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Introduction Where Do Probabilities Come From? Basic Probability Notation The Axioms of Probabilities Unconditional and Cond
Introduction to Probability (1)
John Kelleher & Brian Mac Namee
Introduction Where Do Probabilities Come From? Basic Probability Notation The Axioms of Probabilities Unconditional and Cond
1
Introduction
Summarizing Uncertainty
2
Where Do Probabilities Come From?
3
Basic Probability Notation
Basics
4
The Axioms of Probabilities
5
Unconditional and Conditional Probabilities
6
Summary
Introduction Where Do Probabilities Come From? Basic Probability Notation The Axioms of Probabilities Unconditional and Cond
Introduction
Introduction Where Do Probabilities Come From? Basic Probability Notation The Axioms of Probabilities Unconditional and Cond
So far we have looked at similarity-based and
information-theory-based approaches to machine learning.
Probability theory offers us another way to tackle the same
sorts of problems.
Probability theory offers us a formally sophisticated way in
which to perform reasoning based on an assessment of
the likelihood of various outcomes performed across a
historical dataset.
Probaility theory is used widely in artificial intelligence in
particular because it solves the qualification problem.
Introduction Where Do Probabilities Come From? Basic Probability Notation The Axioms of Probabilities Unconditional and Cond
Let’s consider a dental patient’s toothache as an example
of uncertain reasoning.
Consider a rule that attempts to capture the relationship
between patients having a cavity and patients having a
toothache.
Toothache =⇒ Cavity
But this doesn’t work as every toothache is not caused by
a cavity.
Toothache =⇒ Cavity OR GumProblem OR Abscess . . .
We could try to go the other way around and make a rule
such as
Cavity =⇒ Toothache
This has the same problem, though, as not all cavities
cause toothaches.
Introduction Where Do Probabilities Come From? Basic Probability Notation The Axioms of Probabilities Unconditional and Cond
Summarizing Uncertainty
Trying to use exact rules to cope with a domain like medical
diagnosis fails for three main reasons.
Laziness
it is too much work to list an exceptionless rule-set and actually
too difficult to use such a rule-set.
Theoretical ignorance
Medical science has no complete theory for the domain.
Practical ignorance
Even if we know all the rules, we might be uncertain about
particular patient because not all the necessary tests have
been or can be run.
This is essentially the qualification problem.
Introduction Where Do Probabilities Come From? Basic Probability Notation The Axioms of Probabilities Unconditional and Cond
Summarizing Uncertainty
Probability provides a way of summarizing the uncertainty that
comes from our laziness and ignorance.
We might not know for sure what afflicts a particular
patient, but we believe that there is, say, an 80% chance that is, a probability of 0.8 - that the patient has a cavity if
they have a toothache.
The 80% summarises those cases in which all the factors
needed for a cavity to cause a toothache are present and
other cases in which the patient has both toothache and
cavity but the two are unconnected.
The missing 20% summarizes all the other possible
causes of tootache that we are too lazy or ignorant to
confirm or deny.
Introduction Where Do Probabilities Come From? Basic Probability Notation The Axioms of Probabilities Unconditional and Cond
Where Do Probabilities Come
From?
Introduction Where Do Probabilities Come From? Basic Probability Notation The Axioms of Probabilities Unconditional and Cond
Frequentist position is that the numbers can come only
from experiments (i.e. they are based on the relative
frequency of an event within a sample)
Objectivist view is that probabilities are real aspect of the
universe – propensities of objects to behave in a certain
way.
Subjectivist view describes probabilities as a way of
characterising beliefs about the world, rather than as
having any external physical significance
Introduction Where Do Probabilities Come From? Basic Probability Notation The Axioms of Probabilities Unconditional and Cond
Relative Frequency
The relative frequency for any particular category is the
proportion of all responses in the data set that fall in that
category:
relative frequency =
frequency of event
total number of sample points
Introduction Where Do Probabilities Come From? Basic Probability Notation The Axioms of Probabilities Unconditional and Cond
Example
Suppose we take a coin and begin to toss it over and over.
After each toss, we compute the relative frequency of
heads observer so far, i.e., (number of heads)/(number of
tosses).
Table: Results of Coin Toss Experiment
Toss Number
Outcome
Cum. Number of Heads
Relative Frequency of Heads
1
T
0
0
2
H
1
0.5
3
H
2
.667
4
H
3
.75
5
T
3
.6
6
T
3
.5
7
H
4
.571
8
H
5
.625
9
T
5
.556
10
T
5
.5
Much empirical evidence suggests that as the number of
tosses increases, the relative frequency of heads does not
continue to fluctuate wildly but instead stablizes and
approaches some fixed number.
Introduction Where Do Probabilities Come From? Basic Probability Notation The Axioms of Probabilities Unconditional and Cond
Basic Probability Notation
Introduction Where Do Probabilities Come From? Basic Probability Notation The Axioms of Probabilities Unconditional and Cond
Basics
Chance Experiment
We use the phrases chance experiment or random experiment
to refer to any activity or situation in which the outcome is
uncertain.
Example
A chance experiment might involve: rolling a dice, tossing a
coin, running a survey to ascertain whether or not each person
in a sample supports the death penalty, or it could involve an
investigation carried out in a laboratory to study how varying the
amount of a certain chemical input affects the yield of a product.
Introduction Where Do Probabilities Come From? Basic Probability Notation The Axioms of Probabilities Unconditional and Cond
Basics
Random Variables
A random variable associates a single number with each
outcome of an experiment.
A random variable can be thought of as referring to ”part”
of the world whose ”status” is initially unknown.
We will always capitalize the names of random variables.
Example
For example, imagine we are about to roll two dice. We might
have a random variable called Die1 to hold the value of the first
dice after we roll it, another random variable called Die2 to hold
the value of the second dice after we roll it, and another random
variable called Total to hold the summation of the two dice
values after we roll them.
In our dentistry domain we might have a random variable called
Cavity which would refer to whether my lower left wisdom tooth
has a cavity.
Introduction Where Do Probabilities Come From? Basic Probability Notation The Axioms of Probabilities Unconditional and Cond
Basics
Random Variables and Domains
Each random variable has a domain of values that it can take
on: for example, the domain of Cavity might be htrue, falsei.
Random variables are typically divided into three kinds,
depending on the type of the domain:
1
Boolean random variables have a domain < true, false >
2
Discrete random variables take values from a countable
domain. The values in the domain must be mutually
exclusive and exhaustive. For example, the domain of
Weather might be < sunny , rainy , cloudy , snowy >.
3
Continuous random variables take on values from the real
numbers. For example, the time it takes a car to travel a
quarter of a kilometer.
With some exceptions, we will be concentrating on the discrete
case.
Introduction Where Do Probabilities Come From? Basic Probability Notation The Axioms of Probabilities Unconditional and Cond
Basics
In probability theory the set of all possible worlds (i.e. the set of
all possible results of a chance experiment) is called the
sample space
Sample Space Ω
We use the phrase sample space - denoted by the Greek
symbol Ω (upper case omega) - to describe the results of a set
of chance experiments; e.g., the results of 6 rolls of a dice.
Introduction Where Do Probabilities Come From? Basic Probability Notation The Axioms of Probabilities Unconditional and Cond
Basics
Possible worlds are also known as atomic events or sample
points
A sample point/possible world/atomic event (denoted by the
symbol ω) is one element of a sample space: ω ∈ Ω.
Atomic events ω
An atomic event is a complete specification of the state of the
world. It can be thought of as an assignment of particular
values to all the variables of which the world is composed.
Example
For example, if my world consists of only the Boolean variables
Cavity and Toothache, then there are just four distinct atomic
events; the proposition Cavity = false ∧ Toothache = true is
one such event.
Introduction Where Do Probabilities Come From? Basic Probability Notation The Axioms of Probabilities Unconditional and Cond
Basics
Atomic events have some important properties:
They are mutually exclusive – at most one can actually be
the case.
The set of all possible atomic events is exhaustive – at
least one of them must be the case. That is, the disjunction
of all atomic events is logically equivalent to true.
Any particular atomic event entails the truth or falsehood of
every proposition, whether simple or complex.
Any proposition is logically equivalent to the disjunction of
all atomic events that entail the truth of the proposition.
Introduction Where Do Probabilities Come From? Basic Probability Notation The Axioms of Probabilities Unconditional and Cond
Basics
Probability Space/Model
A probability space or probability model is a sample space with
an assignment P(ω) for every ω ∈ Ω s.t.
0 ≤ P(ω) ≤ 1
P
ω
P(ω) = 1
Example
For example, if we are about to roll a dice, there are 6
possible worlds/atomic events/sample points to consider:
d=1, d=2, d=3, d=4, d=5, d=6.
If we assume that the dice is fair and the the rolls don’t
interfere with each other then each of the possible
worlds/atomic events/sample points has an equal
probability P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1/6.
Introduction Where Do Probabilities Come From? Basic Probability Notation The Axioms of Probabilities Unconditional and Cond
Basics
Probabilistic assertions and queries are not usually about
particular possible worlds, but about sets of them; e.g., we
might be interested in the cases where two dice add up to 11.
Event and Propositions
An event is any collection of possible outcomes of a
chance experiment. In other words, an event A is any
subset of Ω
In AI, these sets are always described by propositions in a
formal language. For each proposition, the corresponding
set contains just those possible worlds in which the
proposition holds.
Introduction Where Do Probabilities Come From? Basic Probability Notation The Axioms of Probabilities Unconditional and Cond
Basics
Propositions
A proposition asserts that such-and-such is the case;
Example
For example, the simplest kind of proposition asserts that a
random variable has a particular value drawn from its domain.
(i.e. Cavity = true) might represent the proposition that I do in
fact have a cavity in my lower left wisdom tooth.
In the slides we may sometimes abbreviate a proposition such
as Cavity = true by cavity and Cavity = false by ¬cavity
Introduction Where Do Probabilities Come From? Basic Probability Notation The Axioms of Probabilities Unconditional and Cond
Basics
The Probability of an Event/Proposition
Is denoted by P(E)
Is defined to be the sum of the probabilities of the worlds in
which it holds (i.e. the probability of an event E is the sum
of the probabilities of all outcomes contained in E).
P(E) =
X
P(ω)
{ω∈E}
Example
For example,
P(die roll < 4) = P(1) + P(2) + P(3) = 1/6 + 1/6 + 1/6 = 1/2
is the value approached by the relative frequency of occurrence
of E in a very long series of replications of a chance
experiment.
Introduction Where Do Probabilities Come From? Basic Probability Notation The Axioms of Probabilities Unconditional and Cond
Basics
Basics summary:
Chance Experiments
Random Variables and Domains
Sample Space Ω
Sample point/Possible world/Atomic Event ω e.g. die roll =
4.
Probability Space/Model Ω with assignment P(ω) for every
ω∈Ω
0 ≤ P(ω) ≤ 1
P
ω P(ω) = 1
Events and Propositions, e.g. P(dieroll > 4)
P
P(E) = {ω∈E} P(ω)
Introduction Where Do Probabilities Come From? Basic Probability Notation The Axioms of Probabilities Unconditional and Cond
The Axioms of Probabilities
Introduction Where Do Probabilities Come From? Basic Probability Notation The Axioms of Probabilities Unconditional and Cond
Andrei Kolmogorov, showed how that you can build up the rest
of probability theory from three simple axioms (Kolmogorov’s
axioms).
(Ax.1) 0 ≤ P(a) ≤ 1
All probabilities are between 0 and 1. For any proposition a
(Ax.2) P(true) = 1
P(false) = 0
Necessarily true (i.e. valid) propositions have probability 1, and
necessarily false (i.e., unsatisfiable) propositions have
probability 0.
(Ax.3) P(a ∨ b) = P(a) + P(b) − P(a ∧ b)
Inclusion-exclusion principle: the cases where a holds together
with the cases where b holds cover all the cases where a ∨ b
holds; but summing the two sets of cases counts the
intersection twice, so we need to subtract P(a ∧ b).
Introduction Where Do Probabilities Come From? Basic Probability Notation The Axioms of Probabilities Unconditional and Cond
We can derive a variety of useful facts from the basic axioms:
Recall that, any proposition a is equivalent to the
disjunction of all the atomic events in which a holds; call
this set of events e(a).
Recall also that atomic events are mutually exclusive, so
the probability of any conjunction of atomic events is zero,
by axiom 2.
Hence, from axiom 3, we can derive the following
relationship:
P
P(a) = ei ∈e(a) P(ei )
The probability of a proposition is equal to the sum of the
probabilities of the atomic events in which it holds
Introduction Where Do Probabilities Come From? Basic Probability Notation The Axioms of Probabilities Unconditional and Cond
Unconditional and Conditional
Probabilities
Introduction Where Do Probabilities Come From? Basic Probability Notation The Axioms of Probabilities Unconditional and Cond
Unconditional/Prior Probabilities
The unconditional or prior probability associated with a
proposition a is the degree of belief accorded to it in the
absence of any other information; it is written as
P(a)
Example
For example, if the prior probability that I have a cavity is 0.1,
then we would write:
P(Cavity = true) = 0.1 or P(cavity ) = 0.1
It is important to remember that P(a) can be used only when
there is no other information!
Introduction Where Do Probabilities Come From? Basic Probability Notation The Axioms of Probabilities Unconditional and Cond
Conditional/Posterior Probabilities
Once we obtain some evidence concerning the previously
unknown random variables making up the domain, prior
probabilities are no longer applicable. Instead we use
Conditional or posterior probabilities; written as:
P(a|b) where the | is prounced as given
Example
For example, P(cavity |toothache) = 0.8 i.e., given that
toothache is all I know the probability of cavity is 0.8
Introduction Where Do Probabilities Come From? Basic Probability Notation The Axioms of Probabilities Unconditional and Cond
If we know more, e.g. swelling is also given, then we have
P(cavity |toothache, swelling) = 0.95
Note: the less specific belief remains valid after more
evidence arrives, but is not always useful
New evidence may be irrelevant, allowing simplification,
e.g.,
P(cavity |toothache, IrelandWins) = P(cavity |toothache) =
0.8
This kind of inference, sanctioned by domain knowledge, is
crucial
Introduction Where Do Probabilities Come From? Basic Probability Notation The Axioms of Probabilities Unconditional and Cond
Practice
Calculate P(fineGael) given the following dataset
Name
ADAMS, Gerry
BANNON, James
BARRETT, Sean
BARRY, Tom
BROUGHAN, Tommy
BROWNE, John
BRUTON, Richard
BUTLER, Ray
BURTON, Joan
BYRNE, Eric
CALLEARY, Dara
COVENEY, Simon
DEENIHAN, Jimmy
DURKAN, Bernard
FITZGERALD, Frances
GILMORE, Eamon
KIRK, Seamus
KITT, Michael
LYNCH, Kathleen
LYONS, John
MALONEY, Eamonn
MARTIN, Micheal
MATHEWS, Peter
Party
Sinn Fein
Fine Gael
Fine Gael
Fine Gael
Labour
Fianna Fail
Fine Gael
Fine Gael
Labour
Labour
Fianna Fail
Fine Gael
Fine Gael
Fine Gael
Fine Gael
Labour
Fianna Fail
Fianna Fail
Labour
Labour
Labour
Fianna Fail
Fine Gael
Min?
No
No
No
No
No
No
Yes
No
Yes
No
No
Yes
Yes
No
Yes
Yes
No
No
No
No
No
No
No
Name
MING
NOONAN, Michael
O’DEA, Willie
QUINN Ruairi
RABBITTE, Pat
REILY, James
RING, Michael
ROSS, Shane
RYAN, Brendan
SHATTER, Alan
SHERLOCK, Sean
SHORTALL, Roisin
SMITH, Brendan
SPRING, Arthur
STAGG, Emmet
TROY, Robert
TUFFY, Joanna
TWOMEY, Liam
VARADKAR, Leo
WALL, Jack
WALLACE, Mick
WALSH, Brian
WHITE, Alex
Party
Ind
Fine Gael
Fianna Fail
Labour
Fine Gael
Fine Gael
Fine Gael
Independent
Labour
Fine Gael
Labour Cork
Labour
Fianna Fail
Labour Kerry
Labour
Fianna Fail
Labour
Fine Gael
Fine Gael
Labour
Ind
Fine Gael
Labour
Min?
No
Yes
No
Yes
Yes
Yes
No
No
No
Yes
No
No
No
No
No
No
No
No
Yes
No
No
No
No
Introduction Where Do Probabilities Come From? Basic Probability Notation The Axioms of Probabilities Unconditional and Cond
Practice
Calculate P(minister |fineGael) given the following dataset
Name
ADAMS, Gerry
BANNON, James
BARRETT, Sean
BARRY, Tom
BROUGHAN, Tommy
BROWNE, John
BRUTON, Richard
BUTLER, Ray
BURTON, Joan
BYRNE, Eric
CALLEARY, Dara
COVENEY, Simon
DEENIHAN, Jimmy
DURKAN, Bernard
FITZGERALD, Frances
GILMORE, Eamon
KIRK, Seamus
KITT, Michael
LYNCH, Kathleen
LYONS, John
MALONEY, Eamonn
MARTIN, Micheal
MATHEWS, Peter
Party
Sinn Fein
Fine Gael
Fine Gael
Fine Gael
Labour
Fianna Fail
Fine Gael
Fine Gael
Labour
Labour
Fianna Fail
Fine Gael
Fine Gael
Fine Gael
Fine Gael
Labour
Fianna Fail
Fianna Fail
Labour
Labour
Labour
Fianna Fail
Fine Gael
Min?
No
No
No
No
No
No
Yes
No
Yes
No
No
Yes
Yes
No
Yes
Yes
No
No
No
No
No
No
No
Name
MING
NOONAN, Michael
O’DEA, Willie
QUINN Ruairi
RABBITTE, Pat
REILY, James
RING, Michael
ROSS, Shane
RYAN, Brendan
SHATTER, Alan
SHERLOCK, Sean
SHORTALL, Roisin
SMITH, Brendan
SPRING, Arthur
STAGG, Emmet
TROY, Robert
TUFFY, Joanna
TWOMEY, Liam
VARADKAR, Leo
WALL, Jack
WALLACE, Mick
WALSH, Brian
WHITE, Alex
Party
Ind
Fine Gael
Fianna Fail
Labour
Fine Gael
Fine Gael
Fine Gael
Independent
Labour
Fine Gael
Labour Cork
Labour
Fianna Fail
Labour Kerry
Labour
Fianna Fail
Labour
Fine Gael
Fine Gael
Labour
Ind
Fine Gael
Labour
Min?
No
Yes
No
Yes
Yes
Yes
No
No
No
Yes
No
No
No
No
No
No
No
No
Yes
No
No
No
No
Introduction Where Do Probabilities Come From? Basic Probability Notation The Axioms of Probabilities Unconditional and Cond
Mathematical definition of conditional probability
Conditional probabilities are defined in terms of unconditional
probabilities as follows:
For any propositions a and b we have:
P(a|b) =
P(a∧b)
P(b)
if P(b) 6= 0
The definition makes sense if you remember that observing b
rules out all those possible worlds where b is false, leaving a
set whose total probability is just P(b). Within that set, the
a-worlds satisfy a ∧ b and constitute a fraction P(a∧b)
P(b) .
Introduction Where Do Probabilities Come From? Basic Probability Notation The Axioms of Probabilities Unconditional and Cond
Product rule
The definition of conditional probability can be rewritten in a
different form called the Product Rule.
P(a ∧ b) = P(a|b)P(b) = P(b|a)P(a)
The product rule comes from the fact that, for a and b to be
true, we need b to be true, and we also need a to be true given
b.
Introduction Where Do Probabilities Come From? Basic Probability Notation The Axioms of Probabilities Unconditional and Cond
Unconditional/Conditional Probabilities summary:
Unconditional/Prior Probabilities
Conditional/Posterior Probabilities
P(a|b) =
P(a∧b)
P(b)
if P(b) 6= 0
P(a ∧ b) = P(a|b)P(b) = P(b|a)P(a) Product Rule
Introduction Where Do Probabilities Come From? Basic Probability Notation The Axioms of Probabilities Unconditional and Cond
Summary
Introduction Where Do Probabilities Come From? Basic Probability Notation The Axioms of Probabilities Unconditional and Cond
1
Introduction
Summarizing Uncertainty
2
Where Do Probabilities Come From?
3
Basic Probability Notation
Basics
4
The Axioms of Probabilities
5
Unconditional and Conditional Probabilities
6
Summary