Asymptotic Developments

from math import log, pi
import poisson_approval as pa

Asymptotic developments are mostly used internally by Poisson Approval in order to compute pivot events and best responses. However, they can also be used directly.

Define an asymptotic developement (manually):

asymptotic = pa.Asymptotic(mu=-1 / 9, nu=-1 / 2, xi=-1 / 2 * log(8 * pi / 9))
asymptotic

exp(- 0.111111 n - 0.5 log n - 0.513473 + o(1))

Access its coefficients:

asymptotic.mu

-0.1111111111111111

asymptotic.nu

-0.5

asymptotic.xi

-0.5134734250965083

Multiply or add two asymptotic developments:

asymptotic2 = pa.Asymptotic(mu=-1 / 2, nu=-1, xi=-1)
asymptotic2

exp(- 0.5 n - log n - 1 + o(1))

asymptotic * asymptotic2

exp(- 0.611111 n - 1.5 log n - 1.51347 + o(1))

asymptotic + asymptotic2

exp(- 0.111111 n - 0.5 log n - 0.513473 + o(1))

We now consider two random variables \(X_1 \sim \text{Poisson}(\tau_1 n)\) and \(X_2 \sim \text{Poisson}(\tau_2 n)\).

tau_1, tau_2 = 1 / 9, 8 / 9
print('P(X_1 = 0) = ', pa.Asymptotic.poisson_value(tau_1, 0))
print('P(X_1 = X_2)  =', pa.Asymptotic.poisson_eq(tau_1, tau_2))
print('P(X_1 = X_2 + 1)  =', pa.Asymptotic.poisson_one_more(tau_1, tau_2))
print('P(X_1 >= X_2) =', pa.Asymptotic.poisson_ge(tau_1, tau_2))
print('P(X_1 > X_2)  =', pa.Asymptotic.poisson_gt(tau_1, tau_2))
print('P(X_1 > X_2 + 1)  =', pa.Asymptotic.poisson_gt_one_more(tau_1, tau_2))

P(X_1 = 0) =  exp(- 0.111111 n + o(1))
P(X_1 = X_2)  = exp(- 0.371461 n - 0.5 log n - 0.68676 + o(1))
P(X_1 = X_2 + 1)  = exp(- 0.371461 n - 0.5 log n - 1.72648 + o(1))
P(X_1 >= X_2) = exp(- 0.371461 n - 0.5 log n - 0.250496 + o(1))
P(X_1 > X_2)  = exp(- 0.371461 n - 0.5 log n - 1.29022 + o(1))
P(X_1 > X_2 + 1)  = exp(- 0.371461 n - 0.5 log n - 2.32994 + o(1))

Like profiles and tau-vectors, asymptotic developments accept the parameter symbolic=True:

from fractions import Fraction
tau_1 = Fraction(1, 9)
tau_2 = Fraction(8, 9)
print('P(X_1 = 0) = ', pa.Asymptotic.poisson_value(tau_1, 0, symbolic=True))
print('P(X_1 = X_2)  =', pa.Asymptotic.poisson_eq(tau_1, tau_2, symbolic=True))
print('P(X_1 = X_2 + 1)  =', pa.Asymptotic.poisson_one_more(tau_1, tau_2, symbolic=True))
print('P(X_1 >= X_2) =', pa.Asymptotic.poisson_ge(tau_1, tau_2, symbolic=True))
print('P(X_1 > X_2)  =', pa.Asymptotic.poisson_gt(tau_1, tau_2, symbolic=True))
print('P(X_1 > X_2 + 1)  =', pa.Asymptotic.poisson_gt_one_more(tau_1, tau_2, symbolic=True))

P(X_1 = 0) =  exp(- n/9 + o(1))
P(X_1 = X_2)  = exp(n*(-1 + 4*sqrt(2)/9) - log(n)/2 - 7*log(2)/4 - log(pi)/2 + log(3) + o(1))
P(X_1 = X_2 + 1)  = exp(n*(-1 + 4*sqrt(2)/9) - log(n)/2 - 13*log(2)/4 - log(pi)/2 + log(3) + o(1))
P(X_1 >= X_2) = exp(n*(-1 + 4*sqrt(2)/9) - log(n)/2 + log(3*2**(1/4)/(4*sqrt(pi)*(1 - sqrt(2)/4))) + o(1))
P(X_1 > X_2)  = exp(n*(-1 + 4*sqrt(2)/9) - log(n)/2 + log(3*2**(3/4)/(16*sqrt(pi)*(1 - sqrt(2)/4))) + o(1))
P(X_1 > X_2 + 1)  = exp(n*(-1 + 4*sqrt(2)/9) - log(n)/2 + log(3*2**(1/4)/(32*sqrt(pi)*(1 - sqrt(2)/4))) + o(1))