emcee - Speaker Deck

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emcee danfm.ca/emcee p(⇥) I have a function

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emcee danfm.ca/emcee p(⇥) I have a function I can Evaluate it

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emcee danfm.ca/emcee p(⇥) I have a function I can Evaluate it I can't Calculate the functional form

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emcee danfm.ca/emcee p(⇥) I have a function I can Evaluate it I can't Calculate the functional form Markov chain Monte Carlo (MCMC)

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emcee danfm.ca/emcee Metropolis-Hastings

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emcee danfm.ca/emcee min ✓ 1 , p ( x 0) p ( x ) Q ( x ; x 0) Q ( x 0; x ) ◆ Metropolis-Hastings

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emcee danfm.ca/emcee min ✓ 1 , p ( x 0) p ( x ) Q ( x ; x 0) Q ( x 0; x ) ◆ Metropolis-Hastings Proposal D (D-1) parameters

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emcee danfm.ca/emcee Metropolis-Hastings x y

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emcee danfm.ca/emcee Metropolis-Hastings x y (in an ideal world)

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emcee danfm.ca/emcee Metropolis-Hastings x y (in an ideal world)

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emcee danfm.ca/emcee Metropolis-Hastings x y (in an ideal world)

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emcee danfm.ca/emcee Metropolis-Hastings x y (in an ideal world) min ✓ 1, p (x 0 ) p (x ) Q (x ;x 0 ) Q (x 0 ;x ) ◆ ?

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emcee danfm.ca/emcee Metropolis-Hastings x y (in an ideal world)

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emcee danfm.ca/emcee Metropolis-Hastings x y (in an ideal world)

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emcee danfm.ca/emcee Metropolis-Hastings x y (in an ideal world)

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emcee danfm.ca/emcee Metropolis-Hastings x y (in an ideal world) min ✓ 1 , p( x 0 ) p( x) Q( x; x 0 ) Q( x 0; x) ◆ ?

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emcee danfm.ca/emcee Metropolis-Hastings x y (in an ideal world)

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emcee danfm.ca/emcee Metropolis-Hastings x y (in an ideal world)

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emcee danfm.ca/emcee Metropolis-Hastings x y (in an ideal world)

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emcee danfm.ca/emcee Metropolis-Hastings x y (in an ideal world)

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emcee danfm.ca/emcee Metropolis-Hastings (in the REAL world) x y

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emcee danfm.ca/emcee Metropolis-Hastings (in the REAL world) x y

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emcee danfm.ca/emcee Metropolis-Hastings (in the REAL world) x y

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emcee danfm.ca/emcee Metropolis-Hastings (in the REAL world) x y

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emcee danfm.ca/emcee Metropolis-Hastings (in the REAL world) x y

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emcee danfm.ca/emcee Metropolis-Hastings (in the REAL world) x y SMALL ACCEPTANCE FRACTION the problem

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emcee danfm.ca/emcee Metropolis-Hastings (in the REAL world) x y LARGE ACCEPTANCE FRACTION the problem

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x y emcee danfm.ca/emcee Metropolis-Hastings (in the REAL world)

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x y emcee danfm.ca/emcee Metropolis-Hastings (in the REAL world)

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x y emcee danfm.ca/emcee Metropolis-Hastings (in the REAL world)

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x y emcee danfm.ca/emcee Metropolis-Hastings (in the REAL world) positive-definite symmetric Proposal D (D-1) parameters

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x y emcee danfm.ca/emcee Metropolis-Hastings (in the REAL world) positive-definite symmetric Proposal D (D-1) parameters This is the Dimension of your parameter space!

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emcee danfm.ca/emcee Metropolis-Hastings (in the REAL world) Scientific Awesomeness how hard is MCMC Metropolis Hastings how things Should be (~number of parameters)

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emcee danfm.ca/emcee Why does all this matter?

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emcee danfm.ca/emcee How do you calculate the optimal proposal?

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emcee danfm.ca/emcee Temperature

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emcee danfm.ca/emcee Temperature

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emcee danfm.ca/emcee Temperature

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emcee danfm.ca/emcee Temperature

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emcee danfm.ca/emcee Temperature

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emcee danfm.ca/emcee Temperature

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emcee danfm.ca/emcee Temperature

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emcee danfm.ca/emcee Temperature

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emcee danfm.ca/emcee Temperature

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emcee danfm.ca/emcee Temperature time

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emcee danfm.ca/emcee Temperature that should be spent interpreting your results writing papers finding bugs in your code time

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emcee danfm.ca/emcee Luckily I have a solution!

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emcee danfm.ca/emcee Luckily I have a solution! HINT: it's up here...

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emcee danfm.ca/emcee bit.ly/mcmc-gw10 "Ensemble samplers with affine invariance" Jonathan Goodman Jonathan Weare Mustaches courtesy: mustachify.me

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emcee danfm.ca/emcee bit.ly/mcmc-gw10 "Ensemble samplers with affine invariance" Jonathan Goodman Jonathan Weare Mustaches courtesy: mustachify.me

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emcee danfm.ca/emcee bit.ly/mcmc-gw10 "Ensemble samplers with affine invariance" Jonathan Goodman Jonathan Weare Mustaches courtesy: mustachify.me

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emcee danfm.ca/emcee affine invariance

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emcee danfm.ca/emcee affine invariance y A x + b Affine Transformation

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emcee danfm.ca/emcee affine invariance The sampler performs Equally well on X and Y y A x + b Affine Transformation

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emcee danfm.ca/emcee Easy to sample Hard to sample

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emcee danfm.ca/emcee Easy to sample Hard to sample y A x + b Affine Transformation

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emcee danfm.ca/emcee Easy to sample Hard to sample y A x + b Affine Transformation easy!

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emcee danfm.ca/emcee Ensemble Samplers (in the REAL world) x y with affine invariance

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emcee danfm.ca/emcee Ensemble Samplers (in the REAL world) x y with affine invariance

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emcee danfm.ca/emcee Ensemble Samplers (in the REAL world) x y with affine invariance

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emcee danfm.ca/emcee Ensemble Samplers (in the REAL world) x y with affine invariance this is a walker

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emcee danfm.ca/emcee Ensemble Samplers (in the REAL world) x y with affine invariance this is a walker

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emcee danfm.ca/emcee Ensemble Samplers (in the REAL world) x y with affine invariance

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emcee danfm.ca/emcee Ensemble Samplers (in the REAL world) x y with affine invariance

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emcee danfm.ca/emcee Ensemble Samplers (in the REAL world) x y with affine invariance

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emcee danfm.ca/emcee Ensemble Samplers (in the REAL world) x y with affine invariance

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emcee danfm.ca/emcee Ensemble Samplers (in the REAL world) x y with affine invariance

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emcee danfm.ca/emcee Ensemble Samplers (in the REAL world) x y with affine invariance min ✓ 1,Z D 1 p (x 0 ) p (x ) ◆

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emcee danfm.ca/emcee Ensemble Samplers (in the REAL world) x y with affine invariance min ✓ 1,Z D 1 p (x 0 ) p (x ) ◆

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emcee danfm.ca/emcee Ensemble Samplers (in the REAL world) x y with affine invariance

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emcee danfm.ca/emcee Ensemble Samplers (in the REAL world) x y with affine invariance

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emcee danfm.ca/emcee Ensemble Samplers (in the REAL world) x y with affine invariance

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emcee danfm.ca/emcee Ensemble Samplers (in the REAL world) x y with affine invariance

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emcee danfm.ca/emcee Ensemble Samplers (in the REAL world) x y with affine invariance Aside: this looks nice and parallel, eh? * * not quite as trivial as you might hope—but possible!

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emcee danfm.ca/emcee +

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emcee danfm.ca/emcee it's hammer time! emceethe MCMC Hammer introducing arxiv.org/abs/1202.3665

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emcee danfm.ca/emcee pip install emcee get it:

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emcee danfm.ca/emcee import numpy as np import emcee def lnprob(x): return -0.5 * np.sum(x ** 2) ndim, nwalkers = 10, 100 p0 = [np.random.rand(ndim) for i in range(nwalkers)] sampler = emcee.EnsembleSampler(nwalkers, ndim, lnprob) sampler.run_mcmc(p0, 1000) use it:

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emcee danfm.ca/emcee DOES IT WORK? obviously it does.

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emcee danfm.ca/emcee 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 exp ✓ (x1 x2) 2 2 ✏ (x1 + x2) 2 2 ◆

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emcee danfm.ca/emcee github.com/dfm/acor Autocorrelation Function the (covariance)

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emcee danfm.ca/emcee 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 exp ✓ (x1 x2) 2 2 ✏ (x1 + x2) 2 2 ◆ Metropolis-Hastings Emcee Autocorrelation Function the

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emcee danfm.ca/emcee 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 exp ✓ (x1 x2) 2 2 ✏ (x1 + x2) 2 2 ◆ Metropolis-Hastings Emcee Autocorrelation Function the

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emcee danfm.ca/emcee 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 exp ✓ (x1 x2) 2 2 ✏ (x1 + x2) 2 2 ◆ Metropolis-Hastings Emcee Autocorrelation Function the

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emcee danfm.ca/emcee Metropolis-Hastings Boom!

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emcee danfm.ca/emcee 4 2 0 2 4 6 0 5 10 15 20 25 30 exp ✓ 100 (x2 x 2 1) 2 + (1 x1) 2 20 ◆

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4 2 0 2 4 6 0 5 10 15 20 25 30 exp ✓ 100 (x2 x 2 1) 2 + (1 x1) 2 20 ◆ emcee danfm.ca/emcee Metropolis-Hastings Emcee Autocorrelation Function the

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emcee isn't always The Right Choice™ emcee danfm.ca/emcee Mustache courtesy: mustachify.me Brendon Brewer Remember:

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emcee danfm.ca/emcee Mustache courtesy: mustachify.me continuous parameters in a vector space emcee needs highly multimodal problems and it is not good at

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emcee danfm.ca/emcee Mustache courtesy: mustachify.me continuous parameters in a vector space emcee needs highly multimodal problems and it is not good at what is?

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emcee danfm.ca/emcee Mustache courtesy: mustachify.me continuous parameters in a vector space emcee needs highly multimodal problems and it is not good at what is? maybe Dnest github.com/eggplantbren/DNest3

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emcee danfm.ca/emcee it's still been pretty useful... Lang & Hogg (2011) Bovy et al. (2011) Dorman et al. (2012) Foreman-Mackey & Widrow (in prep) Mustaches courtesy: mustachify.me ... ... ...

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emceethe MCMC Hammer arxiv.org/abs/1202.3665 danfm.ca/emcee github.com/dfm/emcee paper documentation issues/contributions Check it out: Dustin Lang (Princeton) David W. Hogg (NYU) Jonathan Goodman (NYU)