Probability meditations


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Probability meditations

  1. 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. 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