Bayesian workflow

This tutorial highlights some ideas from https://arxiv.org/abs/1709.01449, including:

  • Making a flip book of the prior predictions

  • posterior predictive checks

Model

We again consider the sine model with gaussian measurement errors.

\[y = A_1 \sin\left(2 \pi \left(\frac{t}{P_1} + t_1\right)\right) + B + \epsilon\]

where \(\epsilon \sim \mathrm{Normal}(0, \sigma)\)

[1]:
import numpy as np
from numpy import pi, sin

def sine_model1(t, B, A1, P1, t1):
    return A1 * sin((t / P1 + t1) * 2 * pi) + B

The model has four unknown parameters per component:

  • the signal offset \(B\)

  • the amplitude \(A\)

  • the period \(P\)

  • the time offset \(t_0\)

Generating data

Lets generate some data following this model:

[2]:
np.random.seed(42)

n_data = 50

# time of observations
t = np.random.uniform(0, 5, size=n_data)
# measurement values
yerr = 1.0
y = np.random.normal(sine_model1(t, B=1.0, A1=0.9, P1=3, t1=0), yerr)

y[0] += 3  # add outlier

Visualise the data

Lets plot the data first to see what is going on:

[3]:
%matplotlib inline
import matplotlib.pyplot as plt

plt.figure()
plt.xlabel('x')
plt.ylabel('y')
plt.errorbar(x=t, y=y, yerr=yerr,
             marker='o', ls=' ', color='orange')

[3]:
<ErrorbarContainer object of 3 artists>
_images/example-sine-bayesian-workflow_6_1.svg

A beautiful noisy data set, with some hints of a modulation.

Given this data set, we should come up with a sensible model.

[4]:
parameters = ['B', 'A1', 'P1', 't1']

def prior_transform(cube):
    # the argument, cube, consists of values from 0 to 1
    # we have to convert them to physical scales

    params = cube.copy()
    # let background level go from -10 to +10
    params[0] = cube[0] * 20 - 10
    # let amplitude go from 0.1 to 100
    params[1] = 10**(cube[1] * 3 - 1)
    # let period go from 0.3 to 30
    params[2] = 10**(cube[2] * 2)
    # let time go from 0 to 1
    params[3] = cube[3]
    return params

def log_likelihood(params):
    # unpack the current parameters:
    B, A1, P1, t1 = params

    # compute for each x point, where it should lie in y
    y_model = sine_model1(t, B=B, A1=A1, P1=P1, t1=t1)
    # compute likelihood
    loglike = -0.5 * (((y_model - y) / yerr)**2).sum()

    return loglike


Lets take a few samples from the prior and look if they look anything like data we would expect.

[5]:
for i in range(8):
    plt.figure()
    cube = np.random.uniform(size=len(parameters))
    params = prior_transform(cube)
    B, A1, P1, t1 = params
    y_model = sine_model1(t, B=B, A1=A1, P1=P1, t1=t1)
    y_sim = np.random.normal(y_model, yerr)

    plt.xlabel('x')
    plt.ylabel('y')
    plt.errorbar(x=t, y=y_sim, yerr=yerr,
                 marker='o', ls=' ')
_images/example-sine-bayesian-workflow_10_0.svg
_images/example-sine-bayesian-workflow_10_1.svg
_images/example-sine-bayesian-workflow_10_2.svg
_images/example-sine-bayesian-workflow_10_3.svg
_images/example-sine-bayesian-workflow_10_4.svg
_images/example-sine-bayesian-workflow_10_5.svg
_images/example-sine-bayesian-workflow_10_6.svg
_images/example-sine-bayesian-workflow_10_7.svg

That looks quite OK actually – the y-ranges are of the right magnitude, the periods are sometimes short, sometimes long.

Make posteriors

[6]:
import ultranest

sampler = ultranest.ReactiveNestedSampler(parameters, log_likelihood, prior_transform)
result = sampler.run()

[ultranest] Sampling 400 live points from prior ...
Z=-40.5(29.83%) | Like=-33.63..-27.78 [-33.8121..-32.7049] | it/evals=3406/237562 eff=1.4361% N=400
/home/user/.local/lib/python3.6/site-packages/ultranest/integrator.py:1243: UserWarning: Sampling from region seems inefficient. You can try increasing nlive, frac_remain, dlogz, dKL, decrease min_ess). [0/40 accepted, it=2500]
  " [%d/%d accepted, it=%d]" % (accepted.sum(), ndraw, nit))
[ultranest] Explored until L=-3e+01
[ultranest] Likelihood function evaluations: 274592
[ultranest]   logZ = -39.02 +- 0.09239
[ultranest] Effective samples strategy satisfied (ESS = 3394.6, need >400)
[ultranest] Posterior uncertainty strategy is satisfied (KL: 0.45+-0.05 nat, need <0.50 nat)
[ultranest] Evidency uncertainty strategy is satisfied (dlogz=0.18, need <0.5)
[ultranest]   logZ error budget: single: 0.14 bs:0.09 tail:0.01 total:0.09 required:<0.50
[ultranest] done iterating.

Posterior predictive checks

The idea of posterior predictive checks (PPC) is to generate new data from each posterior sample, and then verify if they look like the data.

This is a bit circular, but if the model is too rigid, then some data points cannot be reproduced.

[7]:
from ultranest.plot import PredictionBand

t_range = np.linspace(0, 5, 100)
band = PredictionBand(t_range)

plt.figure()

for params in result['samples']:
    B, A1, P1, t1 = params
    y_model = sine_model1(t_range, B=B, A1=A1, P1=P1, t1=t1)
    y_sim = np.random.normal(y_model, yerr)
    band.add(y_sim)

band.line(color='g')
band.shade(color='g', alpha=0.5)
band.shade(q=0.49, color='g', alpha=0.5)

plt.scatter(t, y, marker='o')
plt.xlabel('x')
plt.ylabel('y');

_images/example-sine-bayesian-workflow_15_0.svg

We nicely see that most data points follow the PPC 99.5% probability range (light green).

Except for one point (near x=2, y=4). This is the outlier I snuck in when generating the data.

At this point we could adjust the model.

Plot the parameter posterior probability distribution

A classic corner plot:

[8]:
from ultranest.plot import cornerplot
cornerplot(result)
_images/example-sine-bayesian-workflow_18_0.svg