Linear Bivariate BMM with SAMBA toy models : cdf mixing

Contents

2.2. Linear Bivariate BMM with SAMBA toy models : cdf mixing#

The best way to learn Taweret is to use it. You can run, modify and experiment with this notebook on GitHub Codespaces.

This notebook shows how to use the Bayesian model mixing package Taweret for a toy problem.

Author : Dan Liyanage

Date : 11/10/2022

More about SAMBA toy models can be found in Uncertainties here, there, and everywhere: interpolating between small- and large-g expansions using Bayesian Model Mixing

import sys
import os

# You will have to change the following imports depending on where you have 
# the packages installed

# ! pip install Taweret    # if using Colab, uncomment to install

cwd = os.getcwd()

# Get the first part of this path and append to the sys.path
tw_path = cwd.split("Taweret/")[0] + "Taweret"
samba_path = tw_path + "/subpackages/SAMBA"
sys.path.append(tw_path)
sys.path.append(samba_path)

# For plotting
import matplotlib.pyplot as plt

! pip install seaborn    # comment if installed
! pip install ptemcee    # comment if installed

import seaborn as sns
sns.set_context('poster')
# To define priors. (uncoment if not using default priors)
# ! pip install bilby   # uncomment this line if bilby is not already installed
import bilby

# For other operations
import numpy as np

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2.2.1. 1. Get toy models and the pseudo-experimental data#

# Toy models from SAMBA
from Taweret.models import samba_models as toy_models

m1 = toy_models.Loworder(2, 'uninformative')
m2 = toy_models.Highorder(2, 'uninformative')
truth = toy_models.TrueModel()
exp = toy_models.Data()

g = np.linspace(0.1, 0.6, 10)
plot_g = np.linspace(0.01,1,100)
m1_prediction = m1.evaluate(plot_g)
m2_prediction = m2.evaluate(plot_g)
true_output = truth.evaluate(plot_g)
exp_data= exp.evaluate(g,error = 0.01)

fig, ax_f = plt.subplots(figsize=(10,10))
ax_f.plot(plot_g, m1_prediction[0].flatten(), label='low order')
ax_f.plot(plot_g, m2_prediction[0].flatten(), label='high order')
ax_f.plot(plot_g, true_output[0], label='truth')
ax_f.scatter(g,exp_data[0], marker='x', label='experimental data')
ax_f.set_ylim(1.2,3.2)
ax_f.legend()

<matplotlib.legend.Legend at 0x7fb53b266a50>

../../_images/4234480759e548a7906f47817e82721d24b80a02cc4fabaef1d0685f2a262503.png

2.2.2. 2. Choose a Mixing method#

# Mixing method
from Taweret.mix.bivariate_linear import BivariateLinear as BL

models= {'low_order':m1,'high_order':m2}
mix_model = BL(models_dic=models, method='cdf') 

cdf mixing function has 2                   free parameter(s)
Warning : Default prior is set to {'cdf_0': Uniform(minimum=0, maximum=1, name='cdf_0', latex_label='cdf_0', unit=None, boundary=None), 'cdf_1': Uniform(minimum=0, maximum=1, name='cdf_1', latex_label='cdf_1', unit=None, boundary=None)}
To change the prior use `set_prior` method

#uncoment to change the prior from the default
priors = bilby.core.prior.PriorDict()
priors['cdf_0'] = bilby.core.prior.Uniform(-20,20, name="cdf_0")
priors['cdf_1'] = bilby.core.prior.Uniform(-20,20, name="cdf_1")
mix_model.set_prior(priors)

{'cdf_0': Uniform(minimum=-20, maximum=20, name='cdf_0', latex_label='cdf_0', unit=None, boundary=None),
 'cdf_1': Uniform(minimum=-20, maximum=20, name='cdf_1', latex_label='cdf_1', unit=None, boundary=None)}

2.2.3. 3. Train to find posterior#

mix_model.prior

{'cdf_0': Uniform(minimum=-20, maximum=20, name='cdf_0', latex_label='cdf_0', unit=None, boundary=None),
 'cdf_1': Uniform(minimum=-20, maximum=20, name='cdf_1', latex_label='cdf_1', unit=None, boundary=None)}

y_exp = np.array(exp_data[0]).reshape(1,-1)
y_err = np.array(exp_data[1]).reshape(1,-1)
# The parameters are set to minimum values for computational ease.
# You should increase the ntemps, nwalkers and nsamples and see
# if your results are changing. If so keep increasing them
# until convergence of results. 

kwargs_for_sampler = {'sampler': 'ptemcee',
                    'ntemps': 5,
                    'nwalkers': 50,
                    'Tmax': 100,
                    'burn_in_fixed_discard': 50,
                    'nsamples': 2000,
                    'threads': 6}

result = mix_model.train(x_exp=g, y_exp=y_exp, y_err=y_err, outdir = 'outdir/samba_bivariate', 
label='cdf_mix', kwargs_for_sampler=kwargs_for_sampler)

14:42 bilby INFO    : Running for label 'cdf_mix', output will be saved to 'outdir/samba_bivariate'

The following settings were                     provided for sampler 
{'sampler': 'ptemcee', 'ntemps': 5, 'nwalkers': 50, 'Tmax': 100, 'burn_in_fixed_discard': 50, 'nsamples': 2000, 'threads': 6}

14:42 bilby INFO    : Analysis priors:

14:42 bilby INFO    : cdf_0=Uniform(minimum=-20, maximum=20, name='cdf_0', latex_label='cdf_0', unit=None, boundary=None)

14:42 bilby INFO    : cdf_1=Uniform(minimum=-20, maximum=20, name='cdf_1', latex_label='cdf_1', unit=None, boundary=None)

14:42 bilby INFO    : Analysis likelihood class: <class 'Taweret.sampler.likelihood_wrappers.likelihood_wrapper_for_bilby'>

14:42 bilby INFO    : Analysis likelihood noise evidence: nan

14:42 bilby INFO    : Single likelihood evaluation took nan s

14:42 bilby INFO    : Using sampler Ptemcee with kwargs {'ntemps': 5, 'nwalkers': 50, 'Tmax': 100, 'betas': None, 'a': 2.0, 'adaptation_lag': 10000, 'adaptation_time': 100, 'random': None, 'adapt': False, 'swap_ratios': False}

14:42 bilby INFO    : Using convergence inputs: ConvergenceInputs(autocorr_c=5, autocorr_tol=50, autocorr_tau=1, gradient_tau=0.1, gradient_mean_log_posterior=0.1, Q_tol=1.02, safety=1, burn_in_nact=50, burn_in_fixed_discard=50, mean_logl_frac=0.01, thin_by_nact=0.5, nsamples=2000, ignore_keys_for_tau=None, min_tau=1, niterations_per_check=5)

14:42 bilby INFO    : Generating pos0 samples

14:42 bilby INFO    : Starting to sample

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195|0:01:05|nc:2.4e+05|a0:0.61-0.68|swp:0.41-0.95|n:1700<2000|t!2(+nan,+0.01)|q:1.00|0.27ms/ev

196|0:01:06|nc:2.4e+05|a0:0.61-0.68|swp:0.41-0.95|n:1750<2000|t!2(+nan,+0.00)|q:1.00|0.27ms/ev

197|0:01:06|nc:2.5e+05|a0:0.61-0.68|swp:0.41-0.95|n:1800<2000|t!2(+nan,+0.00)|q:1.00|0.27ms/ev

198|0:01:06|nc:2.5e+05|a0:0.61-0.68|swp:0.41-0.95|n:1850<2000|t!2(+nan,+0.00)|q:1.00|0.27ms/ev

199|0:01:07|nc:2.5e+05|a0:0.61-0.68|swp:0.41-0.95|n:1900<2000|t!2(+nan,+0.01)|q:1.00|0.27ms/ev

200|0:01:07|nc:2.5e+05|a0:0.61-0.68|swp:0.41-0.95|n:1950<2000|t!2(+nan,+0.01)|q:1.00|0.28ms/ev

201|0:01:07|nc:2.5e+05|a0:0.61-0.68|swp:0.41-0.95|n:2000<2000|t!2(+nan,+0.01)|q:1.00|0.28ms/ev

202|0:01:08|nc:2.5e+05|a0:0.61-0.68|swp:0.41-0.95|n:2050<2000|t!2(+nan,+0.01)|q:1.00|0.28ms/ev

203|0:01:08|nc:2.5e+05|a0:0.61-0.68|swp:0.41-0.95|n:2100<2000|t!2(+nan,+0.01)|q:1.00|0.27ms/ev

204|0:01:08|nc:2.6e+05|a0:0.61-0.68|swp:0.41-0.95|n:2150<2000|t!2(+nan,+0.01)|q:1.00|0.27ms/ev

205|0:01:09|nc:2.6e+05|a0:0.61-0.68|swp:0.41-0.95|n:2200<2000|t!2(+nan,+0.01)|q:1.00|0.27ms/ev

206|0:01:09|nc:2.6e+05|a0:0.61-0.68|swp:0.41-0.95|n:2250<2000|t!2(+nan,+0.01)|q:1.00|0.27ms/ev

207|0:01:09|nc:2.6e+05|a0:0.61-0.68|swp:0.41-0.95|n:2300<2000|t!2(+nan,+0.01)|q:1.00|0.27ms/ev

208|0:01:10|nc:2.6e+05|a0:0.61-0.68|swp:0.41-0.95|n:2350<2000|t!2(+nan,+0.01)|q:1.00|0.28ms/ev

209|0:01:10|nc:2.6e+05|a0:0.61-0.68|swp:0.41-0.95|n:2400<2000|t!2(+nan,+0.01)|q:1.00|0.28ms/ev

210|0:01:10|nc:2.6e+05|a0:0.61-0.68|swp:0.41-0.95|n:2450<2000|t!2(+nan,+0.01)|q:1.00|0.28ms/ev

211|0:01:11|nc:2.6e+05|a0:0.61-0.68|swp:0.41-0.95|n:2500<2000|t!2(+nan,+0.01)|q:1.00|0.27ms/ev

212|0:01:11|nc:2.6e+05|a0:0.61-0.68|swp:0.41-0.95|n:2550<2000|t!2(+nan,+0.01)|q:1.00|0.27ms/ev

213|0:01:11|nc:2.7e+05|a0:0.61-0.68|swp:0.41-0.95|n:2600<2000|t=2(+0.01,+0.00)|q:1.00|0.27ms/ev

14:43 bilby INFO    : Finished sampling

14:43 bilby INFO    : Writing checkpoint and diagnostics

14:43 bilby INFO    : Finished writing checkpoint

14:43 bilby INFO    : Sampling time: 0:01:11.955844

14:43 bilby INFO    : Summary of results:
nsamples: 2600
ln_noise_evidence:    nan
ln_evidence:  0.888 +/-  2.140
ln_bayes_factor:    nan +/-  2.140

# Posterior of the mixing parameters. 
result.plot_corner()

../../_images/51306fe83f59f3d8383f2129640d130ff256b6923c9e2139a3ea35817d1318e4.png

#corner plots
import corner
fig, axs = plt.subplots(2,2, figsize=(6,6), dpi=200)
corner.corner(mix_model.posterior,labels=['cdf_0','cdf_1'],quantiles=[0.16, 0.5, 0.84],fig=fig)
plt.show()

../../_images/efbdb4afb3b83926e28a7a55ba9e7d5ad59321e321e95d6f46417cbe715819f3.png

2.2.4. 4. Predictions#

_,mean_prior,CI_prior, _ = mix_model.prior_predict(plot_g, CI=[5,20,80,95])
_,mean,CI, _ = mix_model.predict(plot_g, CI=[5,20,80,95])

(10000, 2)
using provided samples instead of posterior

per5, per20, per80, per95 = CI
prior5, prior20, prior80, prior95 = CI_prior

# Map value prediction for the step mixing function parameter
map_prediction = mix_model.evaluate(mix_model.map, plot_g)

%matplotlib inline
sns.set_context('poster')
fig, ax = plt.subplots(figsize=(10,10))
ax.plot(plot_g, mean.flatten(), label='posterior mean')
ax.fill_between(plot_g,per5.flatten(),per95.flatten(),color=sns.color_palette()[4], alpha=0.2, label='90% C.I.')
ax.fill_between(plot_g,per20.flatten(),per80.flatten(), color=sns.color_palette()[4], alpha=0.3, label='60% C.I.')
ax.fill_between(plot_g,prior20.flatten(),prior80.flatten(),color=sns.color_palette()[2], alpha=0.2, label='60% C.I. Prior')
ax.scatter(g,exp_data[0], marker='x', label='experimental data')
ax.plot(plot_g, mean_prior.flatten(), label='prior mean')
ax.plot(plot_g, map_prediction.flatten(), label='MAP prediction')
ax.set_ybound(1,3)
ax.legend()

<matplotlib.legend.Legend at 0x7fb51f376b10>

../../_images/ce918012ff8d8a35a0d2c940d817e263c8677cd5b523e9308e6279f4529528cb.png