Basic Concepts and Definitions in ProbabilityBased Reasoning

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Basic Concepts and Definitions in
Probability-Based Reasoning

• Robert J. Mislevy
• University of Maryland
• September 14, 2006

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A quote from Glenn Shafer

• Probability is not really about numbers
• it is about the structure of reasoning.
• Glenn Shafer, quoted in Pearl, 1988, p. 77

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Views of Probability

• Two conceptions of probability
• Aleatory (chance)
• Long-run frequencies, mechanisms
• Probability is a property of the world
• Degree of belief (subjective)
• Probability is a property of Your state of
  knowledge (de Finetti) and model of the situation
• Same formal definitions machinery
• Aleatory paradigms as analogical basis for degree
  of belief (Glenn Shafer)

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Frames of discernment

• Frame of discernment is all the possible
  combinations of values of the variables your are
  working with. (Shafer, 1976)
• Discern detect, recognize, distinguish
• Property of you as much as property of world
• Depends on what you know and what your purpose is
• Frame of discernment can evolve over time
• Medical diagnosis
• Document literacy example (more information)

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Frames of Discernment in Assessment

• In Student Model, determining what aspects of
  skill knowledge to use as explicit SM
  variables--psych perspective, grainsize,
  reporting requirements
• In Evidence Model, evidence identification
  (task scoring), evaluation rules map from unique
  work product to common observed variables.
• In Task Model, which aspects of situations are
  important in task design to keep track of and
  manipulate, to achieve assessments purpose?
• Features vs. Values of Variables

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(Random) Variables

• We will start on variables with a finite number
  of possible values.
• Denote random variable by upper case, say X.
• Denote particular values and generic values by
  lower case, x.
• Y is the outcome of a coin flip yÎ h,t.
• Xi is the answer to Item i xi Î 0,1.

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Finite Probability Distributions

• Finite set of possible values x1,xn
• Prob(Xxj), P(Xxj), or more simply p(xj), is the
  probability that X takes the value xj.
• 0 p(xj) 1.
• P(Xxj or Xxm) p(xj) p(xm).

Kolmogorovs axioms (special case of probability
and finite distributions)
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Continuous Probability Distributions

• Infinitely many possible values eg, x
  xÎ0,1, x xÎ(-,)
• Events A1,Am are sets of possible values
• A1 x xlt0, A2 x xÎ(0,1), A3 x
  xgt0,
• P(Aj) is the probability that X takes a value in
  Aj
• 0 p(Aj) 1.
• If A1 Am are disjoint events that exhaust all
  possible values of x, then
• If Aj and Ak are disjoint events, P(Aj È Ak)
  P(Aj) P(Ak).

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Jensens Icy Road Example
Police Inspector Smith is impatiently awaiting
the arrival of Mr. Holmes and Dr. Watson. They
are late, and Inspector Smith has another
important appointment (lunch). Looking out the
window he wonders whether the roads are icy.
Both are notoriously bad drivers, so if the roads
are icy they are likely to crash. His secretary
enters and tells him that Dr Watson has had a car
accident. Watson? OK. It could be worse icy
roads! Then Holmes has most probably crashed
too. Ill go for lunch now. Icy roads? the
secretary replies. It is far from being that
cold, and furthermore all of the roads are
salted. Inspector Smith is relieved. Bad luck
for Watson. Let us give Holmes ten minutes
more. (Jensen, 1996, p. 7)

Jensen, F.V. (1996). An introduction to Bayesian
networks. New York Springer-Verlag.
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From the Icy Road Example

• Ice Is there an icy road?
• Values Yes, No
• Initial Probabilities (.7, .3)
• (Note choice of values for variable icy road.)

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Icy Road Probabilities
Ice
P(Iceyes)
Yes
.7
.3
No
P(Iceno)
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Graph representation
X
the variable
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Hypergraph representation
X
p(x)
the probability distribution
the variable
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Joint probability distributions

• Two random variables, X and Y
• P(Xxj,Yyk), or p(xj, yk), is the probability
  that X takes the value xj and Y takes the value
  yk .
• 0 p(xj , yk) 1.

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Marginal probability distributions 1

• Two discrete random variables, X and Y
• Recall P(Xxj,Yyk), or p(xj, yk), is the
  probability that X takes the value xj and Y takes
  the value yk
• The marginal probability of a value xj of X is
  the sum over all the possible joint probabilities
  p(xj, yk) with that value of X

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Conditional probability distributions

• Two random variables, X and Y
• P(XxjYyk), or p(xj yk), is the probability
  that X takes the value xj given that Y takes the
  value yk .
• This is how we express relationships among
  real-world phenomena
• Coin flip p(heads) vs. p(headsBobReport)
• P(heart attackage, family history, blood
  pressure)
• P(February 10 high temperature geographical
  location, February 9 high temperature)
• IRT P(Xj1) vs. P(Xj1q)

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Conditional probability distributions

• Two discrete random variables, X and Y
• P(XxjYyk), or p(xj yk), is the probability
  that X takes the value xj given that Y takes the
  value yk . As if is an important idea here.

• A picture will make clear what this means

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Conditional probability distributions
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Conditional probability distributions
Probabilities for X conditional on
Yy2 p(x1y2) p(x1,y2) / p(y2)
.3 / .4 .75 p(x2y2)
p(x2,y2) / p(y2) .1 / .4
.25
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Conditional probability distributions
Probabilities for Y conditional on
Xx1 p(y1x1) p(x1,y1) / p(x1)
.1 / .6 .167 p(y2x1)
p(x1,y2) / p(x1) .3 / .6
. 50
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Conditional probability distributions

• Kolmogorovs axioms hold for conditional
  probabilities too, fixing what is on the right
  side of the conditioning bar
• 0 p(xj yk) 1 for each given yk.
• for each given yk
• P(Xxj or Xxm Yyk) p(xj yk) p(xm yk).

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Marginal probability distributions 2

• Two discrete random variables, X and Y
• Recall p(xj yk), is the probability that X xj
  given Y yk .
• The marginal probability of a value of X is the
  sum of its conditional probabilities given all
  possible values of Y, with each weighted by its
  probability

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Graph representation
X
Y
the parent variable
the child variable
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Hypergraph representation
X
Y
p(x)
p(yx)
Marginal probability distribution for parent
Conditional probability distribution for child
given parent
Parent variable
Child variable
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Hypergraph representation
X
p(x)
Y
p(yx,z)
Z
p(z)
Conditional probability distribution for child
given parents
Child variable
Marginal probability distributions for parents
Parent variables
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A relationship between joint and conditional
probability distributions

• p(xj, yk) p(xj yk) p(yk)
• p(yk xj) p(xj) .
• Basis of Bayes Theorem
• p(yk xj) p(xj) p(xj yk) p(yk)
• Þ p(yk xj) p(xj yk)
  p(yk) / p(xj)
• Þ p(yk xj) µ p(xj yk)
  p(yk).

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Bayes Theorem

• The setup, with two random variables, X and Y
• You know conditional probabilities, p(xj yk),
  which tell you what to believe about X if you
  knew the value of Y.
• You learn Xx what should you believe about Y?
• You combine two things
• Relative conditional probabilities (the
  likelihood)
• Previous probabilities about Y values

posterior likelihood
prior
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From the Icy Road Example

• Ice Is there an icy road?
• Values Yes, No
• Initial Probabilities (.7, .3)
• Watson Does Watson have a car crash?
• Values Yes, No
• Probabilities conditional on Icy Road
• (.8, .2) if IceYes, (.1, .9) if IceNo.

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Icy Road Conditional Probabilities
Watson
No
Yes
Ice
.2
Yes
.8
.9
.1
No
p(WatsonnoIceyes)
p(WatsonyesIceyes)
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Icy Road Conditional Probabilities
Watson
No
Yes
Ice
.2
Yes
.8
.9
.1
No
p(WatsonnoIceno)
p(WatsonyesIceno)
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Icy Road Likelihoods
Note 2/9 ratio
Watson
No
Yes
Ice
p(WatsonnoIceyes)
.2
Yes
.8
.9
.1
No
p(WatsonnoIceno)
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Icy Road Likelihoods
Note 8/1 ratio
Watson
No
Yes
Ice
p(WatsonyesIceyes)
.2
Yes
.8
.9
.1
No
p(WatsonyesIceno)
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Icy Road Bayes TheoremIf Watson yes
Prior Likelihood µ Posterior
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Icy Road Bayes TheoremIf Watson yes
Prior Likelihood µ Posterior
Note Sum .59, not 1.00. These arent
probabilities.
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Icy Road Bayes TheoremIf Watson yes
Prior Likelihood µ Posterior
Yes
.95
.05
Divide through by normalizing constant .59 to get
posterior probabilities.
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Independence

• Independence
• The probability of the joint occurrence of
  values of two variables is always equal to the
  product of the probabilities individually
• P(Xx,Yy) P(Xx) P(Yy).
• Equivalent to saying that learning the value of
  one of the variables does not change your belief
  about the other.

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Independence
X and Y independent
X and Y not independent X and Y are dependent
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Conditional independence

• Conditional independence
• The conditional probability of the joint
  occurrence given the value of another variable is
  always equal to the product of the conditional
  probabilities
• P(Xx,YyZz) P(Xx Zz) P(Yy Zz).

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Conditional independence

• Conditional independence is not a grace of
  nature for which we must wait passively, but
  rather a psychological necessity which we satisfy
  actively by organizing our knowledge in a
  specific way.
• An important tool in such organization is the
  identification of intermediate variables that
  induce conditional independence among
  observables if such variables are not in our
  vocabulary, we create them.
• In medical diagnosis, for instance, when some
  symptoms directly influence one another, the
  medical profession invents a name for that
  interaction (e.g., syndrome, complication,
  pathological state) and treats it as a new
  auxiliary variable that induces conditional
  independence dependency between any two
  interacting systems is fully attributed to the
  dependencies of each on the auxiliary variable.
  (Pearl, 1988, p. 44)

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Example Icy Road