Posterior

Given a Prior and a data set \(\mathcal{N}=\{x_i\,:\,i=1..n\}\), pybalonor comes with the Posterior class that allows to evaluate the posterior predictive distribution, its quantiles, and the posterior distribution of the distribution mean.

The formulae evaluated for the posterior ar given in Scientific Background, and [Z2023] details how these equations are evaluated.

Posterior

class Posterior(sample, prior, n_chebyshev=100)

The posterior distribution of the log-normal distribution parameters given a sample and prior.

Parameters

• sample (array_like) – The sample for which to compute the posterior distribution.

• prior (FlatPrior) – The prior distribution.

• n_chebyshev (int, optional) – The number of Chebyshev points used in the barycentric Lagrange interpolation of the posterior predictive CDF for the posterior predictive quantiles. Defaults to 100.

density(l0, l1)

Posterior probability density.

Parameters

• l0 (array_like) – Vector of parameters \(l_0\) at which to evaluate the posterior distribution.

• l1 (array_like) – Vector of parameters \(l_1\) at which to evaluate the posterior distribution.

Returns

density – NumPy array of the posterior density evaluated at l0 and l1.

Return type

array_like

log_density(l0, l1)

Logarithm of the posterior probability density.

Parameters

• l0 (array_like) – Vector of parameters \(l_0\) at which to evaluate the posterior distribution.

• l1 (array_like) – Vector of parameters \(l_1\) at which to evaluate the posterior distribution.

Returns

density – NumPy array of the logarithm of the posterior density evaluated at l0 and l1.

Return type

array_like

log_mean_pdf(mu)

Logarithm of the posterior density of the distribution mean \(\mu\).

Parameters

mu (array_like) – Vector of the distribution mean \(\mu\) at which to evaluate its posterior distribution.

Returns

log_pdf – NumPy array of the logarithm of the posterior density evaluated at mu.

Return type

array_like

log_predictive_pdf(x)

Logarithm of the posterior predictive density.

Parameters

x (array_like) – Vector of \(x\) at which to evaluate the posterior predictive distribution.

Returns

pdf – NumPy array of the logarithm of the posterior predictive density evaluated at x.

Return type

array_like

mean_pdf(mu)

Posterior density of the distribution mean \(\mu\).

Parameters

mu (array_like) – Vector of the distribution mean \(\mu\) at which to evaluate its posterior distribution.

Returns

pdf – NumPy array of the posterior density evaluated at mu.

Return type

array_like

predictive_ccdf(x)

Posterior predictive complementary distribution function (or survivor function).

Parameters

x (array_like) – Vector of \(x\) at which to evaluate the posterior predictive distribution.

Returns

ccdf – NumPy array of the complementary cumulative posterior predictive distribution evaluated at x.

Return type

array_like

predictive_cdf(x)

Posterior predictive cumulative distribution function.

Parameters

x (array_like) – Vector of \(x\) at which to evaluate the posterior predictive distribution.

Returns

cdf – NumPy array of the cumulative posterior predictive distribution evaluated at x.

Return type

array_like

predictive_pdf(x)

Posterior predictive density.

Parameters

x (array_like) – Vector of \(x\) at which to evaluate the posterior predictive distribution.

Returns

pdf – NumPy array of the posterior predictive density evaluated at x.

Return type

array_like

predictive_quantiles(q)

Quantiles of the posterior predictive distribution.

Parameters

q (array_like) – Vector of quantiles \(q\) of the posterior predictive distribution to compute.

Returns

x – NumPy array of the arguments \(x\) of the posterior predictive distribution corresponding to the quantiles \(q\).

Return type

array_like

predictive_tail_quantiles(q)

Tail quantiles of the posterior predictive distribution (or quantiles of the complementary posterior predictive distribution).

Parameters

q (array_like) – Vector of quantiles \(q\) of the complementary cumulative posterior predictive distribution to compute.

Returns

x – NumPy array of the arguments \(x\) of the posterior predictive distribution corresponding to the quantiles \(1-q\).

Return type

array_like