Bayesian Fitting Probabilistic Morphable Models Summer School, June - - PowerPoint PPT Presentation

> DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE PROBABILISTIC MORPHABLE MODELS | JUNE 2017 | BASEL

Bayesian Fitting

Probabilistic Morphable Models Summer School, June 2017 Sandro Schönborn University of Basel

> DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE PROBABILISTIC MORPHABLE MODELS | JUNE 2017 | BASEL

Uncertainty: Probability Distributions

• Probabilistic Models
• Uncertain Observation (noise, outlier, occlusion, …)
• Fitting: Model explanation of observed data – probabilistic?

2

Tells us about the outcome’s certainty!

Observations Fit & Certainty Ground truth

Bishop PRML, 2006

> DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE PROBABILISTIC MORPHABLE MODELS | JUNE 2017 | BASEL

Probability: An Example

• Dentist example: Does the patient have a cavity?

3

Bu But t th the pati tient t eith ther has a cavity ty or does not

• There is no 80% cavity!
• Having a cavity should not depend on whether the

patient has a toothache or gum problems

All these statements do not contradict each other, they summarize th the denti tist’s know

• wledge about the patient

Certainty 𝑄 cavity = 0.1 𝑄 cavity toothache) = 0.8 𝑄 cavity toothache, gum problems) = 0.4

AIMA: Russell & Norvig, Artificial Intelligence. A Modern Approach, 3rd edition,

> DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE PROBABILISTIC MORPHABLE MODELS | JUNE 2017 | BASEL

Uncertainty: Bayesian Probability

• How are probabilities to be interpreted?

They are sometimes contradictory: Why does the distribution change when we have more data? Shouldn’t there be a real distribution of 𝑄 𝜄 ?

• Bayesian probabilities rely on a subjective perspective:

Probability is used to express our current knowledge. It can change when we learn or see more: With more data, we are more certain about our result.

• Not subjective in the sense that it is arbitrary!

There are quantitative rules to follow mathematically

• Probability expresses an observers certainty, often called be

belie lief

4

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.

> DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE PROBABILISTIC MORPHABLE MODELS | JUNE 2017 | BASEL

Towards Bayesian Inference

5

Posterior Models: Gaussian Process Regression Probabilistic Fit: Probabilistic interpretation of data Observed Points Posterior Model Update of prior to posterior model: Bayesian Inference

> DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE PROBABILISTIC MORPHABLE MODELS | JUNE 2017 | BASEL

Belief Updates

6

Mo Model Face distribution Ob Obser ervation Concrete points Possibly uncertain Po Posterior Face distribution consistent with observation Prior belief More knowledge Posterior belief Consistency: Laws of probability calculus!

> DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE PROBABILISTIC MORPHABLE MODELS | JUNE 2017 | BASEL

Joint Distribution

Marginal

Distribution of certain points only

Conditional

Distribution of points conditioned on known values of others

7

Probabilistic model: joint distribution of points

𝑄 𝑦>|𝑦@ = 𝑄 𝑦>, 𝑦@ 𝑄 𝑦@ 𝑄 𝑦> = A 𝑄(𝑦>, 𝑦@)

• DE

𝑄 𝑦>, 𝑦@

Both can be easily calculated for Gaussian models

> DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE PROBABILISTIC MORPHABLE MODELS | JUNE 2017 | BASEL

Certain Observation

• Observations are known values
• Distribution of 𝑦> after
• bserving 𝑦@, … , 𝑦G:

𝑄 𝑦>|𝑦@ … 𝑦G = 𝑄 𝑦>, 𝑦@, … , 𝑦G 𝑄 𝑦@, … , 𝑦G

• Conditional probability

8

> DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE PROBABILISTIC MORPHABLE MODELS | JUNE 2017 | BASEL

Towards Bayesian Inference

• Update belief about 𝑦> by observing 𝑦@, … , 𝑦G

𝑄 𝑦> → 𝑄 𝑦> 𝑦@ … 𝑦G

• Factorize joint distribution

𝑄 𝑦>, 𝑦@, … , 𝑦G = 𝑄 𝑦@, … , 𝑦G|𝑦> 𝑄 𝑦>

• Rewrite conditional distribution

𝑄 𝑦>|𝑦@ … 𝑦G = 𝑄 𝑦>, 𝑦@, … , 𝑦G 𝑄 𝑦@, … , 𝑦G = 𝑄 𝑦@, … , 𝑦G|𝑦> 𝑄 𝑦> 𝑄 𝑦@, … , 𝑦G

• General: Query (𝑅) and Evidence (𝐹)

𝑄 𝑅|𝐹 = 𝑄 𝑅, 𝐹 𝑄 𝐹 = 𝑄 𝐹|𝑅 𝑄 𝑅 𝑄 𝐹

9

> DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE PROBABILISTIC MORPHABLE MODELS | JUNE 2017 | BASEL

Uncertain Observation

• Observations with uncertainty

Model needs to describe how

• bservations are distributed

with joint distribution 𝑄 𝑅, 𝐹

• Still conditional probability

But joint distribution is more complex

• Joint distribution factorized

𝑄 𝑅, 𝐹 = 𝑄 𝐹|𝑅 𝑄 𝑅

• Likelihood 𝑄 𝐹|𝑅
• Prior 𝑄 𝑅

10

> DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE PROBABILISTIC MORPHABLE MODELS | JUNE 2017 | BASEL

Likelihood

𝑄 𝑅, 𝐹 = 𝑄 𝐹|𝑅 𝑄 𝑅

• Likelihood x prior: factorization is more flexible than full joint
• Prior: distribution of core model without observation
• Likelihood: describes how observations are distributed
• Common example: Gaussian distributed points

11

Pr Prior Li Likelihood Joi Joint 𝑄 𝑅 𝑄 𝐹|𝑅 𝑹 𝑭

> DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE PROBABILISTIC MORPHABLE MODELS | JUNE 2017 | BASEL

Bayesian Inference

• Conditional/Bayes rule: method to update beliefs

𝑄 𝑅|𝐹 = 𝑄 𝐹|𝑅 𝑄 𝑅 𝑄 𝐹

• Each observation updates our belief (changes knowledge!)

𝑄 𝑅 → 𝑄 𝑅 𝐹 → 𝑄 𝑅 𝐹, 𝐺 → 𝑄 𝑅 𝐹, 𝐺, 𝐻 → ⋯

• Bayesian Inference: How beliefs evolve with observation
• Recursive: Posterior becomes prior of next inference step

12

Pr Prior Li Likelihood Po Posterior Ma Marginal Likelihood

> DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE PROBABILISTIC MORPHABLE MODELS | JUNE 2017 | BASEL

Marginalization

• Models contain irrelevant/hidden variables

e.g. points on chin when nose is queried

• Marginalize over hidden variables (𝐼)

13

𝑄 𝑅 𝐹 = A 𝑄 𝑅, 𝐼 𝐹

• Q

= A 𝑄 𝐹, 𝐼|𝑅 𝑄 𝑅 𝑄 𝐹, 𝐼

• Q

> DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE PROBABILISTIC MORPHABLE MODELS | JUNE 2017 | BASEL

General Bayesian Inference

• Observation of additional variables
• Common case, e.g. face rendering, landmark locations
• Coupled to core model via likelihood factorization
• General Bayesian inference case:
• Distribution of data 𝐸 (formerly Evidence)
• Parameters 𝜄 (formerly Query)

𝑄 𝜄|𝐸 = 𝑄 𝐸|𝜄 𝑄 𝜄 𝑄 𝐸 𝑄 𝜄|𝐸 ∝ 𝑄 𝐸|𝜄 𝑄 𝜄

14

> DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE PROBABILISTIC MORPHABLE MODELS | JUNE 2017 | BASEL

Example: Bayesian Curve Fitting

• Curve Fitting: Data interpretation with a model
• Posterior distribution expresses certainty
• in parameter space
• in the predictive distribution

15

> DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE PROBABILISTIC MORPHABLE MODELS | JUNE 2017 | BASEL

Posterior of Regression Parameters

16

No data N=1 N=2 N=19

𝑄 𝑥 𝐸>

Bishop PRML, 2006

𝑄 𝑥 𝑄 𝑥 𝐸>, 𝐸@ 𝑄 𝑥 𝐸>, 𝐸@, …

𝑄 𝐸>|𝑥 𝑄 𝐸@|𝑥

> DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE PROBABILISTIC MORPHABLE MODELS | JUNE 2017 | BASEL

More Bayesian Inference Examples

17

Bishop PRML, 2006

Classification

e.g. Bayes classifier

Non-Linear Curve Fitting

e.g. Gaussian Process Regression

> DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE PROBABILISTIC MORPHABLE MODELS | JUNE 2017 | BASEL

Summary: Bayesian Inference

• Belief: formal expression of an observer’s knowledge
• Subjective state of knowledge about the world
• Beliefs are expressed as probability distributions
• Formally not arbitrary: Consistency requires laws of probability
• Observations change knowledge and thus beliefs
• Bayesian inference formally updates prior beliefs to posteriors
• Conditional Probability
• Integration of observation via likelihood x prior factorization

𝑄 𝜄|𝐸 = 𝑄 𝐸|𝜄 𝑄 𝜄 𝑄 𝐸

18