1. Philosophical and Mathematical Meditations on
Probability, A Personal Diary
N N Taleb
MEDITATION ON GENERALIZED UNCERTAINTY
PRINCIPLES (JUNE 2014)
(Amioun, June 14, 2014)
We meditate on integral transforms and the dual of the
function: how we can control a function f but not its integral
transform ˆf, and vice versa, with the aim to find a rigorous
way to discuss natural trade-offs.
The general idea of the uncertainty principle is a compensa-
tion effect of gains of precision in one domain with offsetting
loss in the other one, and vice-versa. Let us ignore the well
known application in physics, and focus on the core idea in
order to generalize to other dimensions.
Consider the simple "box" (real valued) function. With h 2
R+
, a 2 R,
f(x) =
(
h if x 2 [ a, a]
0 elsewhere
(1)
We have ⇤ ⌘
R
x2R
f(x) dx simplifying to
R a
a
h dx hence
⇤ = 2ah.
The Fourier Transform (simplifying by omitting a constant
⇠ ⇡):
ˆf(t) =
Z a
a
eitx
h dx =
2 h sin(a t)
t
(2)
1) ⇤ Constant: When we set ⇤ = 1, with a variable, the
function f becomes a Dirac delta at the limit of a ! 0, with h
going to 1. But its Fourier transform maintains a maximum
height, with the area (integral
R
ˆf(t)dt exploding to ⇡
a .
2) h Constant: When we fix h the height of the box, but
let ⇤ vary, the area ⇤ its Fourier transform becomes constant
equals 2⇡.
-10 -5 5 10
0.1
0.2
0.3
0.4
0.5
-10 -5 5 10
-0.2
0.2
0.4
0.6
0.8
1.0
Figure 1. On the left, f(x) and ˆf(t) on the right, with ⇤ contant, h variable
to get an area ⇤ = 1: when f(x) converges to a Dirac Delta, the Fourier
Transform becomes flatter and flatter, with its integral heading to infinite.
Remark 1. With ⇤ = 1, the "box" function becomes a
Uniform centered at 0. We get similar results with the Beta
distribution or any bounded flat distribution.
-10 -5 0 5 10
0.2
0.4
0.6
0.8
1.0
-10 -5 5 10
-2
2
4
6
8
Figure 2. On the left, f(x) and ˆf(t) on the right, with h constant, ⇤ variable,
but the Fourier Transform converges to (sort of) a Dirac Delta function, or a
stick.
-1.0 -0.5 0.5 1.0
5
10
15
20
25
-10 -5 0 5 10
0.2
0.4
0.6
0.8
1.0
Figure 3. On the left, f(x) and ˆf(t) on the right. We show the limit where f
becomes a Dirac Delta stick, the Fourier Transform becomes flat with infinite
integral.
Some Generalizations to Real Life
What we saw was a sketch of situations where functions
and their Fourier transforms were Fourier transform of each
other. The Gaussian is (again, sort of) the Fourier Transform
of itself. When f(x) = K1
1
e
x2
2
, ˆf(t) = K2 e
2
t2
, so we
can see the tradoff as the scale of the transformation is 1/ .
Remarks
• Provided we can figure out a functional form for an
exposure, the question becomes: Can you gain depth at
the expense of breadth, and vice versa? Or, under which
conditions is there a necessary trade-off?
• Note the problem with the Gaussian is that it is the trans-
form of itself. But it is the case with power laws? These
have transforms with a power-law exponent... I can’t see
the implications clearly. For now, for a Pareto distribution
bounded on the left at 1, ˆf(t) = ↵( it)↵
( ↵, it).
ROBUSTNESS (JUNE 2014)
Thomas S. mentioned that "robustness" is the ability to "fare
well" under different probability distributions.
Now how about the different probability distributions?
1

2. In our definition of fragility, we posit that for a function
of a r.v. it is measured by the changes of
R
x<K
z(x) dFs(x)
under perturbation of the scale s of the distribution F, even
more precisely s the left side (or lower) semi-deviation.
We can show by counterexample that it is not possible to
make the universal claim that the expectation of a function
of a variable under a given probability distributions can be
attained by perturbating the scale of another distribution.
In the case of the Gaussian we cannot get all densities by
perturbating of (x) ⌘ e
x2
2 2
= u where u 2 [0,
p
2⇡) .
The space of solutions is even more restricted:
⇤
= sgn(x) ⇥
ix
p
W ( x2u2)
, x 6= 0
But the good news is that fragility is always measurable.
Next we show how there is always a response on the real line,
hence possibility of measuring fragility.
g =
1
p
2⇡
Z 1
K
x (x) dx
Consider the partial expectation:
@g
@
=
e
K2
2 2
K2
+ 2
p
2⇡ 2
6= 0
For all nondegenerate distributions ( > 0).
More general, where is a general density, the same
can be obtained... Which can be proved via measures twice
differentiated. Where is a density, the sensitivity of ( ) ⌘R 1
K
x (x, ) dx to the scale, using a slight modification to
the Leibnitz rule:
d
d
=
Z 1
K
x
@ (x, )
@
dx
2