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
, a 2 R,
h if x 2 [ a, a]
We have ⇤ ⌘
f(x) dx simplifying to
h dx hence
⇤ = 2ah.
The Fourier Transform (simplifying by omitting a constant
h dx =
2 h sin(a t)
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
ˆf(t)dt exploding to ⇡
2) h Constant: When we ﬁx h the height of the box, but
let ⇤ vary, the area ⇤ its Fourier transform becomes constant
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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.
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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
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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
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
, ˆf(t) = K2 e
, so we
can see the tradoff as the scale of the transformation is 1/ .
• 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?
In our deﬁnition of fragility, we posit that for a function
of a r.v. it is measured by the changes of
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
= u where u 2 [0,
The space of solutions is even more restricted:
= sgn(x) ⇥
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.
x (x) dx
Consider the partial expectation:
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
x (x, ) dx to the scale, using a slight modiﬁcation to
the Leibnitz rule:
@ (x, )