Probabilistic Programming
Miguel Lázaro-Gredilla
[email protected]
January 2013
Machine Learning Group
http://www.tsc.uc3m.es/~miguel/MLG/
Contents
Towards a modern machine learning
Probabilistic programming languages
Infer.NET
The software stack: A digression
The modeling language
References
Towards a modern machine learning
Probabilistic programming languages
Infer.NET
References
Classical machine learning
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Large number of tools with diverse backgrounds
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k-means
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Principal Component Analysis
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Independent Component Analysis
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Classical Neural Networks
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Support Vector Machines
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Density estimation via Parzen windows
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Recursive Least Squares
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Least Mean Squares
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(just an arbitrary sample, we could go on and on...)
Pragmatic, unsystematic, non-probabilistic
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Third generation machine learning
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Data sets described as instances of a probabilistic model
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Gaussian Process regression/classification
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Latent Dirichlet Allocation
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Bayes Point Machine
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...
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We can infer unknowns in a principled way
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Features
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Each tool can be expressed as a Bayesian network
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Systematic, modular, standardized approach
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Proposals are models, not algorithms
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Detach inference from model
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Probabilistic programming languages
Infer.NET
References
Updating classical machine learning (I/V)
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Should we just dump classical ML and jump on the Bayesian
bandwagon?
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Most classical ML tools can be written as some type of
inference on some Bayesian network
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So just update classical ML using a Bayesian interpretation
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Gain insight on the model behind the algorithm
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See overlaps emerge
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Use other types of inference
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It may become obvious how to enhance them
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Probabilistic programming languages
Infer.NET
References
Updating classical machine learning (II/V)
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Classical algorithm: k-means
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Bayesian model (assuming normalized data)
p(xn |zn , {µk }) = N (xn |µzn , v I)
p(v ) = InvGamma(v |1, 1)
p(µk ) = N (µk |0, 10I)
p(zn |w) = Discrete(zn |w)
p(w) = Dirichlet(w|1k×1 )
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Classical model corresponds to
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Maximum likelihood for p({xn }|{zn }, {µk }),
obtained using hard-EM optimization
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Particular case of Gaussian mixture model
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Probabilistic programming languages
Infer.NET
References
Updating classical machine learning (III/V)
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Classical algorithm: Extended Recursive Least Squares
(adaptive)
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Bayesian model
p(xn |wn ) = N (xn |w>
n un , v )
p(wn |wn−1 , α, β) = N (wn |(1 − β)wn−1 , βαI)
p(v ) = InvGamma(v |1, 1)
p(α) = InvGamma(α|1, 1)
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p(β/(1 − β)) = InvGamma(β/(1 − β)|1, 1)
Classical model corresponds to
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Posterior mean for wn , for some magically selected α and v
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Particular case of Kalman filter
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Contrived assumptions needed to represent exponentially
weighted RLS in this framework.
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Hints that it might not make sense, see [KRLST].
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References
Updating classical machine learning (IV/V)
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Classical algorithm: Principal Component Analysis
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Bayesian model
p(xn |yn , W, v ) = N (xn |W> yn , v I)
p(yn ) = N (yn |0, I)
p([W]d-th col ) = N (wd |0, diag([α1 , . . . , αD ])
p(v ) = InvGamma(v |1, 1)
p(αd ) = InvGamma(v |1, 1)
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Classical model corresponds to
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Maximum likelihood for p(xn |W, v ) when v → 0,
with W product of orthogonal and ordered diagonal matrix
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Restrictions on W make it unique, but don’t change the model
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New possibilities: What if we do max. lik. for p(xn |{yn }, v )?
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Infer.NET
References
Updating classical machine learning (V/V)
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Classical algorithm: Support Vector Machines
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Bayesian model
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See: Sollich, P. (2002). Bayesian Methods for Support Vector
Machines: Evidence and Predictive Class Probabilities.
Machine Learning, 46:21-52.
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Ingenious approach, but not very natural
(involves using three classes to solve a binary problem)
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Contents
Towards a modern machine learning
Probabilistic programming languages
Infer.NET
The software stack: A digression
The modeling language
References
Towards a modern machine learning
Probabilistic programming languages
Infer.NET
References
Some observations
According to the previous slides
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Most ML tools have a Bayesian model description
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Full description takes a few lines
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The type of inference (ML, MAP, point estimates, full
Bayesian posterior) is independent of the model
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Though the tractability of each does depend on the model
In the process of creating new ML tools
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What makes a new ML tool worth is:
A novel model
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What we spend most time working on is:
Making inference tractable on the new model
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The idea
Probabilistic programming
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Define a language to describe Bayesian models (“programs”)
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Create some “compiler” that understands those programs and
generates inference engines for them
The new workflow (emphasis is on model design)
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Spend more time thinking about the model
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Program it (just a few lines!)
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Optional: Sample data from the model
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Feed data to the inference engine and assess the results
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Model wasn’t that good, do it over
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Programming paradigms (I/II)
A random assortment of them:
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Imperative (Matlab)
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Procedural (C)
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Object oriented (C++)
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Declarative (SQL)
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Functional (Ocaml, F#)
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Metaprogramming (LISP)
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Domain specific language (Spice)
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References
Programming paradigms (II/II)
For probabilistic programming:
I A domain specific language may be simpler for the user, but
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Doesn’t integrate well with existing codebases
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Doesn’t interface well with DB access, plotting capabilities,
etc.
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Functional languages with metaprogramming can be useful to
write a “guest probabilistic program” within a “host
programming language” (we’ll see F# examples)
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This can be done with imperative style, but looks uglier
(we’ll see Python examples)
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References
An incomplete list of probabilistic programming languages
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BUGS: Bayesian inference using Gibbs sampling
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HANSEI: Extends OCaml, discrete distributions only
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Hierarchical Bayesian Compiler (HBC): Large-scale models
and non-parametric process priors
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PyMCMC: MCMC algorithms for Python classes
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Church: Extends Scheme to describe Bayesian models
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Infer.NET: Provides a probabilistic language within the .NET
platform
See more on http://probabilistic-programming.org
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Contents
Towards a modern machine learning
Probabilistic programming languages
Infer.NET
The software stack: A digression
The modeling language
References
Towards a modern machine learning
Probabilistic programming languages
Infer.NET
References
The Java case
Three big operating systems
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Linux, OSX, Windows
...and one language to rule them all
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Sun Microsystems designed Java to run on the JVM
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...and implemented the JVM to run on
linux, mac, and windows
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platform independence, bliss for programmers
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Infer.NET
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The .NET case
Microsoft reacted and created the JVM counterpart: The CLR
Which language was used to target the CLR?
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The .NET languages: VB.NET, C#, F#, IronPython...
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Different languages with a common set of libraries,
it is easy to interface them
Which operating systems did the CLR run on?
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Microsoft released the specification and standardized it
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CLR-like implementations arose for linux/mac: Mono
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...but the Windows version is always ahead, more complete,
with additional tools, better tested, etc.
Microsoft tries to win in the cross-platform territory (sweet spot
for developers) while still favoring its flagship product, Windows:
Opposing objectives
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Probabilistic programming languages
Infer.NET
References
Infer.NET’s interoperability
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Infer.NET targets all .NET languages, with a focus on
C#, F#, and IronPython
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It can be used on Mono (Mac/Linux) or the CLR (Windows)
(better experience/less buggy on the latter)
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The .NET languages have a growing set of tools for scientific
computing, but nowhere near Matlab yet
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IronPython cannot use Numpy/Scipy/Matplotlib natively
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A new tool called Sho provides an IronPython shell with
Matlab-like capabilities (but Windows only)
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Contents
Towards a modern machine learning
Probabilistic programming languages
Infer.NET
The software stack: A digression
The modeling language
References
Towards a modern machine learning
Probabilistic programming languages
Infer.NET
References
Using Infer.NET
Infer.NET provides:
I A probabilistic modeling language embedded in other
languages
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F# allows a more natural embedding
Compilation to three inference engines
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EP: Approximation includes all non-zero probability points
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VB: Approximation avoids zero probability points
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MCMC: Gibbs sampling, slower
Let’s browse Microsoft Research examples and create a new model
http://research.microsoft.com/en-us/um/cambridge/
projects/infernet/
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Probabilistic programming languages
Infer.NET
References
Two coins (F#)
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Infer.NET
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Two coins (IronPython)
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Infer.NET
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Learning a Gaussian (F#)
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Learning a Gaussian (IronPython)
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Truncated Gaussian (F#)
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Truncated Gaussian (IronPython)
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Gaussian Mixture (F#)
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Gaussian Mixture (IronPython)
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k-means (F#)
[Demo]
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Conclusions
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Probabilistic programming is in its infancy
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We might produce alternative language
definitions/implementations
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We can leverage it to test new models faster
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We can build custom models on the fly
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References
References
I [BayesSVM] Sollich, P. (2002). Bayesian Methods for Support Vector
Machines: Evidence and Predictive Class Probabilities. Machine Learning,
46:21-52.
I [ProbProg] The probabilistic programming wiki
http://probabilistic-programming.org/wiki/Home
I [InferNET] T. Minka, J. Winn, J. Guiver, and D. Knowles Infer.NET 2.5,
Microsoft Research Cambridge, 2012.
http://research.microsoft.com/infernet
I [KRLST] M. Lázaro-Gredilla, S. Van Vaerenbergh, and I. Santamarı́a. “A
Bayesian approach to tracking with kernel recursive least-squares”, MLSP
2011.
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