def inverse_sample_rvs(log_f, a, b, mesh_size=100):
'''
Sample from a continuous univariate density using the inverse transform method.
Args:
log_f : (function) A function which computes the unnormalised of the density.
a : (float) Left side of the support for the denstiy.
b : (float) Right side of the support for the density.
Kwargs:
mesh_size : (int) How many points to use to approximate the integral for computing CDF.
Returns:
x : (float) Sampled value
log_q : (float) The value of the density at x.
'''
u = uniform_rvs(0, 1)
log_u = log(u)
step_size = (b - a) / mesh_size
log_step_size = log(b - a) - log(mesh_size)
knots = [i * step_size + a for i in range(0, mesh_size + 1)]
mid_points = [(x_l + x_r) / 2 for x_l, x_r in zip(knots[:-1], knots[1:])]
log_likelihood = [log_f(x) for x in mid_points]
log_riemann_sum = []
for y in log_likelihood:
log_riemann_sum.append(y + log_step_size)
log_norm_const = log_sum_exp(log_riemann_sum)
log_cdf = None
for x, y in zip(mid_points, log_likelihood):
log_q = y - log_norm_const
log_partial_pdf_riemann_sum = log_q + log_step_size
if log_cdf is None:
log_cdf = log_partial_pdf_riemann_sum
else:
log_cdf = log_sum_exp([log_cdf, log_partial_pdf_riemann_sum])
if log_u < log_cdf:
break
return x, log_q
def bernoulli_rvs(p):
'''
Return a Bernoulli distributed random variable.
Args:
p : (float) Probability of success.
Returns:
x : (int) Binary indicator of success/failure.
'''
u = uniform_rvs(0, 1)
if u <= p:
return 1
else:
return 0
def poisson_rvs(l):
u = uniform_rvs(0, 1)
log_u = log(u)
i = 0
log_l = log(l)
log_p = -l
log_F = log_p
while True:
if log_u < log_F:
return i
log_p += log_l - log(i + 1)
log_F = log_sum_exp([log_F, log_p])
i += 1
def discrete_rvs(p):
'''
Sample a discrete (Categorical) random variable.
Args:
p : (list) Probabilities for each class from 0 to len(p) - 1
Returns:
i : (int) Id of class sampled.
'''
total = 0
u = uniform_rvs(0, 1)
for i, p_i in enumerate(p):
total += p_i
if u < total:
break
return i
def binomial_rvs(n, p):
'''
Sample a binomial distributed random variable.
Args:
n : (int) Number of trials performed.
p : (int) Probability of success for each trial.
Returns:
x : (int) Number of successful trials.
'''
if p > 0.5:
return n - binomial_rvs(n, 1 - p)
if p == 0:
return 0
u = uniform_rvs(0, 1)
log_u = log(u)
log_c = log(p) - log(1 - p)
i = 0
log_prob = n * log(1 - p)
log_F = log_prob
while True:
if log_u < log_F:
return i
log_prob += log_c + log(n - i) - log(i + 1)
log_F = log_sum_exp([log_F, log_prob])
i += 1