# Crash Course¶

dynesty requires three basic ingredients to sample from a given distribution:

• the likelihood (via a loglikelihood() function),
• the prior (via a prior_transform() function that transforms samples from the unit cube to the target prior), and
• the dimensionality of the parameter space.

As an example, let’s define our likelihood to be a 3-D correlated multivariate Normal (Gaussian) distribution and our prior to be uniform in each dimension from [-10, 10):

import numpy as np

# Define the dimensionality of our problem.
ndim = 3

# Define our 3-D correlated multivariate normal likelihood.
C = np.identity(ndim)  # set covariance to identity matrix
C[C==0] = 0.95  # set off-diagonal terms
Cinv = np.linalg.inv(C)  # define the inverse (i.e. the precision matrix)
lnorm = -0.5 * (np.log(2 * np.pi) * ndim +
np.log(np.linalg.det(C)))  # ln(normalization)

def loglike(x):
"""The log-likelihood function."""

return -0.5 * np.dot(x, np.dot(Cinv, x)) + lnorm

# Define our uniform prior.
def ptform(u):
"""Transforms samples u drawn from the unit cube to samples to those
from our uniform prior within [-10., 10.) for each variable."""

return 10. * (2. * u - 1.)


Estimating the evidence and posterior is as simple as:

import dynesty

# "Static" nested sampling.
sampler = dynesty.NestedSampler(loglike, ptform, ndim)
sampler.run_nested()
sresults = sampler.results

# "Dynamic" nested sampling.
dsampler = dynesty.DynamicNestedSampler(loglike, ptform, ndim)
dsampler.run_nested()
dresults = dsampler.results


Combining the results from multiple (independent) runs is easy:

from dynesty import utils as dyfunc

# Combine results from "Static" and "Dynamic" runs.
results = dyfunc.merge_runs([sresults, dresults])


We can visualize our results using several of the built-in plotting utilities. For instance:

from dynesty import plotting as dyplot

# Plot a summary of the run.
rfig, raxes = dyplot.runplot(results)

# Plot traces and 1-D marginalized posteriors.
tfig, taxes = dyplot.traceplot(results)

# Plot the 2-D marginalized posteriors.
cfig, caxes = dyplot.cornerplot(results)


We can post-process these results using some built-in utilities. For instance:

from dynesty import utils as dyfunc

# Extract sampling results.
samples = results.samples  # samples
weights = np.exp(results.logwt - results.logz[-1])  # normalized weights

# Compute 10%-90% quantiles.
quantiles = [dyfunc.quantile(samps, [0.1, 0.9], weights=weights)
for samps in samples.T]

# Compute weighted mean and covariance.
mean, cov = dyfunc.mean_and_cov(samples, weights)

# Resample weighted samples.
samples_equal = dyfunc.resample_equal(samples, weights)

# Generate a new set of results with statistical+sampling uncertainties.
results_sim = dyfunc.simulate_run(results)