Uncertainty Handling

1
Uncertainty Handling

• This is a traditional AI topic, but we need to
  cover it in at least a little detail here
• prior to covering machine learning approaches
• There are many different approaches to handling
  uncertainty
• Formal approaches based on mathematics
  (probabilities)
• Formal approaches based on logic
• Informal approaches
• Many questions arise
• How do we combine uncertainty values?
• How do we obtain uncertainty values?
• How do we interpret uncertainty values?
• How do we add uncertainty values to our knowledge
  and inference mechanisms?

2
Why Is Uncertainty Needed?

• We will find none of the approaches to be
  entirely adequate so the natural question is why
  even bother?
• Input data may be questionable
• to what extent is a patient demonstrating some
  symptom?
• do we rely on their word?
• Knowledge may be questionable
• is this really a fact?
• Knowledge may not be truth-preserving
• if I apply this piece of knowledge, does the
  conclusion necessarily hold true? associational
  knowledge for instance is not truth preserving,
  but used all the time in diagnosis
• Input may be ambiguous or unclear
• this is especially true if we are dealing with
  real-world inputs from sensors, or dealing with
  situations where ambiguity readily exists
  (natural languages for instance)
• Output may be expected in terms of a
  plausibility/probability such as what is the
  likelihood that it will rain today?
• The world is not just T/F, so our reasoners
  should be able to model this and reason over the
  shades of grey we find in the world

3
Methods to Handle Uncertainty

• Fuzzy Logic
• Logic that extends traditional 2-valued logic to
  be a continuous logic (values from 0 to 1)
• while this early on was developed to handle
  natural language ambiguities such as you are
  very tall it instead is more successfully
  applied to device controllers
• Probabilistic Reasoning
• Using probabilities as part of the data and using
  Bayes theorem or variants to reason over what is
  most likely
• Hidden Markov Models
• A variant of probabilistic reasoning where
  internal states are not observable (so they are
  called hidden)
• Certainty Factors and Qualitative Fuzzy Logics
• More ad hoc approaches (non formal) that might be
  more flexible or at least more human-like
• Neural Networks
• We will skip these in this lecture as we want to
  talk about NNs more with respect to learning

4
Fuzzy Logic

• Logic and sets are thought of as crisp
• An item is T or F, an item is in the set or not
  in the set
• Fuzzy logic, based on fuzzy set theory, says that
  an item is in a set by f(a) amount
• where a is the item
• and f is the membership function (which returns a
  real number from 0 to 1)
• Consider the figure on the right that compares
  the crisp and fuzzy membership functions for
  Tall

Membership to set A is often written like this
5
Fuzzy Logic as Process

• First, inputs are translated into fuzzy values
• This process is sometimes referred to as
  fuzzification
• For instance, assume Sue is 21, Jim is 42 and
  Frank is 53, we want to know who is old and who
  is young
• Young Sue / .7, Jim / .2, Frank / .1
• Old Sue / .1, Jim / .4, Frank / .6
• Next, we want to infer over our members
• We might have rules, for instance that you cannot
  be a country club member unless you are OLD and
  WEALTHY
• The rules, when applied, will give us conclusions
  such as Frank can be a member at .8 and Sue at .5
• Finally, given our conclusions, we need to
  defuzzify them
• There is no single, accepted method for
  defuzzification
• how do we convert Frank / .8 and Sue / .5 into
  actions?
• Often, methods compute centers of gravity or a
  weighted average of some kind,
• The result is then used to determine what
  conclusions are acceptable (true) and which are
  not

6
Fuzzy Rules

• Fuzzy logic is often applied in rule based
  formats
• If tall(x) and athletic(x) then
  basketball_player(x)
• If tall(x) and athletic(x) then soccer_player(x)
• If basketball_player(x) and poor_grades(x) then
  skip_college(x)
• where tall, athletic, etc are membership
  functions which return real values
• We will see how to deal with and, or, not, and
  implies on the next slide
• Fuzzy logic can be used to supplement a
  rule-based approach to KBS as seen above
• However, because fuzzy membership functions are
  not necessarily easy to define for ideas like
  nausea, flu, and because the logic begins to
  break down when there are lengthy chains of
  rules, we dont see this approach being used very
  often in modern KBS
• Instead, fuzzy logic is used in many controller
  devices where there are few rules
• If fast_speed(x) and approaching_red_light(x)
  then decelerate(x)
• the amount of deceleration is determined by
  defuzzifying the value derived by the implication
  in the rule

7
Fuzzy Operators
Operator Fuzzy Set Fuzzy Logic
Member (element of) A(x) N/A
AND / Intersection Min Min or
OR / Union Max Max or ab-ab
XOR N/A Max(Min(a, 1-b), Min(1-a, b)) Or ab-3abaababb-aabb
NOT 1 A(x) 1 A(x)
Complement x / 1 - a(x) x in A N/A
? (Implication) N/A Max (1 a, b) or 1 a a b
8
Hedges

• Imagine that we have a function for tall such
  that tall(a) returns as membership in the
  category
• For instance, tall(52) .1, tall(511) .6
  and tall(67) .9
• A hedge is a fuzzy term that is used to convert
  the membership functions output
• What does very tall mean? somewhat tall?
  incredibly tall?
• Common hedges
• Very ? f(x)2
• Not very ? 1 f(x)2
• Somewhat ? f(x)1/2
• About (or around) ? f(x) /- delta
• Nearly ? f(x) delta
• So for example
• if our membership function for Old says that 52
  is a old / .6
• then 52 would be very old / .36, not very old
  .64, somewhat old / .77
• We would need to define a reasonable delta for
  things like around

9
Example

• Consider as an example a controller for an engine
• The job of the controller is to make sure that
  the engine temperature stays within a reasonable
  range
• There are three memberships cold, warm, hot
• In this case, all three membership functions can
  be denoted in one figure (see to the right)

• The controller has 4 simple rules
• IF temperature IS very cold THEN stop fan
• IF temperature IS cold THEN turn down fan
• IF temperature IS normal THEN maintain level
• IF temperature IS hot THEN speed up fan

Turn down and Speed up need to be defined,
possibly through defuzzification for instance,
the extent of cold or hot determines the
degree to turn up or down the fan speed
10
Advantages and Drawbacks

• Advantages
• Handles fuzzy terms in language quite easily
• Logic is simple to implement and compute
• Very appropriate for device controllers
• used in anti-lock brakes, auto-focus in cameras,
  Japans automated subway, the Space Shuttle, etc
• Disadvantages
• How do we define the membership functions?
• there are no learning mechanisms in fuzzy logic
• what if we have membership functions provided
  from two different people
• for instance, what a 611 basketball player
  defines as tall will differ from a 410 gymnast
• How do we reconcile the two different fuzzy
  logics?
• Membership values begin to move away from
  expectations when chains of logic are lengthy so
  this approach is not suitable for many KBS
  problems (e.g., medical diagnosis)

11
Bayesian Probabilities

• Bayes Theorem is given below
• P(H0 E) probability that H0 is true given
  evidence E (the conditional probability)
• P(E H0) probability that E will arise given
  that H0 has occurred (the evidential probability)
• P(H0) probability that H0 will arise (the prior
  probability)
• P(E) probability that evidence E will arise
• Usually we normalize our probabilities so that
  P(E) 1
• The idea is that you are given some evidence E
  e1, e2, , en and you have a collection of
  hypotheses H1, H2, , Hm
• Using a collection of evidential and prior
  probabilities, compute the most likely hypothesis

12
Independence of Evidence

• Note that since E is a collection of some
  evidence, but not all possible evidence, you will
  need a whole lot of probabilities
• P(E1 E2 H0), P(E1 E3 H0), P(E1 E2 E3
  H0),
• If you have n items that could be evidence, you
  will need 2n different evidential probabilities
  for every hypothesis
• In order to get around the problem of needing an
  exponential number of probabilities, one might
  make the assumption that pieces of evidence are
  independent
• Under such an assumption
• P(E1 E2 H) P(E1 H) P(E2 H)
• P(E1 E2) P(E1) P(E2)
• Is this a reasonable assumption?

13
Continued

• Example a patient is suffering from a fever and
  nausea
• Can we treat these two symptoms as independent?
• one might be causally linked to the other
• the two combined may help identify a cause
  (disease) that the symptoms separately might not
• A weaker form of independence is conditional
  independence
• If hypothesis H is known to be true, then whether
  E1 is true should not impact P(E2 H) or P(H
  E2)
• Again, it this a reasonable assumption?
• Consider as an example
• You want to run the sprinkler system if it is not
  going to rain and you base your decision on
  whether it will rain or not on whether it is
  cloudy
• the grass is wet, we want to know the probability
  that you ran the sprinkler versus if it rained
• evidential probabilities P(sprinkler wet) and
  P(rain wet) are not independent of whether it
  was cloudy or not

14
Bayesian Networks

• We can avoid the assumption of independence by
  including causality in our knowledge
• For this, we enhance our previous approach by
  using a network where directed edges denote some
  form of dependence or causality

• An example of a causal network is shown to the
  right along with the probabilities (evidential
  and prior)
• we cannot use Bayes theorem directly because the
  evidential probabilities are based on the prior
  probability of cloudy
• However, a propagation algorithm can be applied
  where the prior probability for cloudiness will
  impact the evidential probabilities of sprinkler
  and rain
• from there, we can finally compute the likelihood
  of rain versus sprinkler

15
Propagation Algorithm

• The idea behind computing probabilities in a
  Bayesian network is the chain rule
• Before describing the chain rule, what we now
  want to compute in a network is the probability
  of a particular path through the network
• If we have a network as shown to the right, we
  might want to compute the probability of P(A, B,
  C, D, E) that is, the probability of visiting
  the nodes A, B, C, D, E in that order

• The chain rule says that P(A, B, C, D, E) P(A)
  P(B A) P(C A, B) P(D A, B, C) P(E
  A, B, C, D)
• In our previous example, we want to know the
  probability that it rained versus the probability
  that we ran the sprinkler given that the grass is
  wet
• P(C, S, W) P(C) P(SC) P(WC,S)
• P(C, R, W) P(C) P(RC) P(WC,R)

16
Example Continued

• First, we compute P(Wet) which is the sum of all
  possible paths through the network
• The summation tries c 0, c 1, s 0, s 1, r
  0 and r 1 (8 possibilities)
• Now we know the denominator to apply in Bayes
  theorem, the numerators will be P(S W) and P(R
  W)
• that is, the probability that the grass is wet
  and the sprinkler was on, and the probability
  that the grass is wet and it rained

So it is more likely that it rained than we ran
the sprinkler Notice the two probabilities do not
add up to 1
17
A Problem

• The example we considered is of a simply
  connected graph
• A multiply connected cause graph is one where
  cycles exist
• The problem here is that the chain rule does not
  work in a multiply connected graph
• consider a graph of four nodes where A ? B ? D
  and also A ? C ? D
• P(A, B, C, D) has two paths of P(D )
• we need some mechanism to allow us to compute
  P(A, B, C, D) with both P(D A, B) and P(D A,
  C)
• By collapsing nodes we can remove the cycles
• Collapse B and C into a single node
• We dont have probabilities for B C so we need
  to come up with a strategy to handle this
  collapsed node
• The strategy is to instantiate all possible
  values of B and C and test out the Bayesian
  network for each of these instantiations
• For B and C, we would have True/True, True/False,
  False/True and False/False
• We run our propagation algorithm on all four
  combinations and see which resulting P(A, B, C,
  D) value is the highest, this tells us not only
  the probability of P(D) but also what values of B
  and C lead us to that conclusion

18
More

• Our previous example was fairly easily reduced to
  a singly connected graph
• We would have to run our propagation algorithm 4
  times, but that isnt too bad a price to pay to
  get around the various problems
• However, how realistic is it for a real world
  problem to have such a simple way to reduce from
  a multiply connected graph to a simply connected
  one?
• Consider the network on the next slide, it is
  very complicatedly multiply connected
• to reduce such a graph, we would have to collapse
  a great number of nodes and then instantiate the
  collapsed node(s) to all possible combinations
• for instance, if we can get around our problem by
  reducing several pathways to three collapsed
  nodes, one with 6 nodes, one with 5 nodes and one
  with 8 nodes, we would have to run our
  propagation algorithm 26 25 28 224 times
• this quickly leads to intractability
• Note A good summary and example of Bayesian
  nets is provided at http//www4.ncsu.edu/bahler/c
  ourses/csc520f02/bayes1.html, you might want to
  check it out

19
Real World Example
Here is a Bayesian net for classification of
intruders in an operating system Notice that it
contains cycles The probabilities for the edges
are learned by sorting through log data
20
Where do the Probabilities Come From?

• Recall we need P(Hi) for every hypothesis and
  P(EiHj) for every piece of evidence for each
  hypothesis
• If we use statistics, can we guarantee that the
  statistics are unbiased?
• I poll 100 doctors offices to find out how many
  of their patients have the flu (to obtain the
  prior probability P(flu))
• This statistic is biased because I gathered the
  information in the summer (the probability would
  probably differ if I gathered the data in the
  winter)
• A prior probability is supposed to be entirely
  independent of evidence such as the season, yet
  gathering the statistics may introduce the bias
• An alternative approach is to rank probabilities
  with respect to each other and then supply values
  (the argument being that probabilities dont have
  to be exact, just relatively correct)
• Bayesian networks can also be trained with data
  sets to find reasonable probabilities
• this is in fact the approach commonly used today

21
HMMs

• A Markov model is a state transition diagram with
  probabilities on the edges
• We use a Markov model to compute the probability
  of a certain sequence of states
• see the figure to the right
• In many problems, we have observations to tell us
  what states have been reached, but observations
  may not show us all of the states
• Intermediate states (those that are not
  identifiable from observations) are hidden
• In the figure on the right, the observations are
  Y1, Y2, Y3, Y4 and the hidden states are Q1, Q2,
  Q3, Q4
• The HMM allows us to compute the most probable
  path that led to a particular observable state
• This allows us to find which hidden states were
  most likely to have occurred

• This is extremely useful for recognition problems
  where we know the end result but not how the end
  result was produced
• we know the patients symptoms but not the
  disease that caused the symptoms to appear
• we know the speech signal that the speaker
  uttered, but not the phonemes that made up the
  speech signal

22
Forward Algorithm

• There are two central HMM algorithms
• The Forward algorithm computes, given a series of
  states, the probability of achieving that state
• this is merely the product of each transition

Here is an example (from Wikipedia) The
probability of two rainy days is P(Rainy)
P(Rainy Rainy) .6.7 .42 The probability
of a sunny day followed by a rainy day
followed by a sunny day P(Sunny) P(Rainy
Sunny) P(Sunny Rainy) .4.4.3
.048
states ('Rainy', 'Sunny') observations
('walk', 'shop', 'clean') start_probability
'Rainy' 0.6, 'Sunny' 0.4 transition_probabili
ty 'Rainy' 'Rainy' 0.7, 'Sunny'
0.3, 'Sunny' 'Rainy' 0.4, 'Sunny' 0.6,
emission_probability 'Rainy'
'walk' 0.1, 'shop' 0.4, 'clean' 0.5,
'Sunny' 'walk' 0.6, 'shop' 0.3, 'clean'
0.1,
23
Viterbi Algorithm

• Of more interest, and more interestingly, is
  determining what sequence of hidden states
  probably led to a given result
• This is accomplished by using the Forward
  algorithm to compute the probabilities of all
  possible hidden state paths leading to the
  observation of interest from the starting point
• The best path is the one with the highest
  probability
• From the previous example, we might want to know
  what the chance was that it rained given that a
  friends activity was to walk
• P(rainy walk) P(rainy) P(walk rainy) .6
  .1 .06
• P(sunny walk) P(sunny) P(walk sunny) .4
  .6 .24
• So it is far more likely that it was sunny since
  your friend walked today
• More complex is when you have a sequence of
  observations in which case the probability is
  P(condition1, condition2 action1, action2)
  P(condition1) P(action1 condition1)
  P(condition2 condition1) P(action2
  condition2)
• Here, we include the probability of the
  transition from condition 1 to condition 2 in our
  calculation

24
Example Continued

• Now consider that we want to know what the
  weather was like three days in a row given that
  your friend walked the first day, shopped the
  second day and cleaned the third day
• We have many 8 probabilities to compute
• P(sunny, sunny, sunny walk, shop, clean)
  P(sunny) P(walk sunny) P(sunny sunny)
  P(shop sunny) P(sunny sunny) P(clean
  sunny) .48
• P(sunny, sunny, rainy walk, shop, clean) .62
• P(sunny, rainy, sunny walk, shop, clean)
  P(sunny) P(walk sunny) P(rainy sunny)
  P(shop rainy) P(sunny rainy) P(clean
  sunny) .43
• P(sunny, rainy, rainy walk, shop, clean) .75
• P(rainy, sunny, sunny walk, shop, clean) .21
• P(rainy, sunny, rainy walk, shop, clean) .35
• P(rainy, rainy, sunny walk, shop, clean) .37
• P(rainy, rainy, rainy walk, shop, clean) .69
• So we find that the most likely (hidden) sequence
  is sunny, rainy, rainy

25
Some Comments

• Aside from the Forward and Viterbi algorithms,
  there is also a Forward-Backward algorithm for
  learning (or adjusting) the transition
  probabilities
• This will be beneficial when it comes to speech
  recognition where transition values may not be
  available
• we will briefly examine this algorithm when we
  cover learning in the next lecture
• Both the Forward and Viterbi algorithms require a
  great number of computations
• This is reduced by using dynamic programming and
  recursion but the number of computations still
  grows exponentially as the number of transitions
  increases
• HMMs are used extensively for modern speech
  recognition systems, leading to the best
  performance (by far) for such automated systems
• But HMMs are not used very often for other
  reasoning problems

26
Certainty Factors

• This form of uncertainty was introduced in the
  MYCIN expert system (1971)
• The idea was to annotate a rule with how certain
  a conclusion might be
• If A then B (.6)
• if A is true, B can be concluded with .6 where .6
  is a plausibility (rather than a fuzzy membership
  or a probability)
• certainty factors are provided by the domain
  expert
• There are numerous questions that arise with CFs
• How do we combine CFs?
• How does an expert provide a CF for a given rule?
• Will CFs be consistent across all rules?
• CFs are informal and so are not mathematically
  sound
• On the other hand,
• CFs are more in the language of the expert,
  unlike probabilities
• CFs can denote belief (when positive) and
  disbelief (when negative)
• CFs have been used in many other systems since
  MYCIN

27
Combining CFs

• We need mechanisms to handle AND, OR, NOT and
  Implications
• If A and B Then C (.8)
• CF(A) .5, CF(B) .7, what is the CF for C?
• AND minimum
• OR maximum
• NOT 1 CF
• Implications multiply CFs
• so the CF for C as a conclusion above is min(.5,
  .7) .8 .4
• What if we have two rules, both of which suggest
  C?
• If D or E Then C (.7)
• We also need to combine the CFs of the same
  conclusion (combining evidence)
• if CF(D) .8 and CF(E) .3, then CF(C)
  max(.8, .3) .7 .56
• so now we have CF(C) .4 and CF(C) .56
• Combine the conclusions using CF1 CF2 CF1
  CF2
• so our new CF(C) .4 .56 .4 .56 .736
• CF is now greater than the two individual CFs
  while remaining lt 1

28
MYCIN Example
if (site culture is blood) (gram organism is
neg) (morphology organism is rod) (burn
patient is serious) then (identity organism is
pseudomonas) (.4) if (gram organism is neg)
(morphology organism is rod)
(compromised-host patient is yes) then
(identity organism is pseudomonas) (.6) if (gram
organism is pos) (morphology organism is
coccus) then (identity organism is pseudomonas)
(-.4) We know culture-1s site is blood, gram
is negative, morphology is most likely (.8) rod,
burn patient is semi-serious (serious at .5) and
the patient has been compromised (i.e., a virus)
CFpseudomonas min(1, 1, .8, .5) .4 .2
from rule 1 CFpseudomonas min(1, 1, 1) .6
.6 from rule 2 CFpseudomonas min(0, 0) -.4
0 from rule 3 CFpseudomonas .2 .6 - .2
.6 .68 (suggestive evidence)

• Translating CFs to English
• 1.0 certain evidence
• gt .8 strongly suggestive ev.
• gt .5 suggestive ev.
• gt 0 weakly suggestive ev.
• 0 no evidence

29
Fuzzier Approaches

• There are several problems with CFs
• Experts may not feel comfortable giving values
  like .6 for one rule and .7 for another
• If you are getting knowledge from two experts,
  they may provide inconsistent CFs
• Expert 1 is more confident in his rules than
  expert 2, so expert 1s CFs are consistently
  higher than expert 2
• After a while, the CFs for the experts rules
  might become more uniform
• After the 500th rule, the expert starts using .5
  for every CF!
• Doctors in particular are likely to use fuzzy
  vocabulary in place of numeric values
• Terminology includes likely, plausible,
  highly unlikely, ruled out, confirmed, etc
• Why not use values more consistent with the
  human?
• Especially since it is very inhuman to compute
  massive amounts of numeric calculations when
  reasoning such that the combining rules for fuzzy
  logic, CFs, probabilities are not very human-like

30
Example

• Simple matching logic to determine if a patient
  has a common viral infection
• Uses a 9 valued vocabulary
• Confirmed
• Very likely
• Likely
• Somewhat likely
• Neutral (dont know)
• Somewhat unlikely
• Unlikely
• Very unlikely
• Ruled Out
• Other matchers will decide how to combine these
  fuzzy values
• For instance, if we want to know whether to call
  the doctor, we might have features that ask is
  it a common viral infection, does the patient
  have nausea, and is the fever abnormally high
• If at least somewhat likely, yes and yes, then
  return yes otherwise return no

Features 1. Achy-eyes This-illness ? 2.
Fever This-illness ? 3. Achiness This-illness
? 4. Runny-nose This-illness ? 5.
Scratchy-throat This-illness ? 6.
Slightly-upset-stomach This-illness ? 7.
Tiredness This-illness ? 8. Malaise
This-illness ? 9. Headache This-illness
? Patterns Y ? ? ? ? ? ? ? ? ? Likely
? Y Y Y Y Y Y Y Y (5) ? Very-likely ?
Y Y Y Y Y Y Y Y (4) ? Likely ? Y ? Y ? ? ? ? ?
? Likely ? Y Y Y Y Y Y Y Y (3) ?
Somewhat-likely ? Y Y Y Y Y Y Y Y (2) ?
Neutral Otherwise ? Ruled Out
31
Advantages and Drawbacks

• Advantages of CFs and Fuzzier Values
• No need to use statistics, which might be biased
• No need for massive computations as seen with
  Bayes and HMMs
• Combining conditions and conclusions is
  permissible without violating any logic regarding
  independence of hypotheses or evidence
• The values do not have to be as accurate as in
  probabilistic methods where the statistics should
  be as accurate as possible
• Although there is no formal learning mechanism
  available, learning algorithms can be constructed
• Its easy!
• Disadvantages
• Not formal, many people dislike informal
  techniques for AI
• We still need to get the values from somewhere,
  domain experts are more likely to provide needed
  values but they still may not be highly accurate

32
Conclusions

• Many AI systems require some form of uncertainty
  handling
• Issues
• Where do the values come from?
• Can they be learned?
• yes for HMMs, Bayesian networks, maybe for CF and
  fuzzier values, no for fuzzy logic
• Is the approach computationally expensive?
• yes for HMMs and Bayesian networks, yes for
  Bayesian probabilities if you do not assume
  independence of values
• Is the approach formal?
• yes for all but CFs/fuzzier values
• Applications
• Speech recognition HMMs primarily
• Expert Systems Bayesian networks, fuzzy logic,
  CF/fuzzier values
• Device controllers fuzzy logic
• There is also non-monotonic logic and
  non-monotonic reasoning having multiple belief
  states