribution
P[M = k|y ] ∝ P[M = k] L(θk |y )πk (θk )dθk
Robin J. Ryder Bayesian Case Studies, week 2
5. Bayes factor
The evidence for or against a model given data is summarized in
the Bayes factor:
P[M = 2|y ]/P[M = 1|y ]
B21 (y ) = ]
P[M = 2]/P[M = 1]
m2 (y )
=
m1 (y )
where
mk (y ) = L(θk |y )πk (θk )dθk
Θk
Robin J. Ryder Bayesian Case Studies, week 2
6. Bayes factor
Note that the quantity
mk (y ) = L(θk |y )πk (θk )dθk
Θk
corresponds to the normalizing constant of the posterior when we
write
π(θk |y ) ∝ L(θk |y )πk (θk )
Robin J. Ryder Bayesian Case Studies, week 2
7. Interpreting the Bayes factor
Jeffrey’s scale of evidenc states that
If log10 (B21 ) is between 0 and 0.5, then the evidence in favor
of model 2 is weak
between 0.5 and 1, it is substantial
between 1 and 2, it is strong
above 2, it is decisive
(and symmetrically for negative values)
Robin J. Ryder Bayesian Case Studies, week 2
8. Analytical value
Remember that ∞
Γ(a)
λa−1 e −bλ dλ =
0 ba
Robin J. Ryder Bayesian Case Studies, week 2
9. Analytical value
Remember that ∞
Γ(a)
λa−1 e −bλ dλ =
0 ba
Thus
b a Γ(a + yi )
m1 (y ) =
Γ(a) (b + n)a+ yi
and
b 2a Γ(a + yiH ) Γ(a + yiF )
m2 (y ) =
Γ(a)2 (b + nH )a+ yiH (b + nF )a+ yiF
Robin J. Ryder Bayesian Case Studies, week 2
10. Monte Carlo
Let
I= h(x)g (x)dx
where g is a density. Then take x1 , . . . , xN iid from g and we have
ˆ 1
IMC =
N h(xi )
N
which converges (almost surely) to I.
When implementing this, you need to check convergence!
Robin J. Ryder Bayesian Case Studies, week 2
11. Harmonic mean estimator
Take a sample from the posterior distribution π1 (θ1 |y ). Note that
1 1
Eπ 1 |y = π1 (θ1 |y )dθ1
L(θ1 |y ) L(θ1 |y )
1 π1 (θ1 )L(θ1 |y )
= dθ1
L(θ1 |y ) m1 (y )
1
=
m1 (y )
thus giving an easy way to estimate m1 (y ) by Monte Carlo.
However, this method is in general not advised, since the
associated estimator has infinite variance.
Robin J. Ryder Bayesian Case Studies, week 2
12. Importance sampling
I= h(x)g (x)dx
If we wish to perform Monte Carlo but cannot easily sample from
g , we can re-write
h(x)g (x)
I= γ(x)dx
γ(x)
where γ is easy to sample from. Then take x1 , . . . , xN iid from γ
and we have
ˆ 1 h(xi )g (xi )
IIS =
N
N γ(xi )
Robin J. Ryder Bayesian Case Studies, week 2