18.1 the exponential tilt maximizes E[s] - beta KL(pi||pi0) over distributions on a finite set
  objective at the tilt -0.680038 = beta log Z = -0.680038; best of 20,000 random distributions -0.739333

18.2 Gaussian tilts attain gain = sigma sqrt(2 KL) exactly: tilt of N(0, sigma^2) by lambda is N(lambda sigma^2, sigma^2)
  sigma=1.0 lambda=1.0: KL=0.5000, gain=1.0000, sigma*sqrt(2KL)=1.0000
  sigma=2.0 lambda=0.5: KL=0.5000, gain=2.0000, sigma*sqrt(2KL)=2.0000
  sigma=0.5 lambda=3.0: KL=1.1250, gain=0.7500, sigma*sqrt(2KL)=0.7500

18.3 true reward along the exploit family; location of the maximum by derivative sign change
  maximum 0.4813 at beta=1.481, exploit mass there 0.0415, KL 0.388
  derivative changes sign near beta = 1.481

18.4 asymptotics of E[r | s = t] for heavy-tailed errors: Cauchy ~ 2/t, Student-3 ~ 4/t
  t=  30  cauchy 0.06682 (2/t = 0.06667)   student-3 0.13363 (4/t = 0.13333)
  t= 100  cauchy 0.02000 (2/t = 0.02000)   student-3 0.04000 (4/t = 0.04000)
  t= 300  cauchy 0.00667 (2/t = 0.00667)   student-3 0.01331 (4/t = 0.01333)

18.5 standard error of the length-bias estimate, sigma_e / (sqrt(n) sd(l)), versus the simulated spread
  n=   20: predicted 0.0143  (Listing 18.3 measured 0.0164); sd(l) = 7.80
  n=   50: predicted 0.0091  (Listing 18.3 measured 0.0092); sd(l) = 7.80
  n=  200: predicted 0.0045  (Listing 18.3 measured 0.0047); sd(l) = 7.80
  n= 1000: predicted 0.0020  (Listing 18.3 measured 0.0021); sd(l) = 7.80

18.6 false-positive rate on 150 checks of pure noise (audit means with SE 0.1, no change in the truth), 20,000 repetitions
  'drop of 3 SE below the running maximum': 1.000   'fixed-lag z > 3.5': 0.0304

18.7 entropy of the tilt: dH/dlambda = -Cov_pi(log pi0, s) - lambda Var_pi(s), lambda = 1/beta
  lambda=0.0: finite difference -0.272448, formula -0.272448
  lambda=0.5: finite difference -0.345790, formula -0.345790
  lambda=2.0: finite difference -0.247991, formula -0.247991
  rare high-reward output, entropy at lambda = 0,1,2,3,4,6: [2.2313 2.2617 2.2968 2.2804 2.067  0.9292]
