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Probabilistic Models: Understanding Joint Distributions, Inference, and Bayes' Rule, Slides of Robotics and Autonomous Systems

Aligarh Muslim UniversityRobotics and Autonomous Systems

An introduction to probabilistic models, focusing on joint distributions, inference, and bayes' rule. Probabilistic models help us understand complex systems by simplifying the representation of reality and making predictions or drawing conclusions based on evidence. The basics of probabilistic inference, the product and chain rules, and bayes' rule, with examples and applications.

Typology: Slides

2013/2014

Uploaded on 02/01/2014

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Probabilistic Models
ā€¢Models describe how (a portion of) the world works
ā€¢Models are always simplifications
ā€¢May not account for every variable
ā€¢May not account for all interactions between variables
ā€¢ā€œAll models are wrong; but some are useful.ā€
ā€“George E. P. Box
ā€¢What do we do with probabilistic models?
ā€¢We (or our agents) need to reason about unknown variables, given
evidence
ā€¢Example: explanation (diagnostic reasoning)
ā€¢Example: prediction (causal reasoning)
ā€¢Example: value of information
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Download Probabilistic Models: Understanding Joint Distributions, Inference, and Bayes' Rule and more Slides Robotics and Autonomous Systems in PDF only on Docsity!

Probabilistic Models

ā€¢ Models describe how (a portion of) the world works

ā€¢ Models are always simplifications

  • May not account for every variable
  • May not account for all interactions between variables
  • ā€œAll models are wrong; but some are useful.ā€
    • George E. P. Box

ā€¢ What do we do with probabilistic models?

  • We (or our agents) need to reason about unknown variables, given

evidence

  • Example: explanation (diagnostic reasoning)
  • Example: prediction (causal reasoning)
  • Example: value of information

Probabilistic Models

ā€¢ A probabilistic model is a joint distribution over a set of

variables

ā€¢ Inference: given a joint distribution, we can reason about

unobserved variables given observations (evidence)

ā€¢ General form of a query:

ā€¢ This conditional distribution is called a posterior distribution or

the the belief function of an agent which uses this model

Stuff you

care

about

Stuff you

already

know

The Product Rule

ā€¢ Sometimes have conditional distributions but want the joint

ā€¢ Example:

R P

sun 0. rain 0.

D W P

wet sun 0. dry sun 0. wet rain 0. dry rain 0.

D W P

wet sun 0. dry sun 0. wet rain 0. dry rain 0.

The Chain Rule

ā€¢ More generally, can always write any joint distribution as an

incremental product of conditional distributions

Inference with Bayesā€™ Rule

ā€¢ Example: Diagnostic probability from causal probability:

ā€¢ Example:

  • m is meningitis, s is stiff neck
  • Note: posterior probability of meningitis still very small
  • Note: you should still get stiff necks checked out! Why?

Exampl

e

givens

Ghostbusters, Revisited

ā€¢ Letā€™s say we have two distributions:

  • Prior distribution over ghost location: P(G)
    • Letā€™s say this is uniform
  • Sensor reading model: P(R | G)
    • Given: we know what our sensors do
    • R = reading color measured at (1,1)
    • E.g. P(R = yellow | G=(1,1)) = 0.

ā€¢ We can calculate the posterior distribution

P(G|r) over ghost locations given a reading

using Bayesā€™ rule:

Example: Independence?

T W P

warm sun 0. warm rain 0. cold sun 0. cold rain 0.

T W P

warm sun 0. warm rain 0. cold sun 0. cold rain 0.

T P

warm 0. cold 0. W P sun 0. rain 0.

Example: Independence

  • N fair, independent coin flips: H 0. T 0.

H 0.

T 0.

H 0.

T 0.

Conditional Independence

ā€¢ Unconditional (absolute) independence is very rare (why?)

ā€¢ Conditional independence is our most basic and robust form of

knowledge about uncertain environments:

ā€¢ What about this domain:

  • Traffic
  • Umbrella
  • Raining

ā€¢ What about fire, smoke, alarm?

Bayesā€™ Nets: Big Picture

ā€¢ Two problems with using full joint distribution tables as our

probabilistic models:

  • Unless there are only a few variables, the joint is WAY too big to

represent explicitly

  • Hard to learn (estimate) anything empirically about more than a few

variables at a time

ā€¢ Bayesā€™ nets: a technique for describing complex joint

distributions (models) using simple, local distributions

(conditional probabilities)

  • More properly called graphical models
  • We describe how variables locally interact
  • Local interactions chain together to give global, indirect interactions

Example Bayesā€™ Net: Car

Graphical Model Notation

ā€¢ Nodes: variables (with domains)

  • Can be assigned (observed) or unassigned

(unobserved)

ā€¢ Arcs: interactions

  • Indicate ā€œdirect influenceā€ between

variables

  • Formally: encode conditional

independence (more later)

ā€¢ For now: imagine that arrows mean

direct causation (in general, they

donā€™t!)

Example: Traffic

ā€¢ Variables:

ā€¢ R: It rains

ā€¢ T: There is traffic

ā€¢ Model 1 : independence

ā€¢ Model 2 : rain causes traffic

ā€¢ Why is an agent using model 2 better?

R

T

Example: Traffic II

ā€¢ Letā€™s build a causal graphical model

ā€¢ Variables

ā€¢ T: Traffic

ā€¢ R: It rains

ā€¢ L: Low pressure

ā€¢ D: Roof drips

ā€¢ B: Ballgame

ā€¢ C: Cavity