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Reasoning under Uncertainty. Eugene Fink LTI Seminar November 16, 2007. Challenges. The available knowledge about the real world is inherently uncertain. We usually make decisions based on incomplete and partially inaccurate data. Challenges. Representation of uncertainty.

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reasoning under uncertainty
Reasoning under Uncertainty

Eugene Fink

LTI SeminarNovember 16, 2007

challenges
Challenges

The available knowledgeabout the real world isinherently uncertain.

We usually make decisionsbased on incomplete and partially inaccurate data.

challenges3
Challenges
  • Representation of uncertainty
  • Fast reasoning based on uncertain knowledge
  • Elicitation of criticaladditional data
  • Learning of reasonabledefault assumptions
  • Contingency reasoning
projects

RAPID (2007–2011)

“Representation and Analysis of Probabilistic Intelligence Data”

Analysis of uncertain military-intelligence data and planning of future data collection.

Projects

RADAR / Space-Time (2003–2008)

“Reflective Agent with DistributedAdaptive Reasoning”

Scheduling and resource allocation under uncertainty.

outline
Outline
  • Representation of uncertainty
  • Reasoning based on uncertain knowledge
  • Elicitation of missing data
  • Future research challenges

Representation of uncertainty

alternative representations

Probability distributions

Weight:

Phone location:

95% purse 2% home 2% office 1% car

probabilitydensity

140

160

Alternative representations
  • Approximations
  • Mary’s weight is about 150. Mary’s cell phone is probably in her purse.
  • Ranges or sets of possible values
  • Mary’s weight is between 140 and 160. Mary’s cell phone may be in her purse, office, home, or car.
approximations

BUT…

DEFAULT APPROACH

  • We assume that small input changes do not cause large output changes
  • We may need to modify standard algorithms to ensure that they do not violate this assumption

WHICH MAY NOT WORK FOR SOME CASES

Approximations

Simple and intuitive approach, which usually does not require changes to standard algorithms.

approximations8

amount ofmedication

patient weight

Approximations

Example:

Selecting an amount of medication.

Since small input changes translate intosmall output changes, we can use anapproximate weight value.

approximations9

chance ofoverloading

155 LB

140 LB

load weight

Approximations

Example:

Loading an elevator.

We can adapt this procedure to the useof approximate weights by subtracting asafety margin from the weight limit.

approximations10

If your weight isexactly 150 lb,you are a winner!

prize

player weight

Approximations

Example:

Playing the “exact weight” game.

If we use approximate weight values, we cannot determine the chances of winning.

ranges or sets of possible values

BUT…

We may lose the accuracy of computation, and we cannot evaluate the probabilities of different possible values.

Ranges or sets of possible values
  • Explicit representation of a margin of error
  • Moderate changes to standard algorithms
ranges or sets of possible values12

amount ofmedication

patient weight

Ranges or sets of possible values

Example:

Selecting an amount of medication.

We obtain a range that includes the correctamount of medication. If the range width is within the acceptable margin of error, we can use it to select an appropriate amount.

ranges or sets of possible values13

chance ofoverloading

load weight

Ranges or sets of possible values

Example:

Loading an elevator.

We identify the danger of overloading, but we cannot determine its probability.

ranges or sets of possible values14

prize

player weight

Ranges or sets of possible values

Example:

Playing the “exact weight” game.

We still cannot determine the chances of winning.

probability distributions

BUT…

  • Major changes to standard algorithms
  • Major increase of the running time
Probability distributions

Accurate analysis of possible values and their probabilities.

probability distributions16
Probability distributions

Example:

Playing the “exact weight” game.

prize

player weight

We can determine possible outcomes and evaluate their probabilities.

radar rapid approach to uncertainty representation

probabilitydensity

140

150

160

weight

RADAR / RAPID approach to uncertainty representation

ranges or sets of values

ranges or setswith probabilities

probability distributions

We approximate a probability density function by a set of uniform distributions, and represent it as a set of ranges with probabilities.

Weight:

0.1 chance: [140..145] 0.8 chance: [145..155]

0.1 chance: [155..160]

uncertain data
Uncertain data
  • Nominal values

An uncertain nominal value is a set of possible values and their probabilities.

Phone location:

0.95 chance: purse 0.02 chance: home 0.02 chance: office 0.01 chance: car

uncertain data19
Uncertain data
  • Nominal values
  • Integers and reals

An uncertain numeric value is a probability-density function represented by a set of uniform distributions.

Weight:

0.1 chance: [140..145] 0.8 chance: [145..155]

0.1 chance: [155..160]

probabilitydensity

140

150

160

weight

uncertain data20
Uncertain data
  • Nominal values
  • Integers and reals
  • Strings

An uncertain string is a regularexpression with probabilities.

uncertain data21
Uncertain data
  • Nominal values
  • Integers and reals
  • Strings
  • Spatial regions

An uncertain region is a set of rectangular

regions and their probabilities.

y

0.8

0.1

0.1

x

uncertain data22

0.2 chance

0.8 chance

or a set of possible functions and their probabilities.

Uncertain data
  • Nominal values
  • Integers and reals
  • Strings
  • Spatial regions
  • Functions

An uncertain function is apiecewise-linear function with uncertain y-coordinates

amount ofmedication

patient weight

outline23
Outline
  • Representation of uncertainty
  • Reasoning based on uncertain knowledge
  • Elicitation of missing data
  • Future research challenges
uncertainty arithmetic

Arithmetic operations

+ - x

≤ ≠ ¬

  • Logical operations
  • Function application

μσ

  • Analysis of distributions
Uncertainty arithmetic

We have developed a library of basic operations on uncertain data, which input and output uncertain values.

uncertainty arithmetic25

BUT…

  • Approximate and relatively slow
  • Assumes that all probability distributions are independent
Uncertainty arithmetic
  • Allows extension of standard algorithms to reasoning with uncertain values
  • Supports the control of the trade-off between the speed and accuracy
radar application
RADAR application

Scheduling and resource allocation based on uncertain knowledge of scheduling constraints, preferences, and available resources.

  • Uncertain room and event properties
  • Uncertain resource availability and prices
  • Uncertain utility functions

We use an optimization algorithm that searches for a schedule with the greatest expected quality.

radar results

Manual and auto scheduling

Search time

ScheduleQuality

ScheduleQuality

0.83

0.83

0.80

0.78

0.72

Auto

Auto

Auto

0.63

Manual

0.9

Manual

Manual

0.8

0.7

0.6

4

1

3

9

2

5

6

7

8

10

13 rooms

84 events

5 rooms

32 events

9 rooms

62 events

Time (seconds)

13 rooms

84 events

problem size

RADAR results

Scheduling of conference events.

without

uncertainty

with

uncertainty

rapid application
RAPID application

Analysis of military intelligence, which usually includes uncertain and partially inaccurate data.

  • Relational database with uncertain data
  • Retrieval of approximate and probabilistic matches for given queries
  • Automated inferences, verification of given hypotheses, and search for novel patterns
outline29
Outline
  • Representation of uncertainty
  • Reasoning based on uncertain knowledge
  • Elicitation of missing data
  • Future research challenges
elicitation challenge
Elicitation challenge
  • Identification of critical missing data
  • Analysis of the trade-off between the cost of data acquisition and the expected performance improvements
  • Planning of effective data collection
radar rapid approach to elicitation of additional data
RADAR / RAPID approach to elicitation of additional data
  • For each candidate question, estimate the probabilities of possible answers
  • For each possible answer, compute its cost, as well as its impact on the utility of reasoning or optimization
  • For each question, compute its expected impact on the overall utility, and select questions with best expected impacts
radar rapid approach to elicitation of additional data32
RADAR / RAPID approach to elicitation of additional data

Top-Level Control

modelutility andlimitations

ModelConst-ruction

QuestionSelection

ModelEvalu-ation

currentmodel

Reasoning orOptimization

answers

questions

DataCollection

radar application33
RADAR application

Elicitation of additional data about scheduling constraints, preferences, and available resources.

The system identifies critical missing knowledge, sends related questions to the user, and improves the world model based on the user’s answers.

radar application34

Parser

Optimizer

Info elicitor

Updateresourceallocation

Chooseand sendquestions

Graphicaluser interface

User

RADAR application

Elicitation of additional data about scheduling constraints, preferences, and available resources.

Top-level control

and learning

Processnew info

radar example initial schedule

Initial schedule:

Posters

Talk

RADAR example: Initial schedule

Available rooms:

2

1

3

  • Assumptions:
  • Invited talk: – Needs a projector
  • Poster session: – Small room is OK – Needs no projector
  • Requests:
  • Invited talk, 9–10am: Needs a large room
  • Poster session, 9–11am: Needs a room
  • Missing info:
  • Invited talk: – Projector need
  • Poster session: – Room size – Projector need
radar example choice of questions

Useless info: There are no large rooms w/o a projector

×

Useless info: There are no unoccupied larger rooms

×

Potentially useful info

RADAR example: Choice of questions

Initial schedule:

2

1

Posters

3

Talk

  • Candidate questions:
  • Invited talk: Needs a projector?
  • Poster session:Needs a larger room? Needs a projector?
  • Requests:
  • Invited talk, 9–10am: Needs a large room
  • Poster session, 9–11am: Needs a room
radar example improved schedule

Initial schedule:

2

1

Posters

3

Talk

New schedule:

2

1

3

Talk

RADAR example: Improved schedule
  • Requests:
  • Invited talk, 9–10am: Needs a large room
  • Poster session, 9–11am: Needs a room

Info elicitation:

System:

Does the poster sessionneed a projector?

Posters

User:A projector may be useful,but not really necessary.

radar results38

Dependency of the qualityon the number of questions

Manual and auto repair

ScheduleQuality

ScheduleQuality

0.72

0.68

0.72

0.61

Auto withElicitation

0.50

Auto w/oElicitation

ManualRepair

After Crisis

0.68

10

30

40

50

20

Number of Questions

RADAR results

Repairing a conference schedule after a “crisis” loss of rooms.

rapid application39
RAPID application

Proactive collection ofmilitary intelligence.

  • Identification of critical uncertainties, based on given tasks and priorities
  • Planning of intelligence collection, based on the analysis of cost/benefit trade-offs and related risks
rapid application40

Analyst GUI

Goals, queries, andhypotheses

Uncertaininferencerules

RAPID application

Proactive collection ofmilitary intelligence.

Knowledgeentry andediting

Prioritized plans for proactivedata collection

Learnedinferencerules

RAPID

Inference

Engine

RAPID

Proactive

Planner

Criticaluncertainties

Inferredfacts

Uncertainfacts

Evaluation ofhypotheses

Querymatches

outline41
Outline
  • Representation of uncertainty
  • Reasoning based on uncertain knowledge
  • Elicitation of missing data
  • Future research challenges
future work
Future work
  • Learning of defaults and “common-sense” rules
  • Contingency reasoning
  • Theory of proactive learning
defaults assumptions

Example assumptions:

  • Almost all people weigh less than 500 lb
  • Tall people usually weigh more than short people
  • For people under eighteen years old, the expected weight increases with age
Defaults assumptions

Learning to make reasonable common-sense assumptions in the absence of specific data.

defaults assumptions44
Defaults assumptions

Learning to make reasonable common-sense assumptions in the absence of specific data.

  • Representation of general uncertain assumptions, context-based assumptions, and uncertain dependencies
  • Passive and active learning of these assumptions and dependencies
  • Unsupervised learning of relevant contexts
contingency reasoning
Contingency reasoning

Analysis of possible futuredevelopments and preparationto likely developments.

  • Identification of critical uncertainties and their discretization into specific scenarios
  • Compact representation of scenario spaces
  • Construction of related contingency plans
proactive learning
Proactive learning

General theory of the development andanalysis of related learning techniques.

  • Integration of learning with follow-up reasoning

Top-Level Control

Integration of learning algorithms with reasoning engines that use the learned knowledge.

QuestionSelection

ModelConst-ruction

ModelEvalu-ation

modelutility andlimitations

currentmodel

Reasoning orOptimization

answers

questions

DataCollection

proactive learning47
Proactive learning

General theory for the development andanalysis of related learning techniques.

  • Integration of learning with follow-up reasoning

Top-Level Control

  • Automated selection of learning examples

QuestionSelection

ModelConst-ruction

ModelEvalu-ation

modelutility andlimitations

currentmodel

Active selection of examples based on the trade-off among their cost, expected accuracy, and impact on the learned-knowledge utility.

Reasoning orOptimization

answers

questions

DataCollection

proactive learning48
Proactive learning

General theory for the development andanalysis of related learning techniques.

  • Integration of learning with follow-up reasoning

Top-Level Control

  • Automated selection of learning examples

QuestionSelection

ModelConst-ruction

ModelEvalu-ation

modelutility andlimitations

currentmodel

  • Automated selection of high-level strategies

Reasoning orOptimization

answers

questions

Intelligent choice and guidance of learning strategies, with the purpose to reduce the cost and time of learning.

DataCollection

proactive learning49
Proactive learning

General theory for the development andanalysis of related learning techniques.

  • Integration of learning with follow-up reasoning

Top-Level Control

  • Automated selection of learning examples

QuestionSelection

ModelConst-ruction

ModelEvalu-ation

modelutility andlimitations

currentmodel

  • Automated selection of high-level strategies

Reasoning orOptimization

answers

questions

  • Proactive analysis of future needs

DataCollection

Automated evaluation of future needs for the learned knowledge, and adaptation of the learning process to both expected and sudden changes in these needs.

slide50

Reasoning under

Uncertainty