Analysis of uncertain data:Evaluation of Given Hypotheses

Selection of probes for information gathering

Anatole Gershman, Eugene Fink, Bin Fu, and Jaime G. Carbonell

Analysis of uncertain data:Evaluation of Given Hypotheses

Selection of probes for information gathering

Anatole Gershman, Eugene Fink, Bin Fu, and Jaime G. Carbonell

Example

The analyst has to distinguish between two hypotheses:

Retires

JoinsVikings

0.4

0.6

ExampleObservations:

According to many rumors, quarterback Brett Favre has closed on the purchase of a home in Eden Prairie, MN, where the Minnesota Vikings' team facility is located.

Without the tearful public ceremony that accompanied his retirement announcement from the Green Bay Packers just 11 months ago, quarterback Brett Favre has told the New York Jets he is retiring.

Minnesota coach Brad Childress, jilted at the altar Tuesday afternoon by Brett Farve telling him he wasn’t going to play for the Vikings in 2009.

ExampleObservation distributions:

Without the tearful public ceremony that accompanied his retirement announcement from the Green Bay Packers just 11 months ago, quarterback Brett Favre has told the New York Jets he is retiring.

P(says retire | Retires) = 0.9

P(says retire | Joins Vikings) = 0.6

Bayesian induction:P (Retire|says retire) = P (Retire) ∙ P(says retire|Retire) /

(P (Retire) ∙ P(says retire|Retire)

+ P (Joins Vikings) ∙ P(Joins Vikings|Retire))

= 0.5

General problem

We have to distinguish among n mutually exclusive hypotheses, denoted H1, H2,…, Hn.

0.6For every hypothesis, we know its prior;

thus, we have an array of n of priors,

P(H1), P(H2), P(Hn)

0.4

General problemWe base the analysis on m observable features, denoted OBS1, OBS2, …, OBSm. Each observation is a variable that takes one of several discrete values.

OBS1

For every observation, OBSa, we know the number of its possible values, num[a]. Thus, we have num[1..m] with the number of values for each observation.

num[1] = 2

I will RETIRE!

I won’t RETIRE!

For every hypothesis, we know the related probability distribution of each observation. P(oa,j | Hi) represents the probabilities of possible values of OBSa.

0.9

0.1

0.4

0.6

We know a specific value of each observation val [1..m].

General problem

We have to evaluate the posterior probabilities of the n given hypotheses, denoted Post(H1), Post(H2), Post(Hn)

0.5 0.5

Extension #1Prior:

0.6

0.35

Something else H0(“surprise”)

0.05

Extension #1After discovering val, Posterior probability of H0:Post(H0) = P(H0) ∙ P(val | H0) / P(val) = P(H0) ∙ P(val | H0)

/ (P(H0) ∙ P(val | H0) + likelihood(val)).

Bad news: We do not know P(val | H0).

Good news: Post(H0) monotonically depends on P(val | H0); thus, if we obtain lower and upper bounds for P(val | H0), we also get bounds for Post(H0).

Plausibility principleUnlikely events normally do not happen; thus, if we have observed val, then its likelihood must not be too small.

Plausibility threshold: We use a global constant plaus, which must be between 0.0 and 1.0. If we have observed val, we assume that P(val) ≥ plaus / num.

We use it to obtains bounds for P(val | H0), : Lower: (plaus / num − likelihood(val)) / prior[0]. Upper: 1.0.

Plausibility principle

We use it to obtains bounds for P(val | H0): Lower: (plaus / num − likelihood(val)) / P(H0) . Upper: 1.0.

We substitute these bounds into the dependency of Post(H0) on P(val | H0), , thus obtaining the bounds for Post(H0): Lower: 1.0 − likelihood(val) ∙ num / plaus. Upper: P(H0) / (P(H0) + likelihood(val)).

We have derived bounds for the probability that none of the given hypotheses is correct.

Extension #2Multiple observations:Which one(s) to Use?

Independence assumption: usually does not work.

Bayesian analysis: Use their joint distribution? Difficult to get.

We identify the highest-utility observation and do not use other observations to corroborate it.

Extension #2Utility Function

0.4 0.6

Which one is “better”?

0.5 0.5

0.35 0.65

Extension #2Utility Function

Shannon’s Entropy (negation)

0.4 0.6

0.5 0.5

0.35 0.65

Extension #2Utility Function

KL-divergence

0.4 0.6

0.5 0.5

0.35 0.65

Extension #2Utility Function

Self-defined function

0.4 0.6

0.5 0.5

0.35 0.65

Analysis of uncertain data:Evaluation of Given Hypotheses

Selection of probes for information gathering

Anatole Gershman, Eugene Fink, Bin Fu, and Jaime G. Carbonell

Example

The analyst has to distinguish between two hypotheses:

Retires

JoinsVikings

0.4

0.6

Example

0.4

0.6

I will RETIRE!

Ask

Probe: Execute external action and observe its response, to gather more information.

0.5

0.5

ExampleProbe:

ProbeCost

ObservationProbability

Gain (utility function)

Probe Selectionsingle-obs-gain(probej)= visible[i, a, j]· (likelihood(1) · probe-gain(1)+ … + likelihood(num[a]) · probe-gain(num[a]))+ (1.0 − visible[i, a, j]) · cost[j]

gain(probj)= max (single-obs-gain(probj, obs1),…,single-obs-gain(probj, obsm))

Probe Cost

ObservationProbability

UtilityFunction

ExperimentTask: Evaluating hypothesizes (H1, H2, H3, H4).

No Probe, Accuracy of distinguishing between H1 and other hypotheses

Experiment

Probe Selection to distinguish H1 and other hypotheses

Task: Evaluating hypothesizes (H1, H2, H3, H4).

Experiment

Probe Selection to distinguish four hypotheses

Task: Evaluating hypothesizes (H1, H2, H3, H4).

Summary

Use Bayesian inference to distinguish among mutually exclusive hypotheses.

H0 hypothesis Multiple observations

Use Probe to gather more information for better analysis

Cost, Utility function, Observation Probability,...

Thank you