Guides / Prepare your inputs / Likelihood functions and distributions

Likelihood functions and distributions¶

Use this page if you can evaluate each object’s likelihood \(\ell_i(z)=p(d_i\mid z)\) at any true value \(z\). Examples are a Student-\(t\) error, a skewed error from a fit, or a SciPy distribution per object. from_density() averages each function over the bins.

Say what the function is¶

A function can be a likelihood or a posterior, and binference cannot tell them apart. Pass kind="likelihood" for a likelihood. For a posterior, pass the prior it was computed under as interim= (Posteriors under an interim prior).

A vectorised function¶

The function takes the points, shape (M,) in 1-D or (M, ndim) on a grid. It returns an (N, M) array: one row per object. With log=True it returns logarithms.

import numpy as np
from scipy import stats

import binference

rng = np.random.default_rng(1)
z_obs = rng.uniform(0.2, 2.8, size=300)
sigma = rng.uniform(0.05, 0.3, size=300)
bins = binference.Bins.linear(0.0, 3.0, 30)


def loglike(z):
    z = np.asarray(z).reshape(-1)
    return stats.t.logpdf(z[None, :], df=3, loc=z_obs[:, None], scale=sigma[:, None])


lk_t = binference.Likelihoods.from_density(loglike, bins, kind="likelihood", log=True)
print(lk_t.N, lk_t.K)

Define the function at module level, not as a lambda. A lambda cannot be pickled, so saved or copied likelihoods lose their shape within bins.

A list of distributions¶

Pass one frozen SciPy distribution per object, or any object with logpdf, pdf, logcdf or cdf:

dists = [stats.norm(m, s) for m, s in zip(z_obs[:10], sigma[:10])]
lk_d = binference.Likelihoods.from_density(dists, bins, kind="likelihood")
print(lk_d.N)

How the bin averages are computed¶

Likelihoods are averaged over each bin, uniformly in the bin’s measure coordinate. That is the shape the model gives the population within a bin.
Posteriors are integrated over each bin in \(z\), then divided by the interim prior’s mass in the bin.
On 1-D bins, a distribution with a CDF is integrated exactly where that gives the answer: for posteriors, and for likelihoods on linear bins. The integral is taken in log space, so far tails keep their precision.
Everything else uses adaptive quadrature. Each bin is halved along every axis until the error estimate is below tol, relative to the object’s largest bin average. At most max_depth halvings are made.

If some bins do not reach tol, binference raises an InputWarning (warnings). If that happens, raise max_depth or tol, or pass n_sub= for a fixed number of points per bin and axis.