Maximum EntropyMethod for extracting information from noisy data

Finding Compressibilities

Reconstructing 3D densities

Mott Shells?n

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Maybe?

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n

r

Maybe not?

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n

r

How to tell?

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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

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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

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!2 = 1

Not Magic -- but not bad

Better Data

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0.1

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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

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3D Density

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2D Column Density

Abel

Noisy Data

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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

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0

5

10

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

0200

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10

3D Reconstruction

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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]