Download Pr(Amount,Time)

Working with Probability Tables
Recall Discrete Probability
• Sample space is made up if discrete outcomes
• Discrete: Nominal (categories)
Experiment: Did he do it?
• Discrete: Ordinal (orderable somehow)
Experiment: How many arsons will there be in
my neighbourhood this year?
Recall Discrete Probability
• Each outcome is associated with a probability:
Table:
G
I
Pr(G) =57% Pr(I) =43%
Pr(TO)
TO: G
57
I
43
Recall Discrete Probability
• Each outcome is associated with a probability:
•
Time:
We can have one or more discrete random
variates too:
A 3D histogram
Amount:
low
morning
28
noon
3
night
0
medium
3
18
13
high
1
10
24
Pr(Time,Amount)
Recall Discrete Probability
• Each outcome is associated with a probability:
•
Here is what a table with 3 random variates could
look like:
A:
B:
C: c1
c2
c3
c4
a1
b1
5.2
7.5
2.4
0.9
a1
b2
0.3
2.9
8.0
0.2
a1
b3
6.0
1.1
8.7
5.7
a2
b1
3.2
1.2
1.3
6.9
a2
b2
2.1
0.4
4.7
7.3
a2
b3
5.1
5.9
8.0
5.0
Pr(A,B,C)
•
We can’t really make a visual histogram anymore
though…
Recall Discrete Probability
• Each outcome is associated with a probability:
•
Here is what a table with 3 random variates could
look like (or):
This is how R likes a multidimensional table
C = c1
C = c2
B
A
b1
5.2
3.2
a1
a2
C = c3
b2
0.3
2.1
b3
6.0
5.1
a1
a2
A
b1
2.4
1.3
b2
8.0
4.7
b3
8.7
8.0
b1
7.5
1.2
a1
a2
C = c4
B
A
B
b2
2.9
0.4
B
A
a1
a2
b1
0.9
6.9
b2
0.2
7.3
Pr(A,B,C)
•
b3
1.1
5.9
We can’t really make a visual histogram anymore
though…
b3
5.7
5.0
Recall Discrete Probability
• Each outcome is associated with a probability:
•
Here is what a table with 3 random variates could
look like (or):
•
We can’t really make a
visual histogram anymore
though…
B
b1
b1
b1
b1
b2
b2
b2
b2
b3
b3
b3
b3
b1
b1
b1
b1
b2
b2
b2
b2
b3
b3
b3
b3
C
c1
c2
c3
c4
c1
c2
c3
c4
c1
c2
c3
c4
c1
c2
c3
c4
c1
c2
c3
c4
c1
c2
c3
c4
Pr(A,B,C)
5.2
7.5
2.4
0.9
0.3
2.9
8.0
0.2
6.0
1.1
8.7
5.7
3.2
1.2
1.3
6.9
2.1
0.4
4.7
7.3
5.1
5.9
8.0
5.0
Pr(A,B,C)
I find this format most general,
easiest to read and least ambiguous
A
a1
a1
a1
a1
a1
a1
a1
a1
a1
a1
a1
a1
a2
a3
a4
a5
a6
a7
a8
a9
a10
a11
a12
a13
Recall Discrete Probability
• Each outcome is associated with a probability:
•
Conditional probability can be represented as
tables too:
Amount
:
low
medium
high
Time:
morning
noon
night
87.0
8.7
4.3
9.1
59.1
31.8
0.8
34.4
64.9
Pr(Amount|Time) = Pr(Time,Amount)/Pr(Time)
Representation of Probability Tables in R
•
A 1D “prior” table (one RV):
Pr(A
)
A: yes 0.37
no 0.63
library(gRbase)
We’ll use gRbase and
gRain packages a lot when
working with Bayesian
networks
parray
R code:
A <- parray("A",
levels = list(c("yes","no")),
values =
c( 0.37,0.63))
output:
Representation of Probability Tables in R
•
A 2D table, either a joint PMF or conditional PMF:
Pr(Amount,Time)
Pr(Amount,Time)
Amount:
Time:
morning
0.28
noon
0.03
night
0
medium
0.03
0.18
0.13
high
0.01
0.1
0.24
low
Column variable
A.and.T <- parray(“Amount”,”Time”,
Row variable
list(
c("low","medium","high"),
# Amount levels
c("morning","noon","night") # Time levels
),
values=rbind(
You can enter any values
c(28, 3, 0),
and choose to normalize
c( 3, 18, 13),
them.
c( 1, 10, 24)
),
normalize = "all"
)
Representation of Probability Tables in R
•
A 2D table, either a joint PMF or conditional PMF:
Pr(Amount,Time)
Pr(Amount,Time)
Amount:
output:
Time:
morning
0.28
noon
0.03
night
0
medium
0.03
0.18
0.13
high
0.01
0.1
0.24
low
Representation of Probability Tables in R
•
A 2D table, either a joint PMF or conditional PMF:
Pr(Amount|Time)
Pr(Amount | Time)
Amount:
Time:
morning
0.87
noon
0.091
night
0.008
medium
0.09
0.591
0.344
high
0.04
0.319
0.649
low
A.given.T <- parray(c("Amount", "Time"),
list(
c("low","medium","high"),
# Amount levels
c("morning","noon","night") # Time levels
),
values=rbind( c(0.87, 0.091, 0.008),
c(0.09, 0.591, 0.344),
c(0.04, 0.319, 0.649) )
)
•
NOTE: You the user has to remember if it is joint or
conditional
Representation of Probability Tables in R
•
A 2D table, either a joint PMF or conditional PMF:
Pr(Amount|Time)
Pr(Amount | Time)
Amount:
Time:
morning
0.87
noon
0.091
night
0.008
medium
0.09
0.591
0.344
high
0.04
0.319
0.649
low
output:
•
NOTE: You the user has to remember if it is joint or
conditional
Representation of Probability Tables in R
•
Higher nD (n = 3 or more) tables are arrays of 2D
tables.
•
These are really hard to enter by had so let’s just look at one
for now:
Pr(Transfer | Artist, Location)
A 3D table
Representation of Probability Tables in R
•
Higher nD (n = 3 or more) tables are arrays of 2D
tables.
•
Same table as pervious slide, just in an easier to read format:
Pr(Transfer | Artist, Location)
Working with Probability Tables
•
So it’s clear that we can represent lots of discrete
probability distributions (they are probability mass
functions, PMFs actually) as tables: f
•
How can we “combine” them while respecting the laws
of probability?
•
gRbase library makes it easy!
•
Add tables = tableAdd()
•
Multiply tables = tableMult()
•
Divide tables = tableDiv()
•
Marginalize around selected RV = tableMargin()
Example with Probability Tables
Consider two variables, A and B, that can co-occur. A has two
states and B has three states. The probabilities of A and B can
be listed in a table:
Pr(A,B) b1
b2
b3
a1
0.14 0.12 0.12
a2
0.24 0.26 0.08
Compute:
Compute:
Compute:
Compute:
Example with Probability Tables
Compute:
(margin is B)
Compute:
(divide)
Compute:
(marginalize)
Compute:
(multiply and
divide)
# Input: Pr(A,B)
A.and.B <- parray(c("A", "B"),
levels=c(2,3),
values=rbind(c(0.14, 0.16, 0.12),
c(0.24, 0.26, 0.08)
)
)
A.and.B
# Output
sum(A.and.B)
# Check
B <- tableMargin(A.and.B, margin = "B")
B
sum(B)
# Pr(B)
# Output
# Check
A.given.B <- tableDiv(A.and.B, B)
t(A.given.B)
colSums(t(A.given.B))
# Pr(A|B)
# Output
# Check
A <- tableMargin(A.and.B, margin = "A")
A
sum(A)
# Pr(A)
# Output
# Check
B.given.A <- tableDiv(tableMult(A.given.B, B), A) # Pr(B|A)
t(B.given.A)
# Output
colSums(t(B.given.A))
# Check
The Law of Total Probability
•
Suppose a sample space can be partitioned into a set of
disjoint events Bi such that
B
B
4
1
B3
A
B2
The Law of Total Probability
•
Suppose a sample space can be partitioned into a set of
disjoint events Bi such that
•
The probability of an arbitrary event A in Ω can be written as:
Law of total probability
Example: A medical test
• Professor Shenkin LOVES hamburgers. But he’s also a
hypochondriac. He thinks he is infected with “Mad Cow Disease”
(MCD), so he gets himself tested (T).
• The true positive rate of the test is: Pr(T+ | MCD+) = 0.7
• The false positive rate of the test is: Pr(T+ | MCD-) = 0.1
• The background prevalence of MCD in the yummy cow population is:
Pr(MCD+) = 0.02
What is the most likely joint outcome?
What is the probability that Prof. Shenkin tests positive for MCD, Pr(T+)?
Suppose Professor Shenkin is positive for MCD. What is the probability
that he truly has MCD, Pr(MCD+| T+)?
Use probability table arithmetic in R/gRbase
library(gRbase)
# Pr(MCD) table:
MCD <- parray("MCD",
levels = list(c("MCD+","MCD-")),
values =
c(0.02,0.98))
MCD
sum(MCD)
# Pr(T | MCD) table:
T.given.MCD <- parray(c("T", "MCD"),
list(
c("T+","T-"),
c("MCD+","MCD-")
),
values=rbind( c(0.7, 0.1),
c(0.3, 0.9)
)
)
T.given.MCD
colSums(T.given.MCD)
# Pr(T,MCD) = Pr(T | MCD) Pr(MCD)
T.and.MCD <- tableMult(T.given.MCD,MCD)
T.and.MCD
sum(T.and.MCD)
# Pr(T)
TT <- tableMargin(T.and.MCD, margin = "T")
TT
sum(TT)
# Pr(MCD | T) = Pr(T | MCD) Pr(MCD)/Pr(T)
MCD.givenT <- tableDiv(tableMult(T.given.MCD, MCD), TT)
t(MCD.givenT)
colSums(t(MCD.givenT))