Quickstart

Suppose you had the following data set for which you wanted to perform a Bayesian analysis of the log-normal distribution:

import numpy as np
# Log-normally-distributed random values with l0=1.0, l1=0.5:
N = np.array([1.10556883, 5.07527308, 2.60603815, 3.76415609,
              4.14403281, 3.82246619, 3.62519342, 1.5404973 ,
              2.29341138, 2.69632835, 1.47836845, 2.47652787,
              4.51722975])

The data is synthetically generated from a log-normal distribution with \(l_0=1\) and \(l_1=\frac{1}{2}\). Its histogram looks as follows:

Now define a FlatPrior instance within the ranges \(-10\leq l_0 \leq 2\) and \(0 \leq l_1 \leq 3\):

from pybalonor import Posterior, FlatPrior
L0_MIN = -10.0
L0_MAX = 2.0
L1_MIN = 0.0
L1_MAX = 3.0
prior = FlatPrior(L0_MIN, L0_MAX, L1_MIN, L1_MAX)

With this prior and the previously defined sample, the Posterior can be computed:

posterior = Posterior(N, prior)

Posterior density

The posterior distribution in \(l_0\) and \(l_1\) is computed using the posterior’s density() method:

l0 = np.linspace(L0_MIN, L0_MAX, 500)
l1 = np.linspace(L1_MIN, L1_MAX, 201)
p = posterior.density(*np.meshgrid(l0, l1))

The log_density() method can resolve values far off the peak:

lp = posterior.log_density(*np.meshgrid(l0, l1))

Posterior predictive

Similarly, the posterior predictive distribution in \(x\) can be computed using either of the log_predictive_pdf(), predictive_pdf(), predictive_cdf(), predictive_ccdf(), predictive_quantiles(), or predictive_tail_quantiles() methods:

x = np.linspace(0.0, 15.0, 100)
y = posterior.predictive_cdf(x)

Posterior of the mean

The posterior of the distribution’s mean \(\mu\) can be evaluated using either of log_mean_pdf() or mean_pdf()

mu = np.geomspace(mu_min, mu_max, 10000)
posterior = Posterior(N, prior)
y = posterior.mean_pdf(mu)