Maximum Entropy - Cornell...

Maximum EntropyMethod for extracting information from noisy data

Finding Compressibilities

Reconstructing 3D densities

Mott Shells?n

r20 40 60 80 100

1

2

3

4

Maybe?

20 40 60 80 100

1

2

3

4

n

r

Maybe not?

20 40 60 80 100

1

2

3

4

n

r

How to tell?

20 40 60 80 100

1

2

3

4

20 40 60 80 100

1

2

3

4

Monte-Carlo ApproachGenerate smooth random data which is consistent with

experimental data

Fraction of random data that has Mott plateausgives probability that there is a Mott plateau

Extension: Can produce most probable smooth reconstructionbin consistent random data

bin with most elements = most probable

Maximum Entropy

TricksEnforce known symmetries/constraints:

Density is positiveslope is negative

20 40 60 80 100

1

2

3

4

Best Fitwith theseconstraints!2 = 0.87

(slightly overfit)

Statistical Mechanics

Microcanonical:Generate all configurations of fixed

!2plays roll of energy

!2

Canonical:Minimize Free energy

!2 ! TS

1T

=!S

!"2 = Lagrange Multiplier

Choose so fit is appropriate

EntropyCorrect entropy depends on algorithm

used to randomly generate data

Typical Choice: Shannon Entropy

S = !!

j

nj

Nlog

nj

N

N =!

j

nj

(I take nj= slope at position j)

Results

20 40 60 80 100

1

2

3

4

!2 = 1

Not Magic -- but not bad

Better Data

20 40 60 80 100

1

2

3

20 40 60 80 100

0.1

0.2

0.3

0.4

Slope

Not biased againstsharp features

Abstract PictureModel Space

Data Space

si

di

slopes

trapezoid rule

density

nidata:noise: !i

map

want model that describes data

Model Space

Data Space

si

di

slopes

trapezoid rule

density

nidata: noise: !i

Fitting:

!2 =!

i

(ni ! di)2

"2i

finitedifferencederivative

minimize wrt si

min(!2) = 0overfit:

Model Space

Data Space

si

di

slopes

trapezoid rule

density

nidata: noise: !i

Better:

!2 =!

i

(ni ! di)2

"2i

= Npixels

but does not uniquely determine si

Bayesian Approach

!2 =!

i

(ni ! di)2

"2i

= Npixels

siFind

for which

that carries least information

i.e. Maximize Entropy

Inverse Abel

20 40 60 80 100

1

2

3

4

3D Density

20 40 60 80 100

50

100

150

200

2D Column Density

Abel

Noisy Data

20 40 60 80 100

50

100

150

200

0 20 40 60 80 100

1

2

3

4

2D

3D density Column Noise Inverse(Naive)

2D

3D density Column Noise Inverse(Max Ent)

Actual Data

3D -- Unitary Fermi gas -- courtesy of Randy Hulet

Actual Data0 200 400

2 4 6 8

0

5

10

3D -- Unitary Fermi gas -- courtesy of Randy Hulet

0200

400

2 4 6 8

0

5

10

3D Reconstruction

0 100 200 300 400 500

0.05

0.10

0.15

off-axis slice:

ThoughtsNot as good as a real model

Introduces less bias

Needs relatively clean data

Dreams: extract equations of state, phase diagrams, entropy (temp), superfluid density...

[Ho -- last DARPA meeting]