Geoscience Reference
In-Depth Information
APPENDIX
A
Basic Mathematical Reference
This is a “just enough” quick reference. Please consult standard textbooks for details.
PROBABILITY
The probability of a discrete random variable
A
taking the value
a
is
P(A
=
a)
∈[
0
,
1
]
. This is
sometimes written as
P(a)
when there is no danger of confusion.
Normalization:
a
P(A
=
a)
=
1.
Joint
probability:
P(A
=
a, B
=
b)
=
P(a,b)
,
the
two
events
both
happen
at
the
same
time.
Marginalization:
P(A
=
a)
=
b
P(A
=
a, B
=
b)
.
Conditional probability:
P(a
|
b)
=
P (a, b)/P (b)
, the probability of
a
happening given
b
happened.
The product rule:
P(a,b)
=
P(a)P(b
|
a)
=
P(b)P(a
|
b)
.
P(b
|
a)P(a)
P(b)
Bayes rule:
P(a
|
b)
=
. In general, we can condition on one or more random vari-
P(b
|
a,C)P(a
|
C)
P(b
|
C)
ables
C
:
P(a
|
b, C)
=
D
. In the special case when
θ
is the model parameter and
is the
p(
D
|
θ)p(θ)
p(
D
)
observed data, we have
p(θ
|
D
)
=
, where
p(θ)
is called the prior,
p(
D
|
θ)
the likelihood
function of
θ
(it is
not normalized
:
p(
)
=
p(
D
|
θ)dθ
=
1),
p(
D
D
|
θ)p(θ)dθ
the evidence, and
p(θ
|
D
)
the posterior.
Independence: The product rule can be simplified as
P(a,b)
=
P(a)P(b)
, if and only if
A
and
B
are independent. Equivalently, under this condition
P(a
|
b)
=
P(a)
,
P(b
|
a)
=
P(b)
.
A continuous random variable
x
has a probability density function (pdf )
p(x)
≥
0. Unlike
discrete random variables, it is possible for
p(x) >
1 because it is a probability density, not a
probability mass. The probability mass in interval
is
P(x
1
<X<x
2
)
=
x
2
[
x
1
,x
2
]
p(x) dx
,
x
1
which is between
[
0
,
1
]
.
Normalization:
−∞
p(x) dx
=
1.
