Learning and Decision
Making
Lecture 1
Learning and Decision Making
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1
“Anytime that you have
a 50-50 chance of getting something right,
there’s a 90% probability
you’ll get it wrong”
–Andy Rooney
Learning and Decision Making
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What is probability?
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Has roots in games of chance
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Used to compute the likelihood of a given event's
occurrence
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Natural tool to model uncertainty
Learning and Decision Making
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What is probability?
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Classical definition of probability of event A
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N possible events
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NA ways by which A can occur
NA
P[A] =
N
•
Examples:
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What is the probability of drawing an even number in a
die?
Learning and Decision Making
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What is probability?
•
Classical definition of probability of event A
•
N possible events
•
NA ways by which A can occur
NA
P[A] =
N
•
Examples:
•
What is the probability of drawing a ♤ in a card deck?
Learning and Decision Making
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5
What is probability?
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Frequentist definition of probability of event A
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N observed events
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NA times that event A occurred
NA
P[A] = lim
N!1 N
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Relative frequency of event A
Learning and Decision Making
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What is probability?
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Subjective definition of probability of event A
•
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Degree of belief that event A may occur
Example:
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Probability of Benfica winning the I Liga.
Learning and Decision Making
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Formally, …
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Probability space is a triplet (Ω, 𝓕, P) where:
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Ω is the sample space
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𝓕 is the set of events
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P is a probability measure
Learning and Decision Making
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Formally, …
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Sample space Ω
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Space of possible outcomes
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Each and every thing that may happen
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Example: In a die throw, the possible outcomes are {1, 2, 3, 4, 5, 6}
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Example: When drawing a card, the possible outcomes are {A♤,
2♤, …, J♢, Q♢, K♢}
Learning and Decision Making
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Formally, …
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Set of events, 𝓕
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Subsets of Ω that we can “measure”, i.e., assign a
probability
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Includes the empty set ∅ and the full set Ω
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Examples of events (die throw):
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Drawing an even number: {2, 4, 6}
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Drawing a number larger than 3: {4, 5, 6}
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Drawing no number: ∅
Learning and Decision Making
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Formally, …
•
Set of events, 𝓕
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Subsets of Ω that we can “measure”, i.e., assign a probability
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Includes the empty set ∅ and the full set Ω
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Examples of events (card draw):
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Drawing a spade: {A♤, 2♤, 3♤, 4♤, 5♤, 6♤, 7♤, 8♤,
9♤, 10♤, J♤, Q♤, K♤}
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Drawing a figure: {J♤, Q♤, K♤, J♧, Q♧, K♧, J♡, Q♡,
K♡, J♢, Q♢, K♢}
Learning and Decision Making
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Formally, …
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Probability, P
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“Measures” each event in 𝓕
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Axioms of probability:
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P(A) ≥ 0
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P(Ω) = 1
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Given disjoint events A1, …, An
P[A1 [ . . . [ An ] =
n
X
P[Ai ]
i=1
Learning and Decision Making
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Can you show:
•P(A) ≥ 0
•P(Ω) = 1
• P[A1 [ . . . [ An ] =
n
X
P[Ai ]
i=1
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P(∅) = 0
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Given A, B with A ⊂ B, P[A [ B]  P[A] + P[B]
Learning and Decision Making
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Conditional probability
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Conditional probability of event A given B:
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Probability of A occurring, given that B occurred
P[A \ B]
P[A | B] =
P[B]
… assuming P(B) > 0.
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Verifies all axioms of probability
Learning and Decision Making
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Conditional probability
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Important properties:
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Independent events:
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Law of total probability: given disjoint events A1, …, An, with
P[A | B] = P[A]
A1 U … U An = Ω,
P[B] =
n
X
i=1
P[B | Ai ] · P[Ai ]
Learning and Decision Making
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Conditional probability
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Important properties:
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Bayes theorem:
likelihood prior
P[B | A]P[A]
P[A | B] =
P[B]
evidence
hypothesis observations
Learning and Decision Making
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Random variable
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Given a probability space (Ω, 𝓕, P),
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Sometimes Ω is a cumbersome set to work with
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It would be more convenient to instead work in some
other space, more mathematically convenient (for
example, R)
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A random variable X is a map X : Ω ⟶ E, where E is some
convenient space (usually R)
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We usually work with random variables, ignoring the
underlying probability space
Learning and Decision Making
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Random variable
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We write P(X = x) to represent
P[X = x] = P[{! 2 ⌦ : X(!) = x}]
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If the r.v. takes values in a discrete set, we call it a discrete
random variable
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If the r.b. takes values in a continuous set, we call it a
continuous random variable
Learning and Decision Making
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Expectation
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Given a r.v. X and a function f : X ⟶ R, the expectation of f is
the quantity
X
E[f (X)] =
f (x)P[X = x]
x
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We can have conditional expectations:
E[f (X) | Y = y] =
X
x
f (x)P[X = x | Y = y]
Learning and Decision Making
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Properties
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Law of total probability with expectations:
X
E[f (X)] =
E[f (X) | Y = y]P[Y = y]
y
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Conditioning “eliminates” expectation:
E[f (X) | X = x] = f (x)
Learning and Decision Making
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Some useful results
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Law of large numbers: Given i.i.d. real-valued r.v.s X1, …, Xn,
such that
E[X1 ] = E[X2 ] = . . . = E[XN ] = µ
then
N
X
1
a.s.
lim
Xi = µ
N !1 N
i=1
Learning and Decision Making
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Some useful results
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Jensen’s inequality: Given a real-valued r.v. X and a convex
function f, then
f (E[X])  E[f (X)]
x1
x2
Learning and Decision Making
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