 Nomenclature & Abbreviations
General Mathematical Symbols
R Real domain (1, 1)
R
+
Real positive domain (0, 1)
Z Real int ege r domai n ·· , 1, 0, 1, 2, ···}
(0
,
1)
Continuous close interval between 0 and 1,
which incl u de s 0 and 1
(0
,
1]
Continuous op e n i nterval between 0 and 1,
which incl u de s 0 and not 1
lim
n!1
Alimitwhenn tends to inﬁnity
8 For all
:
Such that
ˇx The true value for x
ˆx An approxim at ion of x
P
Sum operation
¬ The negation symbol
Q
Product operation
R
dx Integral operat i on w it h re spe ct to x
dv( x)
dx
⌘r
x
v Derivative or gradient of v(x)withrespecttox
@v(x,y,z)
@x
Partial de r ivative of v(x, y, z)withrespecttox
|x| Absolute values of x
Approx im at el y equ al
/ Proportional to
Equivalent
ln
(
x
)
log
e
(
x
)
Natural logarithm of x
ln(exp(x)) = x
exp
(
x
)
e
x
Exponential function of x,=2.71828
x
,
exp(ln(x)) = x
x An inﬁnitesimal interval for x
A , BAimplies B and B implies A j.-a. goulet xxiv
Linear Algebra
x A scalar variable
x A column vec t or, x =[x
1
x
2
··· x
X
]
|
X A matrix
x
i
[
x
]
i
i
th
element of a vector
x
ij
[
X
]
ij
{i, j}
th
element of a matrix
X
=
diag
(
x
) Square matrix
X
where the terms on the main diagonal are the
element s of x and 0 els ewh er e
x
=
diag
(
X
)
Vector x consisting in th e main diagon al terms of a mat r i x X
I
The identity matrix, i.e., a square matri x with 1 on the main
diagonal and 0 elsewhere
blkdiag
(
A, B
) Block diagonal matrix where matrices
A
and
B
are concate-
nated on the main diagonal of a single matrix
|
Transposition operator : [X]
ij
=[X
|
]
ji
· Scalar product
Matrix multiplication
||x||
p
L
p
-norm of a vector x
det
(
A
)
|A| Determinant of a Matrix A
tr
(
A
)
Sum of the elements on the main diagonal of A
! A transformation from a space to another
J
y,x
The Jacobian matrix so that [J
y,x
]
k,l
=
@y
k
@x
l
@g(x)
@x
i
Partial de r ivative of g(x)withrespecttothei
th
variable x
i
rg
(
x
)
h
@g(x)
@x
1
···
@g(x)
@x
n
i
H
The Hessian matrix containing second-order partial derivatives,
[H[g(x)]]
ij
=
@
2
g (x )
@x
i
@x
j
Probability Theory and Random Variables
A
=
{E
1
,E
2
, ···} Asetisdescribedbyacalligraphic letter
#
A Number of eleme nts i n a set A
S Univer se/s amp l in g sp ace, i . e. , th e en se mble of possi b le r e su lt s
x An elementary event, x 2S
x 2S x belongs to the sampling space S
E An ensemble of eleme ntary events
E S E is a subset of S
E S E is a subset or is equal to S
E
=
S A certain event
E
=
; An impossible event
E The compleme nt of the e vent E
Pr
(
·
)
The probability of an event (2 (0, 1))
probabilistic machine learning for civil engineers xxv
[ Union operation for events, i.e., “or”
\ Inters ec t ion operation for events, i.e., “and”
X A random vari abl e
X A vector of random variables
f
(
x
)
f
X
(
x
)
Probability density function of a random variable X
X f
(
x
)
X
is dis t ri b ut e d as described by its marginal probability density
function f(x)
X f
(
x
)
X
is distri b ut e d as described by its joint probability density
function f(x)
d
! Converges i n dis t ri b ut i on
x, x
i
Realization of a random vari ab l e x : X f(x)
F
(
x
)
F
X
(
x
)
Cumulat i ve distri b ut i on (or mass) function of a random variable
X
(
x
) Cumulative d i st r ib u ti on function of a standard Normal random
variable with mean eq ual to 0 and variance equal to 1
p
(
x
)
p
X
(
x
)
Probability mass function of a random variable X
X ?? Y The random variables X and Y are statistic all y independent
X ?? Y |z
The random variables
X
and
Y
are conditi onal l y independent
given z
X|y The random variable X is conditi on al ly dependent on y
E
[
X
]
Expectation operation for a random variable X
var
[
X
]
Variance operation for a random variable X
cov
(
X, Y
)
Covariance operation for a pair of random variables X, Y
X
Coecient of variation of a random variable X
µ
X
The mean of a random variable X
2
X
The variance of a random variable X
The correlation coecient
µ
X
The mean values for a vector of rand om variables X
X
A covariance matrix for a vector of random variables X
R
X
A correlation matrix for a vector of random variables X
D
X
A standard deviation matrix for a vector of random variables X
N
(
x
;
µ,
2
)
The prob abi l i ty density function of a univariate Normal random
variable X, parameterized by its me an and variance
N
(
x
;
µ
X
,
X
) The j oint probability density function of a multivariate Nor-
mal random variable
X
, parameteri z ed by its mean vect or and
covariance matrix
ln N
(
x
;
,
) The p rob ab il i ty density function of a log-normal random vari-
able
X
, parameteri z ed by its mean and standard deviation
deﬁned in the log space
B
(
x
;
,
) The p rob ab il i ty density function of a Beta random variable
X
,
parameterized by and
U
(
x
;0
,
1) The u ni f orm probability density function for a r an dom variable
X deﬁned for the interval (0, 1)
(
x
)
The dirac-delta function j.-a. goulet xxvi
f
(
y|x
)
L
(
x|y
) The l i kelihoo d , i.e., the prior probability density of observin g
Y = y given x
Optimization
˜
f
(
)
Target function we want maximize
˜
f
0
(
)=
d
˜
f()
d
First derivative of a function
˜
f
00
(
)=
d
2
˜
f()
d
2
Second derivat i ve of a func ti on
arg max
˜
f
(
)
The values of that maximize
˜
f()
max
˜
f
(
)
The maximal value of
˜
f()
An optimal value
d Search d ire c t i on
Scale factor for the search direction
I
(
i
) An in di c at or vector for which all values are equal to 0, except
the i
th
, which is equ al to on e
Sampling
Vector of parameters
q
(
0
|
) Proposal distribution describing the probability of moving to
0
from
˜
f
(
)
Target distributi on fr om which we want to draw samples
Acceptance probability
Acceptance rate
ˆ
R Estimated potential scale reduction
Utility & Decisions
A
=
{
a
1
, ··· ,
a
A
} A set of possible actions
x 2 X Z An outcome from a set of discrete states
L A lottery
U
(
a, x
)
Utility given a state x and an action a
U
(
a
)
E
[
U
(
a, X
)]
Expected utility conditional on an action a
U
(
s
)
U
(
s,
)
E
[
U
(
s,
)]
Long-term expect ed utility conditional on a current stat e
s
and
that a policy is followed
(
s
)
A policy deﬁning an action a to take, conditional on a state s
r
(
s, a, s
0
) The r eward for being in state
s
, taking the action
a
and ending
in state s
0
Q
(
s, a
)
Action-utility function
Discount fact or
The learning rate
probabilistic machine learning for civil engineers xxvii
The probabil i ty that an agent takes a random action rather
than following the policy (s)
An assignment in a recurrent equation, e.g., x 2x
Abbreviations
AI
Artiﬁcial intelligence
BN
Bayesian n etwork
CDF
Cumulat i ve d is t ri b ut i on fu nc t ion
CLT
Central limit theorem
CMF
Cumulat i ve mas s fun ct i on
CPT
Conditional probability table
CV
Cross-validati on
DAG
Directed acyclic graph
DBN
Dynamic Bayesi an network
e.g.
For example
EM
Expectation maximization
EPSR
Estimated potential scale reduction
FCNN
Fully connected feed for ward neural network
GA
GMM
Gaussian mixture model
GP
Gaussian process
GPC
Gaussian process classiﬁcation
GPR
Gaussian process regression
GPU
Graphical processing unit
HMM
Hidden Markov model
i.e.
That is
iid
Independent identically distributed
KF
Kalman ﬁlter
KS
Kalman smoother
LDA
Linear discriminant analysis
LLOCV
Leave-one-out cross-validation
MAP
Maximu m a-poste ri or i
MCMC
Markov chain M onte Carlo
MDP
Markov de ci si on pr ocess
ML
Machine learning
MLE
Maximu m likelihood estimate
NB
Na¨ıve Bayes
NN
Neural network
NR
Newton-Raphson
PCA
Principal component analysis
PDF
Probability density function
j.-a. goulet xxviii
PMF
Probability mass function
POMDP
Partial l y obse r vable Markov decision process
PSD
Positi ve s emi - de ﬁn i te
QDA
RL
Reinforcement learning
R.V.
Random variable
SKF
Switch i ng Kal man ﬁ l te r
SSM
State-space model
sym.
Symmetric
TD
Temporal dierence
VOI
Value of information
VPI
Value of perfect information 