Deductive and Inductive Probabilistic Programming

Fabrizio Riguzzi

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Outline

Probabilistic programmingProbabilistic logic programmingInferenceLearningApplications

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Probabilistic Programming

Users specify a probabilistic model in its entirety (e.g., by writingcode that generates a sample from the joint distribution) andinference follows automatically given the specification.PP languages provide the full power of modern programminglanguages for describing complex distributionsReuse of libraries of modelsInteractive modelingAbstraction

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Probabilistic Programming

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Probabilistic Programming Languages

Name Extends from Host languageVenture Scheme C++Probabilistic-C C CAnglican Scheme ClojureIBAL OCamlPRISM B-PrologInfer.NET .NET Framework .NET Frameworkdimple MATLAB, Javachimple MATLAB, JavaBLOG JavaPSQL SQLBUGSFACTORIE ScalaPMTK MATLAB MATLABAlchemy C++Dyna Prolog

Name Extends from Host languageFigaro ScalaChurch Scheme JavaScript, SchemeProbLog Prolog Python, JythonProBT C++, PythonStan (software) C++Hakaru Haskell HaskellBAli-Phy (software) Haskell C++ProbCog Java, PythonGamble RacketTuffy JavaPyMC Python PythonLea Python PythonWebPPL JavaScript JavaScriptPicture Julia JuliaTuring.jl Julia Julia

Source https://en.wikipedia.org/wiki/Probabilistic_programming_language

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Probabilistic Programming Languages

Only three are logic, the others imperative/functional/objectorientedDARPA released in 2013 the funding call “ProbabilisticProgramming for Advancing Machine Learning” (PPAML)Aim: develop probabilistic programming languages andaccompanying tools to facilitate the construction of new machinelearning applications across a wide range of domains.Focus: functional PP

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Probabilistic Logic Programming

What are we missing?Is logic programming to blame?

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Thesis

Probabilistic logic programming is alive and kicking!

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Strengths

Relationships are first class citizensConceptually easier to liftStrong semanticsInductive systems

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Weaknesses

Handling non-terminationContinuous variables

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Non-termination

Possible when the number of explanations for the query is infinite

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Non-termination: Inducing Arithmetic Functions

Church code http://forestdb.org/models/arithmetic.html

(define (random-arithmetic-fn)(if (flip 0.3)

(random-combination (random-arithmetic-fn)(random-arithmetic-fn))

(if (flip)(lambda (x) x)(random-constant-fn))))

(define (random-combination f g)(define op (uniform-draw (list + -)))(lambda (x) (op (f x) (g x))))

(define (random-constant-fn)(define i (sample-integer 10))(lambda (x) i))

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Non-termination: Inducing Arithmetic Functions

LPAD (cplint) code http://cplint.lamping.unife.it/example/inference/arithm.pl

eval(X,Y):-random_fn(X,0,F),Y is F.

op(L,+):0.5;op(L,-):0.5.random_fn(X,L,F):-comb(L), random_fn(X,l(L),F1), random_fn(X,r(L),F2),op(L,Op), F=..[Op,F1,F2].

random_fn(X,L,F):-\+ comb(L), base_random_fn(X,L,F).

comb(_):0.3.base_random_fn(X,L,X):- identity(L).base_random_fn(_X,L,C):- \+ identity(L), random_const(L,C).identity(_):0.5.random_const(_,C):discrete(C,[0:0.1,1:0.1,2:0.1,3:0.1,4:0.1,5:0.1,6:0.1,7:0.1,8:0.1,9:0.1]).

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Non-termination: Inducing Arithmetic Functions

Aim: given observations of couples input-output for the randomfunction, predict the output for a new inputArbitrarily complex functions have a non-zero probability of beingselectedThe program has a non-terminating execution.Exact inference: infinite number of explanations

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Non-termination: Inducing Arithmetic Functions

(define (sample)(rejection-query(define my-proc (random-arithmetic-fn))(my-proc 2)

(= (my-proc 1) 3)))(hist (repeat 100 sample))

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Solution

Use (T. Sato, P. Meyer, Infinite probability computation by cyclicexplanation graphs, Theor. Pract. Log. Prog. 14 2014) or (A.Gorlin, C. R. Ramakrishnan, S. A. Smolka, Model checking withproba- bilistic tabled logic programming, Theor. Pract. Log. Prog.12 (4-5) 2012)or resort to sampling: with the increase of complexity, theprobability of functions tend to 0 and the probability of the infinitetrace is 0Metropolis Hastings: (Nampally, A., Ramakrishnan, C.: AdaptiveMCMC-based inference in probabilistic logic programs. arXivpreprint arXiv:1403.6036 2014)Monte Carlo sampling is attractive for the simplicity of itsimplementation and because you can improve the estimate asmore time is available.

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Monte Carlo

The disjunctive clauseCr = H1 : α1 ∨ . . . ∨ Hn : αn ← L1, . . . ,Lm.is transformed into the set of clauses MC(Cr )MC(Cr ,1) = H1 ← L1, . . . ,Lm, sample_head(n, r ,VC,NH),NH = 1.. . .MC(Cr ,n) = Hn ← L1, . . . ,Lm, sample_head(n, r ,VC,NH),NH = n.

Sample truth value of query Q:

...(call(Q)-> NT1 is NT+1 ; NT1 =NT),

...

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Metropolis-Hastings MCMC

A Markov chain is built by taking an initial sample and bygenerating successor samples.The initial sample is built by randomly sampling choices so thatthe evidence is true.A successor sample is obtained by deleting a fixed number ofsampled probabilistic choices.Then the evidence is queriedIf the query succeeds, the goal is queried.The sample is accepted with a probability of min{1, N0

N1} where N0

(N1) is the number of choices sampled in the previous (current)sample.

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Solution

In cplint:

?- mc_mh_sample(eval(2,4),eval(1,3),100,100,3,T,F,P).

Probability of eval(2,4) given that eval(1,3) is true

F = 90,T = 10,P = 0.1

You can also try rejection sampling (usually slower)

?- mc_rejection_sample(eval(2,4),eval(1,3),100,T,F,P).

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Solution

You may be interested in the distribution of the outputIn cplint:

?- mc_mh_sample_arg_bar(eval(2,Y),eval(1,3),100,100,3,Y,V).

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Solution

You may be interested in the expected value of the outputIn cplint:

?- mc_mh_expectation(eval(2,Y),eval(1,3),100,100,3,Y,E).

E = 3.21

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Continuous Random Variables

Distributional clauses (B. Gutmann, I. Thon, A. Kimmig, M.Bruynooghe, and L. De Raedt, “The magic of logical inference inprobabilistic programming,” Theory and Practice of LogicProgramming, 2011)Gaussian mixture model in cplint:

heads:0.6;tails:0.4.g(X): gaussian(X,0, 1).h(X): gaussian(X,5, 2).mix(X) :- heads, g(X).mix(X) :- tails, h(X).

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Continuous Random Variables

Inference by samplingWithout evidence or evidence on discrete random variables, youcan reuse the same methodsSampling arguments of goals for building a probability density ofthe arguments.

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Gaussian Mixture Model

heads:0.6;tails:0.4.g(X): gaussian(X,0, 1).h(X): gaussian(X,5, 2).mix(X) :- heads, g(X).mix(X) :- tails, h(X).

?- mc_sample_arg(mix(X),10000,X,L0),histogram(L0,40,Chart).

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Evidence on Continuous Random Variables

You cannot use rejection sampling or Metropolis-Hastings, as theprobability of the evidence is 0You can use likelihood weighting to obtain samples of continuousarguments of a goal. (Nitti, D., De Laet, T., De Raedt, L.:Probabilistic logic programming for hybrid relational domains.Mach. Learn. 103(3), 407-449 2016)

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Likelihood Weighting

For each sample to be taken, likelihood weighting samples thequery and then assigns a weight to the sample on the basis ofevidence.The weight is computed by deriving the evidence backward in thesame sample of the query starting with a weight of oneEach time a choice should be taken or a continuous variablesampled, if the choice/variable has already been taken, thecurrent weight is multiplied by probability of the choice/by thedensity value of the continuous value.

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Bayesian Estimation

Problem fromhttp://www.robots.ox.ac.uk/~fwood/anglican/examples/viewer/?worksheet=gaussian-posteriors

Estimate the true value of a Gaussian distributed random variable,given some observed data.The variance is known and we suppose that the mean has itself aGaussian distribution with mean 1 and variance 5 (prior on theparameter)We take different measurement (e.g. at different times), indexedwith an integer.

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Bayesian Estimation

Anglican code(def dataset [9 8])(defquery gaussian-model [data](let [mu (sample (normal 1 (sqrt 5)))

sigma (sqrt 2)](doall (map (fn [x] (observe

(normal mu sigma) x)) data))mu))

(def posterior((conditional gaussian-model :smc :number-of-particles 10) dataset))

(def posterior-samples (repeatedly 20000 #(sample* posterior)))

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Bayesian Estimation

cplint code http://cplint.lamping.unife.it/example/inference/gauss_mean_est.pl

value(I,X) :-mean(M),value(I,M,X).

mean(M): gaussian(M,1.0, 5.0).value(_,M,X): gaussian(X,M, 2.0).?- mc_sample_arg(value(0,Y),10000,Y,L0),

mc_lw_sample_arg(value(0,X),(value(1,9),value(2,8)),10000,X,L),densities(L0,L,40,Chart).

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Learning

Parameter learningStructure learning more developed for PLP, but(Perov, Yura N., and Frank D. Wood. Learning ProbabilisticPrograms. arXiv preprint arXiv:1407.2646 2014).(Lake, Brenden M., Ruslan Salakhutdinov, and Joshua B.Tenenbaum. Human-level concept learning through probabilisticprogram induction. Science 350.6266 2015).(Gaunt, Alexander L., et al. TerpreT: A Probabilistic ProgrammingLanguage for Program Induction. arXiv preprint arXiv:1608.044282016).

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Parameter Learning

Problem: given a set of interpretations, a program, find theparameters maximizing the likelihood of the interpretations (or ofinstances of a target predicate)Exploit the equivalence with BN to use BN learning algorithmsThe interpretations record the truth value of ground atoms, not ofthe choice variablesUnseen data: relative frequency can’t be used

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Parameter Learning

(Thon et al. ECML 2008) proposed an adaptation of EM forCPT-L, a simplified version of LPADsThe algorithm computes the counts efficiently by repeatedlytraversing the BDDs representing the explanations(Ishihata et al. ILP 2008) independently proposed a similaralgorithmLFI-PROBLOG (Gutamnn et al. ECML 2011) is the adaptation ofEM to ProbLogEMBLEM (Riguzzi & Bellodi IDAJ 2013) adapts (Ishihata et al. ILP2008) to LPADs

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Structure Learning

Given a trivial LPAD or an empty one, a set of interpretations(data)Find the model and the parameters that maximize the probabilityof the data (log-likelihood)SLIPCOVER: Structure LearnIng of Probabilistic logic program bysearching OVER the clause space

1 Beam search in the space of clauses to find the promising ones2 Greedy search in the space of probabilistic programs guided by the

LL of the data.

Parameter learning by means of EMBLEM

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Applications

Link prediction: given a (social) network, compute the probabilityof the existence of a link between two entities (UWCSE)

advisedby(X, Y) :0.3 :-publication(P, X),publication(P, Y),student(X).

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Applications

Classify web pages on the basis of the link structure (WebKB)

coursePage(Page1): 0.3 :- linkTo(Page2,Page1),coursePage(Page2).coursePage(Page1): 0.3 :- linkTo(Page2,Page1),facultyPage(Page2)....coursePage(Page): 0.3 :- has(’abstract’,Page)....

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Applications

Entity resolution: identify identical entities in text or databases

samebib(A,B):0.3 :-samebib(A,C), samebib(C,B).sameauthor(A,B):0.3 :-

sameauthor(A,C), sameauthor(C,B).sametitle(A,B):0.3 :-

sametitle(A,C), sametitle(C,B).samevenue(A,B):0.3 :-

samevenue(A,C), samevenue(C,B).samebib(B,C):0.3 :-

author(B,D),author(C,E),sameauthor(D,E).samebib(B,C):0.3 :-

title(B,D),title(C,E),sametitle(D,E).samebib(B,C):0.3 :-

venue(B,D),venue(C,E),samevenue(D,E).samevenue(B,C):0.3 :-

haswordvenue(B,word_06),haswordvenue(C,word_06).

...

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Applications

Chemistry: given the chemical composition of a substance,predict its mutagenicity or its carcenogenicity

active(A):0.5 :-atm(A,B,c,29,C),gteq(C,-0.003),ring_size_5(A,D).

active(A):0.5 :-lumo(A,B), lteq(B,-2.072).

active(A):0.5 :-bond(A,B,C,2),bond(A,C,D,1),ring_size_5(A,E).

active(A):0.5 :-carbon_6_ring(A,B).

active(A):0.5 :-anthracene(A,B).

...

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Applications

Medicine: diagnose diseases on the basis of patient information(Hepatitis), influence of genes on HIV, risk of falling of elderlypeople (FFRAT)

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Experiments - Area Under the PR Curve

System HIV UW-CSE MondialSLIPCOVER 0.82± 0.05 0.11± 0.08 0.86± 0.07SLIPCASE 0.78± 0.05 0.03± 0.01 0.65± 0.06LSM 0.37± 0.03 0.07± 0.02 -ALEPH++ - 0.05± 0.01 0.87± 0.07RDN-B 0.28± 0.06 0.28± 0.06 0.77± 0.07MLN-BT 0.29± 0.04 0.18± 0.07 0.74± 0.10MLN-BC 0.51± 0.04 0.06± 0.01 0.59± 0.09BUSL 0.38± 0.03 0.01± 0.01 -

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Experiments - Area Under the PR Curve

System Carcinogenesis Mutagenesis HepatitisSLIPCOVER 0.60 0.95± 0.01 0.80± 0.01SLIPCASE 0.63 0.92± 0.08 0.71± 0.05LSM - - 0.53± 0.04ALEPH++ 0.74 0.95± 0.01 -RDN-B 0.55 0.97± 0.03 0.88± 0.01MLN-BT 0.50 0.92± 0.09 0.78± 0.02MLN-BC 0.62 0.69± 0.20 0.79± 0.02BUSL - - 0.51± 0.03

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PLP Online

http://cplint.lamping.unife.it/Inference (knwoledge compilation, Monte Carlo)Parameter learning (EMBLEM)Structure learning (SLIPCOVER)

http://www.cs.kuleuven.be/~dtai/problog/Inference (knwoledge compilation, Monte Carlo)Parameter learning (LFI-ProbLog)

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Conclusions

PLP is still a fertile field but......we must look at other communities and build bridges and......join forces!Much is left to do:

Tractable sublanguages (see following talk)Lifted inferenceStructure/Parameter learning (also for programs with continuousvariables)

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