> DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE
PROBABILISTIC MORPHABLE MODELS | JUNE 2017 | BASEL
BayesianFitting
ProbabilisticMorphableModels
SummerSchool,June2017
SandroSchönborn
UniversityofBasel
> DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE
PROBABILISTIC MORPHABLE MODELS | JUNE 2017 | BASEL
Uncertainty:ProbabilityDistributions
• ProbabilisticModels
• UncertainObservation(noise,outlier,occlusion,…)
• Fitting:Modelexplanationofobserveddata– probabilistic?
Tellsusabouttheoutcome’scertainty!
Observations
Fit&Certainty
Groundtruth
BishopPRML,2006
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> DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE
PROBABILISTIC MORPHABLE MODELS | JUNE 2017 | BASEL
Probability:AnExample
• Dentistexample:Doesthepatienthaveacavity?
𝑃 cavity = 0.1
𝑃 cavity toothache) = 0.8
Certainty
𝑃 cavity toothache, gumproblems) = 0.4
Butthepatienteitherhasacavityordoesnot
• Thereisno80%cavity!
• Havingacavityshouldnotdependonwhetherthe
patienthasatoothacheorgumproblems
Allthesestatementsdonotcontradicteachother,they
summarizethedentist’sknowledge aboutthepatient
AIMA:Russell&Norvig,ArtificialIntelligence.AModernApproach, 3rd edition,
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> DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE
PROBABILISTIC MORPHABLE MODELS | JUNE 2017 | BASEL
Uncertainty:BayesianProbability
• Howareprobabilitiestobeinterpreted?
Theyaresometimescontradictory:Whydoesthedistributionchangewhen
wehavemoredata?Shouldn’ttherebeareal distributionof𝑃 𝜃 ?
• Bayesianprobabilitiesrelyonasubjectiveperspective:
Probabilityisusedtoexpressourcurrentknowledge.Itcanchangewhenwe
learnorseemore:Withmoredata,wearemorecertainaboutourresult.
Subjectivity: There is no single, real underlying distribution. A probability
distribution expresses our knowledge – It is different in different situations
and for different observers since they have different knowledge.
• Notsubjectiveinthesensethatitisarbitrary!
Therearequantitativerulestofollowmathematically
• Probabilityexpressesanobserverscertainty,oftencalledbelief
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> DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE
PROBABILISTIC MORPHABLE MODELS | JUNE 2017 | BASEL
TowardsBayesianInference
PosteriorModels:GaussianProcessRegression
ProbabilisticFit:Probabilisticinterpretationofdata
ObservedPoints
Updateofpriortoposteriormodel:
BayesianInference
PosteriorModel
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> DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE
PROBABILISTIC MORPHABLE MODELS | JUNE 2017 | BASEL
BeliefUpdates
Model
Facedistribution
Observation
Concretepoints
Possiblyuncertain
Posterior
Facedistribution
consistentwithobservation
Priorbelief
Moreknowledge
Posteriorbelief
Consistency:Lawsofprobabilitycalculus!
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> DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE
PROBABILISTIC MORPHABLE MODELS | JUNE 2017 | BASEL
JointDistribution
Probabilisticmodel:jointdistributionofpoints
𝑃 𝑥> , 𝑥@
Marginal
Conditional
Distributionofcertainpointsonly
Distributionofpointsconditionedon
known valuesofothers
𝑃 𝑥> = A 𝑃(𝑥> , 𝑥@ )
DE
𝑃 𝑥> |𝑥@
𝑃 𝑥> , 𝑥@
=
𝑃 𝑥@
Bothcanbeeasilycalculated
forGaussianmodels
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> DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE
PROBABILISTIC MORPHABLE MODELS | JUNE 2017 | BASEL
CertainObservation
• Observationsareknown values
• Distributionof𝑥> after
observing 𝑥@ , … , 𝑥G :
𝑃 𝑥> |𝑥@ … 𝑥G
𝑃 𝑥> , 𝑥@ , … , 𝑥G
=
𝑃 𝑥@ , … , 𝑥G
• Conditionalprobability
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> DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE
PROBABILISTIC MORPHABLE MODELS | JUNE 2017 | BASEL
TowardsBayesianInference
• Updatebeliefabout𝑥> byobserving𝑥@ , … , 𝑥G
𝑃 𝑥> → 𝑃 𝑥> 𝑥@ … 𝑥G
• Factorizejointdistribution
𝑃 𝑥> , 𝑥@ , … , 𝑥G = 𝑃 𝑥@ , … , 𝑥G |𝑥> 𝑃 𝑥>
• Rewriteconditionaldistribution
𝑃 𝑥> |𝑥@ … 𝑥G =
𝑃 𝑥> , 𝑥@ , … , 𝑥G
𝑃 𝑥@ , … , 𝑥G |𝑥> 𝑃 𝑥>
=
𝑃 𝑥@ , … , 𝑥G
𝑃 𝑥@ , … , 𝑥G
• General:Query(𝑄)andEvidence(𝐸)
𝑃 𝑄|𝐸 =
𝑃 𝑄, 𝐸
𝑃 𝐸|𝑄 𝑃 𝑄
=
𝑃 𝐸
𝑃 𝐸
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> DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE
PROBABILISTIC MORPHABLE MODELS | JUNE 2017 | BASEL
UncertainObservation
• Observationswithuncertainty
Modelneedstodescribehow
observationsaredistributed
withjointdistribution𝑃 𝑄, 𝐸
• Stillconditionalprobability
Butjointdistributionismorecomplex
• Jointdistributionfactorized
𝑃 𝑄, 𝐸 = 𝑃 𝐸|𝑄 𝑃 𝑄
• Likelihood𝑃 𝐸|𝑄
• Prior𝑃 𝑄
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> DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE
PROBABILISTIC MORPHABLE MODELS | JUNE 2017 | BASEL
Likelihood
Joint
Likelihood Prior
𝑃 𝑄, 𝐸 = 𝑃 𝐸|𝑄 𝑃 𝑄
• Likelihoodxprior: factorizationismoreflexiblethanfulljoint
• Prior:distributionofcoremodelwithoutobservation
• Likelihood:describeshowobservationsaredistributed
• Commonexample:Gaussiandistributedpoints
𝑸
𝑬
𝑃 𝑄
𝑃 𝐸|𝑄
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> DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE
PROBABILISTIC MORPHABLE MODELS | JUNE 2017 | BASEL
BayesianInference
• Conditional/Bayesrule:methodtoupdatebeliefs
Posterior
Likelihood Prior
𝑃 𝐸|𝑄 𝑃 𝑄
𝑃 𝑄|𝐸 =
𝑃 𝐸
MarginalLikelihood
• Eachobservationupdatesourbelief(changesknowledge!)
𝑃 𝑄 → 𝑃 𝑄 𝐸 → 𝑃 𝑄 𝐸, 𝐹 → 𝑃 𝑄 𝐸, 𝐹, 𝐺 → ⋯
• BayesianInference:Howbeliefsevolvewithobservation
• Recursive:Posteriorbecomespriorofnextinferencestep
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> DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE
PROBABILISTIC MORPHABLE MODELS | JUNE 2017 | BASEL
Marginalization
• Modelscontainirrelevant/hiddenvariables
e.g.pointsonchinwhennoseisqueried
• Marginalizeoverhiddenvariables(𝐻)
𝑃 𝑄 𝐸 = A 𝑃 𝑄, 𝐻 𝐸 = A
Q
Q
𝑃 𝐸, 𝐻|𝑄 𝑃 𝑄
𝑃 𝐸, 𝐻
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> DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE
PROBABILISTIC MORPHABLE MODELS | JUNE 2017 | BASEL
GeneralBayesianInference
• Observationofadditionalvariables
• Commoncase,e.g.facerendering,landmarklocations
• Coupledtocoremodelvialikelihoodfactorization
• GeneralBayesianinferencecase:
• Distributionofdata𝐷 (formerlyEvidence)
• Parameters𝜃 (formerlyQuery)
𝑃 𝐷|𝜃 𝑃 𝜃
𝑃 𝜃|𝐷 =
𝑃 𝐷
𝑃 𝜃|𝐷 ∝ 𝑃 𝐷|𝜃 𝑃 𝜃
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> DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE
PROBABILISTIC MORPHABLE MODELS | JUNE 2017 | BASEL
Example:BayesianCurveFitting
• CurveFitting:Datainterpretationwithamodel
• Posteriordistributionexpressescertainty
• inparameterspace
• inthepredictivedistribution
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> DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE
PROBABILISTIC MORPHABLE MODELS | JUNE 2017 | BASEL
PosteriorofRegressionParameters
𝑃 𝑤
Nodata
𝑃 𝐷> |𝑤
𝑃 𝑤 𝐷>
N=1
𝑃 𝐷@ |𝑤
N=2
𝑃 𝑤 𝐷> , 𝐷@
N=19
𝑃 𝑤 𝐷> , 𝐷@ , …
BishopPRML,2006
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> DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE
PROBABILISTIC MORPHABLE MODELS | JUNE 2017 | BASEL
MoreBayesianInferenceExamples
BishopPRML,2006
Non-LinearCurveFitting
e.g.GaussianProcessRegression
Classification
e.g.Bayesclassifier
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> DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE
PROBABILISTIC MORPHABLE MODELS | JUNE 2017 | BASEL
Summary:BayesianInference
• Belief:formalexpressionofanobserver’sknowledge
• Subjectivestateofknowledgeabouttheworld
• Beliefsareexpressedasprobability distributions
• Formallynotarbitrary:Consistencyrequireslawsofprobability
• Observationschangeknowledgeandthusbeliefs
• Bayesianinferenceformallyupdatespriorbeliefstoposteriors
• ConditionalProbability
• Integrationofobservationvialikelihoodxprior factorization
𝑃 𝐷|𝜃 𝑃 𝜃
𝑃 𝜃|𝐷 =
𝑃 𝐷
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