2.3. Linear Bivariate BMM with SAMBA toy models : switchcos mixing#
The best way to learn Taweret is to use it. You can run, modify and experiment with this notebook GitHub Codespaces.
This notebook try to reproduce the work by A. C. Semposki, R. J. Furnstahl, D. R. Phillips in Uncertainties here, there, and everywhere: interpolating between small- and large-g expansions using Bayesian Model Mixing using the Bayesian model mixing package Taweret.
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
Steps for Bayesian Model Mixing with Taweret
1. Define your models using BayesModel class in `Taweret.core.bayesmodel` (Only if you want to bring in your own models)
2. Choose a Mixing method (Available mixing methods can be found in Taweret/mix)
3. Find the posterior for the mixture weights (parameters in the mixture function) and/or model parameters by training the mixed model.
4. Make predictions!
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"
sys.path.append(tw_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. (uncomment if not using default priors)
! pip install bilby # comment if already installed
import bilby
# For other operations
import numpy as np
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2.3.1. 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 0x7f0004a3d010>
2.3.2. 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='switchcos')
Need to specify a constrained prior for switchcos. g1 < g3 < g2
We follow Bilby documentation on how to specify a prior with constraints https://lscsoft.docs.ligo.org/bilby/prior.html#prior-constraints
# from scipy import stats
# def luniform(theta, a, b):
# if theta > a and theta < b:
# return 0.0
# else:
# return -np.inf
# def lpdf(params):
# if len(params) == 3:
# #g1 truncated between (0, 0.35)
# g1 = luniform(params[0], 0.01, 0.3) + stats.norm.logpdf(params[0], 0.1, 0.05) #0.1 for 2 v 2, #0.1 for 5 v 5
# #g3 truncated between (g1, 0.35)
# g3 = luniform(params[2], params[0], 0.55) + stats.norm.logpdf(params[2], 0.4, 0.05) #0.4 for 2 v 2, #0.25 for 5 v 5
# #g2 truncated between (g3, 0.35)
# g2 = luniform(params[1], params[2], 0.8) + stats.norm.logpdf(params[1], 0.6, 0.05) #0.6 for 2 v 2, #0.4 for 5 v 5
# return (g1 + g2 + g3)
# else:
# print(params)
# raise ValueError('The number of parameters does not match any available switching function.')
from bilby.core.prior.joint import BaseJointPriorDist, JointPrior, JointPriorDistError
from Taweret.utils.priors import Priors as samba_P
samba_prior = samba_P()
class custom_prior_switchcos(BaseJointPriorDist):
def __init__(self, names, bounds=None):
'''
A class defining prior for swithcos mixing method
Parameters
==========
names: list
A list of the parameter names in the prior
'''
super().__init__(names=names, bounds=bounds)
self.names=names
def ln_prob(self,value):
try:
n_sam = value.shape[1]
return [samba_prior.lpdf(x) for x in value]
except:
return samba_prior.lpdf(value)
def _sample(self, size, **kwargs):
"""
Draw, and set, a sample for the switchcos parameter joint distribution
Parameters
==========
size: int
number of samples to generate, defaults to 1
"""
samps = np.zeros((size, len(self)))
samps[:,0] = np.random.uniform(0.12, 0.18, size)
samps[:,2] = np.random.uniform(0.19, 0.24, size)
samps[:,1] = np.random.uniform(0.25, 0.30, size)
return samps
class prior_switchcos(JointPrior):
def __init__(self, dist, name=None, latex_label=None, unit=None):
if not isinstance(dist, custom_prior_switchcos):
raise JointPriorDistError("dist object must be instance of custom_prior_switchcos")
super().__init__(dist=dist, name=name, latex_label=latex_label, unit=unit)
mvg = custom_prior_switchcos(names=['switchcos_0', 'switchcos_1', 'switchcos_2'], bounds=[[0.0,0.35],[0.0,0.35],[0.0,0.35]])
priors = bilby.core.prior.PriorDict()
priors['switchcos_0'] = prior_switchcos(mvg,'switchcos_0')
priors['switchcos_1'] = prior_switchcos(mvg,'switchcos_1')
priors['switchcos_2'] = prior_switchcos(mvg,'switchcos_2')
mix_model.set_prior(priors)
{'switchcos_0': __main__.prior_switchcos(dist=custom_prior_switchcos(names=['switchcos_0', 'switchcos_1', 'switchcos_2'], bounds={'switchcos_0': (0.0, 0.35), 'switchcos_1': (0.0, 0.35), 'switchcos_2': (0.0, 0.35)}), name='switchcos_0', latex_label='switchcos_0', unit=None),
'switchcos_1': __main__.prior_switchcos(dist=custom_prior_switchcos(names=['switchcos_0', 'switchcos_1', 'switchcos_2'], bounds={'switchcos_0': (0.0, 0.35), 'switchcos_1': (0.0, 0.35), 'switchcos_2': (0.0, 0.35)}), name='switchcos_1', latex_label='switchcos_1', unit=None),
'switchcos_2': __main__.prior_switchcos(dist=custom_prior_switchcos(names=['switchcos_0', 'switchcos_1', 'switchcos_2'], bounds={'switchcos_0': (0.0, 0.35), 'switchcos_1': (0.0, 0.35), 'switchcos_2': (0.0, 0.35)}), name='switchcos_2', latex_label='switchcos_2', unit=None)}
2.3.3. 3. Train to find posterior#
mix_model.prior
{'switchcos_0': __main__.prior_switchcos(dist=custom_prior_switchcos(names=['switchcos_0', 'switchcos_1', 'switchcos_2'], bounds={'switchcos_0': (0.0, 0.35), 'switchcos_1': (0.0, 0.35), 'switchcos_2': (0.0, 0.35)}), name='switchcos_0', latex_label='switchcos_0', unit=None),
'switchcos_1': __main__.prior_switchcos(dist=custom_prior_switchcos(names=['switchcos_0', 'switchcos_1', 'switchcos_2'], bounds={'switchcos_0': (0.0, 0.35), 'switchcos_1': (0.0, 0.35), 'switchcos_2': (0.0, 0.35)}), name='switchcos_1', latex_label='switchcos_1', unit=None),
'switchcos_2': __main__.prior_switchcos(dist=custom_prior_switchcos(names=['switchcos_0', 'switchcos_1', 'switchcos_2'], bounds={'switchcos_0': (0.0, 0.35), 'switchcos_1': (0.0, 0.35), 'switchcos_2': (0.0, 0.35)}), name='switchcos_2', latex_label='switchcos_2', unit=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,
'npool':1}
result = mix_model.train(x_exp=g, y_exp=y_exp, y_err=y_err, outdir = 'outdir/samba_bivariate_1', label='switchcos_mix',
kwargs_for_sampler=kwargs_for_sampler)
/home/runner/work/Taweret/Taweret/.tox/book/lib/python3.13/site-packages/bilby/core/likelihood.py:127: FutureWarning: Setting non-trivial parameters for <class 'Taweret.sampler.likelihood_wrappers.likelihood_wrapper_for_bilby'>. This is deprecated behaviour and will be removed in Bilby version 3. See https://bilby-dev.github.io/bilby/parameters for more details.
warnings.warn(msg, FutureWarning)
02:33 bilby INFO : Running for label 'switchcos_mix', output will be saved to 'outdir/samba_bivariate_1'
/home/runner/work/Taweret/Taweret/.tox/book/lib/python3.13/site-packages/bilby/core/sampler/ptemcee.py:134: FutureWarning: The ptemcee sampler interface in bilby is deprecated and will be removed in Bilby version 3. Please use the `ptemcee-bilby`sampler plugin instead: https://github.com/bilby-dev/ptemcee-bilby.
warnings.warn(msg, FutureWarning)
02:33 bilby INFO : Analysis priors:
02:33 bilby INFO : switchcos_0=__main__.prior_switchcos(dist=custom_prior_switchcos(names=['switchcos_0', 'switchcos_1', 'switchcos_2'], bounds={'switchcos_0': (0.0, 0.35), 'switchcos_1': (0.0, 0.35), 'switchcos_2': (0.0, 0.35)}), name='switchcos_0', latex_label='switchcos_0', unit=None)
02:33 bilby INFO : switchcos_1=__main__.prior_switchcos(dist=custom_prior_switchcos(names=['switchcos_0', 'switchcos_1', 'switchcos_2'], bounds={'switchcos_0': (0.0, 0.35), 'switchcos_1': (0.0, 0.35), 'switchcos_2': (0.0, 0.35)}), name='switchcos_1', latex_label='switchcos_1', unit=None)
02:33 bilby INFO : switchcos_2=__main__.prior_switchcos(dist=custom_prior_switchcos(names=['switchcos_0', 'switchcos_1', 'switchcos_2'], bounds={'switchcos_0': (0.0, 0.35), 'switchcos_1': (0.0, 0.35), 'switchcos_2': (0.0, 0.35)}), name='switchcos_2', latex_label='switchcos_2', unit=None)
02:33 bilby INFO : Analysis likelihood class: <class 'Taweret.sampler.likelihood_wrappers.likelihood_wrapper_for_bilby'>
02:33 bilby INFO : Analysis likelihood noise evidence: nan
02:33 bilby INFO : Single likelihood evaluation took 1.417e-04 s
02:33 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}
02:33 bilby INFO : Global meta data was removed from the result object for compatibility. Use the `BILBY_INCLUDE_GLOBAL_METADATA` environment variable to include it. This behaviour will be removed in a future release. For more details see: https://bilby-dev.github.io/bilby/faq.html#global-meta-data
02:33 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)
02:33 bilby INFO : Generating pos0 samples
02:33 bilby INFO : Starting to sample
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506|0:02:56|nc:6.3e+05|a0:0.61-0.65|swp:0.86-1.00|n:-100<2000|t!2(+0.03,+0.29)|q:1.00|0.28ms/ev
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508|0:02:57|nc:6.4e+05|a0:0.61-0.65|swp:0.86-1.00|n:0<2000|t!2(+0.04,+0.32)|q:1.00|0.28ms/ev
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521|0:03:01|nc:6.5e+05|a0:0.61-0.65|swp:0.86-1.00|n:650<2000|t!2(+0.04,+0.46)|q:1.00|0.28ms/ev
522|0:03:02|nc:6.5e+05|a0:0.61-0.65|swp:0.86-1.00|n:700<2000|t!2(+0.01,+0.29)|q:1.00|0.28ms/ev
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528|0:03:04|nc:6.6e+05|a0:0.61-0.65|swp:0.86-1.00|n:1000<2000|t!2(+0.01,+0.24)|q:1.00|0.29ms/ev
529|0:03:04|nc:6.6e+05|a0:0.61-0.65|swp:0.86-1.00|n:1050<2000|t!2(+0.01,+0.40)|q:1.00|0.29ms/ev
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536|0:03:07|nc:6.7e+05|a0:0.61-0.65|swp:0.86-1.00|n:1400<2000|t!2(+0.02,+0.16)|q:1.00|0.28ms/ev
537|0:03:07|nc:6.7e+05|a0:0.61-0.65|swp:0.86-1.00|n:1450<2000|t!2(+0.00,+0.28)|q:1.00|0.28ms/ev
538|0:03:07|nc:6.7e+05|a0:0.61-0.65|swp:0.86-1.00|n:1500<2000|t!2(+0.01,+0.25)|q:1.00|0.28ms/ev
539|0:03:08|nc:6.7e+05|a0:0.61-0.65|swp:0.86-1.00|n:1550<2000|t!2(+0.01,+0.27)|q:1.00|0.28ms/ev
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549|0:03:11|nc:6.9e+05|a0:0.61-0.65|swp:0.86-1.00|n:2050<2000|t!2(+0.01,+0.18)|q:1.00|0.28ms/ev
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554|0:03:13|nc:6.9e+05|a0:0.61-0.65|swp:0.86-1.00|n:2300<2000|t!2(+0.02,+0.32)|q:1.00|0.27ms/ev
555|0:03:13|nc:6.9e+05|a0:0.61-0.65|swp:0.86-1.00|n:2350<2000|t!2(+0.01,+0.13)|q:1.00|0.27ms/ev
556|0:03:14|nc:7.0e+05|a0:0.61-0.65|swp:0.86-1.00|n:2400<2000|t!2(+0.01,+0.23)|q:1.00|0.27ms/ev
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558|0:03:14|nc:7.0e+05|a0:0.61-0.65|swp:0.86-1.00|n:2500<2000|t!2(+0.01,+0.26)|q:1.00|0.27ms/ev
559|0:03:15|nc:7.0e+05|a0:0.61-0.65|swp:0.86-1.00|n:2550<2000|t!2(+0.00,+0.40)|q:1.00|0.27ms/ev
560|0:03:15|nc:7.0e+05|a0:0.61-0.65|swp:0.86-0.99|n:2600<2000|t=2(+0.01,+0.07)|q:1.00|0.27ms/ev
02:37 bilby INFO : Finished sampling
02:37 bilby INFO : Writing checkpoint and diagnostics
02:37 bilby INFO : Finished writing checkpoint
02:37 bilby INFO : Sampling time: 0:03:15.511527
02:37 bilby WARNING : Result.save_to_file called with extension=True. This will default to json, and ignore the extension from the filename. This behaviour is deprecated and will be removed.
02:37 bilby WARNING : Result.save_to_file called with extension=True. This will default to json, and ignore the extension from the filename. This behaviour is deprecated and will be removed.
02:37 bilby INFO : Summary of results:
nsamples: 2600
ln_noise_evidence: nan
ln_evidence: 3.950 +/- 0.010
ln_bayes_factor: nan +/- 0.010
# Posterior of the mixing parameters.
result.plot_corner()
#corner plots
import corner
fig, axs = plt.subplots(3,3, figsize=(10,10), dpi=100)
corner.corner(mix_model.posterior,labels=['switchcos_0','switchcos_1', 'switchcos_2'],quantiles=[0.16, 0.5, 0.84],fig=fig)
plt.show()
2.3.4. 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])
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)
_,_,CI_weights,_=mix_model.predict_weights(plot_g, CI=[5,20, 80, 95])
perw_5, perw_20, perw_80, perw_95 = CI_weights
prior_sample = np.array(list(priors.sample(5000).values())).T
_,_,CI_prior_weights,_=mix_model.predict_weights(plot_g, CI=[5,20, 80, 95], samples=prior_sample)
priorw_5, priorw_20, priorw_80, priorw_95 = CI_prior_weights
%matplotlib inline
fig, ax = plt.subplots(figsize=(10,10))
ax.fill_between(plot_g,perw_5,perw_95,color=sns.color_palette()[4], alpha=0.2, label='90% C.I.')
ax.fill_between(plot_g,perw_20,perw_80, color=sns.color_palette()[4], alpha=0.3, label='60% C.I.')
ax.fill_between(plot_g,priorw_5,priorw_95,color=sns.color_palette()[2], alpha=0.2, label='90% C.I.Prior')
ax.fill_between(plot_g,priorw_20,priorw_80, color=sns.color_palette()[2], alpha=0.3, label='60% C.I. Prior')
w,_= mix_model.evaluate_weights(mix_model.map,plot_g)
ax.vlines(mix_model.map[0],0,1, linestyle='--', color=sns.color_palette()[1], label='swithcos_0')
ax.vlines(mix_model.map[1],0,1, linestyle='--', color=sns.color_palette()[2], label='swithcos_1')
ax.vlines(mix_model.map[2],0,1, linestyle='--', color=sns.color_palette()[3], label='swithcos_2')
ax.plot(plot_g, w, label='MAP')
ax.legend()
ax.set_title('The Mixing Function')
ax.set_xlabel('g')
ax.set_ylabel('w')
Text(0, 0.5, 'w')
%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()
ax.set_xlabel('g')
Text(0.5, 0, 'g')
