
352 Notation
Notation Description
F A probability distribution representation
F
E
Empirical probability distribution representation
F
N
Normal probability distribution representation,
parameterised by mean and variance
F
C,m
m-categorical probability distribution representation
F
Q,m
m-quantile probability distribution representation
⇧
F
A projection onto the probability distribution repre-
sentation F
bzc, dze
Floor and ceiling operations, mapping z
2R
to the
nearest integer that is less or equal (floor), or greater
or equal (ceiling), than z
d
A probability metric, typically used for the purposes
of contraction analysis
L
⌧
(✓)
Quantile regression loss function for target thresh-
old ⌧ 2(0, 1) and location estimate ✓ 2R
{u}
An indicator function that takes the value 1 when u
is true and 0 otherwise; also {·}
J(⇡) Objective function for a control problem
G
Greedy policy operator, which produces a policy
which is greedy with respect to a given action-value
function
T
G
, T
G
The Bellman and distributional Bellman optimality
operators derived from greedy selection rule G
J
⇢
(⇡)
A risk-sensitive control objective function, with risk
measure ⇢
A statistical functional or sketch
⇠
⇡
Steady state distribution under policy ⇡
(x)
State representation for state x, a mapping
:
X!
R
n
⇧
,⇠
Projection onto the linear subspace generated by
with state weighting ⇠
M (R)
Space of signed probability measures over the reals
`
⇠,2
Weighted Cramér distance over return-distribution
functions, with state weighting given by ⇠
⇧
,⇠,`
2
Projection onto the linear subspace generated by
,
minimising the `
⇠,2
distance
L Loss function