Conflicts in Bayesian Networks

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Conflicts in Bayesian Networks

• January 23, 2007
• Marco Valtorta
• mgv_at_cse.sc.edu

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Example Case Study 4Bayesian Network Fragment
Matching
1) Report Date 1 April, 2003. FBI Abdul Ramazi
is the owner of the Select Gourmet Foods shop in
Springfield Mall. Springfield, VA. (Phone number
703-659.2317). First Union National Bank lists
Select Gourmet Foods as holding account number
1070173749003. Six checks totaling 35,000 have
been deposited in this account in the past four
months and are recorded as having been drawn on
accounts at the Pyramid Bank of Cairo, Egypt and
the Central Bank of Dubai, United Arab Emirates.
Both of these banks have just been listed as
possible conduits in money laundering schemes.
ltProtegePerson rdfabout"ProtegeOmniseer_00135
".. ProtegefamilyName"Ramazi"ProtegegivenNa
me"Abdullarrdfslabel"Abdulla Ramazi"/gt
.. ltProtegeBank rdfabout"ProtegeOmniseer_00
614"ProtegealternateName"Pyramid Bank of
Cairo" rdfslabel"Pyramid Bank of
Cairo"gt ltProtegeaddress rdfresource"ProtegeOm
niseer_00594"/gt ltProtegenote rdfresource"Prote
geOmniseer_00625"/gt lt/ProtegeBankgt . ltProtegeR
eport rdfabout"ProtegeOmniseer_00626"
Protegeabstract"Ramazi's deposit in the past 4
months (1)" rdfslabel"Ramazi's deposit in the
past 4 months (1)"gt ltProtegereportedFrom
rdfresource"ProtegeOmniseer_00501"/gt ltProtege
detail rdfresource"ProtegeOmniseer_00602"/gt ltP
rotegedetail rdfresource"ProtegeOmniseer_0061
2"/gt lt/ProtegeReportgt lt/rdfRDFgt
Partially- Instantiated Bayesian Network Fragment
BN FragmentRepository
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Example Case Study 4Bayesian Network Fragment
Composition
. . . . .

Fragments
Situation-Specific Scenario
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Value of Information

• An item of information is useful if acquiring it
  leads to a better decision, that is, to a more
  useful action
• An item of information is useless if the actions
  that are taken after acquiring it are no more
  useful than before acquiring it
• In particular, information is useless if the
  actions that are taken after acquiring it are the
  same as before acquiring it
• In the absence of a detailed model of the utility
  of actions, the decrease in uncertainty about a
  variable of interest is taken to be a proxy for
  the increase in utility the best item of
  information to acquire is the one that reduces
  the most the uncertainty about a variable of
  interest
• Since the value of the new item of information is
  not known, we average over its possible values
• Uncertainty is measured by entropy. Reduction in
  uncertainty is measured by reduction in entropy

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Example Case Study 4Computing Value of
Information and Surprise
This is the output of the VOI program on a
situation-specific scenario for Case Study 4
(Sign of the Crescent). Variable Travel (which
represents suspicious travel) is significant for
determining the state of variable
Suspect (whether Ramazi is a terrorist),
even when it is already known that Ramazi has
performed suspicious banking transactions.
Ramazi performed illegal banking transactions
Yes
Is Ramazi a terrorist?
Would it help to know whether he traveled to
sensitive locations?
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Value of Information Formal Definition

• Let V be a variable whose value affects the
  actions to be taken by an analyst. For example,
  V indicates whether a bomb is placed on a
  particular airliner
• Let p(v) be the probability that variable V has
  value v.
• The entropy of V is
• Let T be a variable whose value we may acquire
  (by expending resources). For example, T
  indicates whether a passenger is a known
  terrorist.
• The entropy of V given that T has value t is
• The expected entropy of V given T is
• The value of information is

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Surprise Detection

• Surprise is the situation in which evidence (a
  set of findings) and a situation-specific
  scenario are incompatible
• Since situation-specific scenarios are Bayesian
  networks, it is very unusual for an outright
  inconsistency to occur
• In some cases, however, the evidence is very
  unlikely in a given scenario this may be because
  a rare case has been found, or because the
  scenario cannot explain the evidence
• To distinguish these two situations, we compare
  the probability of the evidence in the
  situation-specific scenario to the probability of
  the evidence in a scenario in which all events
  are probabilistically independent and occur with
  the same prior probability as in the
  situation-specific scenario

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Example Case Study 4Computing Surprise
The VALUE OF INFORMATION of the test node C for
the target node A is 0.0 Parsing the XMLBIF file
'ssn.xml' ... done! PROBABILITY FOR
JOINT FINDINGS 5.0E-4 Prior probability for
NODE Suspicious Personyes is 0.01 Prior
probability for NODE Unusual Activitiesyes is
0.0656 Prior probability for NODE Stolen
Weaponsyes is 0.05 PROBABILITY FOR
INDIVIDUAL FINDINGS 3.28E-5 No conflict was
detected.
This shows the output of the surprise detection
program. In this case, the user is informed that
no conflict is detected, i.e., the scenario
is likely to be a good interpretive model for the
evidence received
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Surprise Detection Formal Definition

• Let the evidence be a set of findings
• The probability of the evidence in the
  situation-specific scenario is where
  is the distribution represented in the
  situation-specific scenario
• The probability of the evidence in the model in
  which all variables are independent is
• The evidence is surprising if
• The conflict index is defined as
• The probability under that is greater
  than is
• Proof Laskey, 1991
• If the conflict index is high, it is unlikely
  that the findings could have been generated by
  sampling the situation-specific scenario
• It is reasonable to inform the analyst that no
  good explanatory model of the findings exists,
  and we are in the presence of a novel or
  surprising situation

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The Independent Straw Model

• In the absence of conflict, the joint probability
  of all evidence variables is greater than the
  product of the probabilities of each evidence
  variable. This is normally the case, because
  P(xy) gt P(x), and P(x,y) P(xy)P(y).

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Straw Models in Diagnosis
A bipartite straw model is obtained by the
elimination of some variables from a given model.
In diagnosis by heuristic classification, one
can divide variables into three sets Target,
Evidence, and Other
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How to Compute the Conflict Index (I)

• The marginal probability of each finding is the
  normal result of any probability computation
  algorithm

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How to Compute the Conflict Index (II)

• The probability of the evidence is a bi-product
  of probability update computed using the variable
  elimination or junction tree algorithms

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P(e) from the Variable Elimination Algorithm
P(? ?yes, ?yes) ?X\ ? (P(??) P(??)
P(??,?) P(??,?) P(?)P(??)P(??)P(?))
Bucket ?
P(??)P(?), ?yes
Hn(u)?xn?ji1Ci(xn,usi)
Bucket ?
P(??)
Bucket ?
P(??,?), ?yes
Bucket ?
P(??,?)
H?(?)
H?(?,?)
Bucket ?
P(??)
H?(?,?,?)
Bucket ?
P(??)P(?)
H?(?,?,?)
Bucket ?
H?(?,?)
Bucket ?
H?(?)
H?(?)
P (e) 1- k, where k is a normalizing constant
k
P(? ?yes, ?yes)
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Sensitivity Analysis

• Sensitivity analysis assesses how much the
  posterior probability of some event of interest
  changes with respect to the value of some
  parameter in the model
• We assume that the event of interest is the value
  of a target variable. The parameter is either a
  conditional probability or an unconditional prior
  probability
• If the sensitivity of the target variable having
  a particular value is low, then the analyst can
  be confident in the results, even if the analyst
  is not very confident in the precise value of the
  parameter
• If the sensitivity of the target variable to a
  parameter is very high, it is necessary to inform
  the analyst of the need to qualify the conclusion
  reached or to expend more resources to become
  more confident in the exact value of the parameter

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Example Case Study 4Computing Sensitivity
This is the output of the Sensitivity Analysis
program on a situation-specific scenario for Case
Study 4. In the context of the information
already acquired, i.e., travel to dangerous
places, large transfers of money, etc., the
parameter that links financial irregularities to
being a suspect is much more important for
assessing the belief in Ramazi being a terrorist
than the parameter that links dangerous travel to
being a suspect. The analyst may want to
concentrate on assessing the first parameter
precisely.
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Sensitivity Analysis Formal Definition

• Let the evidence be a set of findings
• Let t be a parameter in the situation-specific
  scenario
• Then, Castillo et
  al., 1997 Jensen, 2000
• a and ß can be determined by computing P(e) for
  two values of t
• More generally, if t is a set of parameters, then
  P(e)(t) is a linear function in each parameter in
  t, i.e., it is a multi-linear function of t
• Recall that
• Then,
• We can therefore compute the sensitivity of a
  target variable V to a parameter t by repeating
  the same computation with two values for the
  evidence set, viz. e and