2.5. Comparative study of BMM method in bivariate linear module using Coleman toy models.

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

The models can be found in Coleman Thesis : https://go.exlibris.link/3fVZCfhl

This notebook shows how to use the Bayesian model mixing methods available in bivariate_linear mixing method of package Taweret for a toy problem.

Author : Dan Liyanage

Date : 08/14/2023

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

# Setting Taweret path
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. (uncoment if not using default priors)
# ! pip install bilby      # uncomment if not already installed
import bilby

# For other operations
import numpy as np

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# Import models with a predict method
from Taweret.models import coleman_models as toy_models

m1 = toy_models.coleman_model_1()
m2 = toy_models.coleman_model_2()
truth = toy_models.coleman_truth()

#!pwd

g = np.linspace(0,9,10)
plot_g = np.linspace(0,9,100)
true_output = truth.evaluate(plot_g)
exp_data = truth.evaluate(g)

2.5.1. 1. The models and the experimental data.

Truth

\(f(x) = 2-0.1(x-4)^2\), where \(x \in [-1, 9]\)

Model 1

\(f_1(x,\theta)= 0.5(x+\theta)-2\) , where \(\theta \in [1, 6]\)

Model 2

\(f_2(x,\theta)= -0.5(x-\theta) + 3.7\) , where \(\theta \in [-2, 3]\)

Experimental data

sampled from the Truth with a fixed standard deviation of 0.3

sns.set_context('notebook')
fig, axs = plt.subplots(1,2,figsize=(20,5))
prior_ranges = [(1,6), (-2,3)]
for i in range(0,2):
    ax = axs.flatten()[i]
    ax.plot(plot_g, true_output[0], label='truth', color='black')
    ax.errorbar(g,exp_data[0],exp_data[1], fmt='o', label='experimental data', color='r')
    ax.legend()
    ax.set_ylim(-2,4)
    for value in np.linspace(*prior_ranges[i],10):
        if i==0:
            predict_1 = m1.evaluate(plot_g, value, full_corr=False)
            ax.plot(plot_g, predict_1[0])
            ax.set_ylabel(r'$f_1(x)$')
        if i==1:
            predict_2 = m2.evaluate(plot_g, value, full_corr=False)
            ax.plot(plot_g, predict_2[0])      
            ax.set_ylabel(r'$f_2(x)$')    
    ax.set_xlabel('x') 
    

2.5.2. 2. Choose a Mixing method

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

models= {'model1':m1,'model2':m2}

mix_model_BMMC_mix = BL(models_dic=models, method='addstepasym', nargs_model_dic={'model1':1, 'model2':1},
              same_parameters = False)

mix_model_BMMcor_mix = BL(models_dic=models, method='addstepasym', nargs_model_dic={'model1':1, 'model2':1},
              same_parameters = False, BMMcor=True)

mix_model_mean_mix = BL(models_dic=models, method='addstepasym', nargs_model_dic={'model1':1, 'model2':1},
              same_parameters = False, mean_mix=True)

mix_models = [mix_model_BMMC_mix, mix_model_BMMcor_mix, mix_model_mean_mix]

## uncoment to change the prior from the default
priors = bilby.core.prior.PriorDict()
priors['addstepasym_0'] = bilby.core.prior.Uniform(0, 9, name="addstepasym_0")
priors['addstepasym_1'] = bilby.core.prior.Uniform(0, 9, name="addstepasym_1")
priors['addstepasym_2'] = bilby.core.prior.Uniform(0, 1, name="addstepasym_2")
for mix_model in mix_models:
    mix_model.set_prior(priors)

for mix__model in mix_models:
    print(mix_model.prior)

{'addstepasym_0': Uniform(minimum=0, maximum=9, name='addstepasym_0', latex_label='addstepasym_0', unit=None, boundary=None), 'addstepasym_1': Uniform(minimum=0, maximum=9, name='addstepasym_1', latex_label='addstepasym_1', unit=None, boundary=None), 'addstepasym_2': Uniform(minimum=0, maximum=1, name='addstepasym_2', latex_label='addstepasym_2', unit=None, boundary=None), 'model1_0': Uniform(minimum=1, maximum=6, name='model1_0', latex_label='model1_0', unit=None, boundary=None), 'model2_0': Uniform(minimum=-2, maximum=3, name='model2_0', latex_label='model2_0', unit=None, boundary=None)}
{'addstepasym_0': Uniform(minimum=0, maximum=9, name='addstepasym_0', latex_label='addstepasym_0', unit=None, boundary=None), 'addstepasym_1': Uniform(minimum=0, maximum=9, name='addstepasym_1', latex_label='addstepasym_1', unit=None, boundary=None), 'addstepasym_2': Uniform(minimum=0, maximum=1, name='addstepasym_2', latex_label='addstepasym_2', unit=None, boundary=None), 'model1_0': Uniform(minimum=1, maximum=6, name='model1_0', latex_label='model1_0', unit=None, boundary=None), 'model2_0': Uniform(minimum=-2, maximum=3, name='model2_0', latex_label='model2_0', unit=None, boundary=None)}
{'addstepasym_0': Uniform(minimum=0, maximum=9, name='addstepasym_0', latex_label='addstepasym_0', unit=None, boundary=None), 'addstepasym_1': Uniform(minimum=0, maximum=9, name='addstepasym_1', latex_label='addstepasym_1', unit=None, boundary=None), 'addstepasym_2': Uniform(minimum=0, maximum=1, name='addstepasym_2', latex_label='addstepasym_2', unit=None, boundary=None), 'model1_0': Uniform(minimum=1, maximum=6, name='model1_0', latex_label='model1_0', unit=None, boundary=None), 'model2_0': Uniform(minimum=-2, maximum=3, name='model2_0', latex_label='model2_0', unit=None, boundary=None)}

2.5.3. 3. Train to find posterior

g.shape

(10,)

#from Taweret.utils.utils import normed_mvn_loglike

kwargs_for_sampler = {'sampler':'ptemcee',
                'ntemps':5,
                'nwalkers':40,
                'Tmax':100,
                'burn_in_fixed_discard':500,
                'nsamples':3000,
                'threads':6,
                'verbose':False}
                #'safety':2,
                #'autocorr_tol':5}

import os
import shutil
outdirs = ['outdir/mix_model_1', 'outdir/mix_model_2', 'outdir/mix_model_3']
labels = ['BMMC','BMMcor','BMMmean']
results = []

for i in range(0,3):
    mix_model = mix_models[i]
    label = labels[i]
    outdir = outdirs[i]
    if os.path.isdir(outdir):
        print('removing '+outdir)
        shutil.rmtree(outdir)
    else:
        print('file does not exist '+outdir)
    result = mix_model.train(x_exp=g.reshape(-1,1), y_exp=exp_data[0].reshape(-1,1), y_err=exp_data[1].reshape(-1,1)
                         ,kwargs_for_sampler=kwargs_for_sampler, label=label, outdir=outdir)
    results.append(result)

/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)
19:35 bilby INFO    : Running for label 'BMMC', output will be saved to 'outdir/mix_model_1'

file does not exist outdir/mix_model_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)
19:35 bilby INFO    : Analysis priors:

19:35 bilby INFO    : addstepasym_0=Uniform(minimum=0, maximum=9, name='addstepasym_0', latex_label='addstepasym_0', unit=None, boundary=None)

19:35 bilby INFO    : addstepasym_1=Uniform(minimum=0, maximum=9, name='addstepasym_1', latex_label='addstepasym_1', unit=None, boundary=None)

19:35 bilby INFO    : addstepasym_2=Uniform(minimum=0, maximum=1, name='addstepasym_2', latex_label='addstepasym_2', unit=None, boundary=None)

19:35 bilby INFO    : model1_0=Uniform(minimum=1, maximum=6, name='model1_0', latex_label='model1_0', unit=None, boundary=None)

19:35 bilby INFO    : model2_0=Uniform(minimum=-2, maximum=3, name='model2_0', latex_label='model2_0', unit=None, boundary=None)

19:35 bilby INFO    : Analysis likelihood class: <class 'Taweret.sampler.likelihood_wrappers.likelihood_wrapper_for_bilby'>

19:35 bilby INFO    : Analysis likelihood noise evidence: nan

19:35 bilby INFO    : Single likelihood evaluation took 2.904e-04 s

19:35 bilby INFO    : Using sampler Ptemcee with kwargs {'ntemps': 5, 'nwalkers': 40, 'Tmax': 100, 'betas': None, 'a': 2.0, 'adaptation_lag': 10000, 'adaptation_time': 100, 'random': None, 'adapt': False, 'swap_ratios': False}

19:35 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

19:35 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=500, mean_logl_frac=0.01, thin_by_nact=0.5, nsamples=3000, ignore_keys_for_tau=None, min_tau=1, niterations_per_check=5)

19:35 bilby INFO    : Generating pos0 samples

19:35 bilby INFO    : Starting to sample

19:36 bilby INFO    : Finished sampling

19:36 bilby INFO    : Writing checkpoint and diagnostics

19:36 bilby INFO    : Finished writing checkpoint

19:36 bilby INFO    : Sampling time: 0:01:24.179991

19:36 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. 

19:36 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. 

19:36 bilby INFO    : Summary of results:
nsamples: 3240
ln_noise_evidence:    nan
ln_evidence: -8.957 +/-  2.525
ln_bayes_factor:    nan +/-  2.525

/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)
19:36 bilby INFO    : Running for label 'BMMcor', output will be saved to 'outdir/mix_model_2'

file does not exist outdir/mix_model_2

19:36 bilby INFO    : Analysis priors:

19:36 bilby INFO    : addstepasym_0=Uniform(minimum=0, maximum=9, name='addstepasym_0', latex_label='addstepasym_0', unit=None, boundary=None)

19:36 bilby INFO    : addstepasym_1=Uniform(minimum=0, maximum=9, name='addstepasym_1', latex_label='addstepasym_1', unit=None, boundary=None)

19:36 bilby INFO    : addstepasym_2=Uniform(minimum=0, maximum=1, name='addstepasym_2', latex_label='addstepasym_2', unit=None, boundary=None)

19:36 bilby INFO    : model1_0=Uniform(minimum=1, maximum=6, name='model1_0', latex_label='model1_0', unit=None, boundary=None)

19:36 bilby INFO    : model2_0=Uniform(minimum=-2, maximum=3, name='model2_0', latex_label='model2_0', unit=None, boundary=None)

19:36 bilby INFO    : Analysis likelihood class: <class 'Taweret.sampler.likelihood_wrappers.likelihood_wrapper_for_bilby'>

19:36 bilby INFO    : Analysis likelihood noise evidence: nan

19:36 bilby INFO    : Single likelihood evaluation took 2.268e-04 s

19:36 bilby INFO    : Using sampler Ptemcee with kwargs {'ntemps': 5, 'nwalkers': 40, 'Tmax': 100, 'betas': None, 'a': 2.0, 'adaptation_lag': 10000, 'adaptation_time': 100, 'random': None, 'adapt': False, 'swap_ratios': False}

19:36 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

19:36 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=500, mean_logl_frac=0.01, thin_by_nact=0.5, nsamples=3000, ignore_keys_for_tau=None, min_tau=1, niterations_per_check=5)

19:36 bilby INFO    : Generating pos0 samples

19:36 bilby INFO    : Starting to sample

19:38 bilby INFO    : Finished sampling

19:38 bilby INFO    : Writing checkpoint and diagnostics

19:38 bilby INFO    : Finished writing checkpoint

19:38 bilby INFO    : Sampling time: 0:02:04.865254

19:38 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. 

19:38 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. 

19:38 bilby INFO    : Summary of results:
nsamples: 3720
ln_noise_evidence:    nan
ln_evidence:  7.103 +/-  5.062
ln_bayes_factor:    nan +/-  5.062

/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)
19:38 bilby INFO    : Running for label 'BMMmean', output will be saved to 'outdir/mix_model_3'

file does not exist outdir/mix_model_3

19:38 bilby INFO    : Analysis priors:

19:38 bilby INFO    : addstepasym_0=Uniform(minimum=0, maximum=9, name='addstepasym_0', latex_label='addstepasym_0', unit=None, boundary=None)

19:38 bilby INFO    : addstepasym_1=Uniform(minimum=0, maximum=9, name='addstepasym_1', latex_label='addstepasym_1', unit=None, boundary=None)

19:38 bilby INFO    : addstepasym_2=Uniform(minimum=0, maximum=1, name='addstepasym_2', latex_label='addstepasym_2', unit=None, boundary=None)

19:38 bilby INFO    : model1_0=Uniform(minimum=1, maximum=6, name='model1_0', latex_label='model1_0', unit=None, boundary=None)

19:38 bilby INFO    : model2_0=Uniform(minimum=-2, maximum=3, name='model2_0', latex_label='model2_0', unit=None, boundary=None)

19:38 bilby INFO    : Analysis likelihood class: <class 'Taweret.sampler.likelihood_wrappers.likelihood_wrapper_for_bilby'>

19:38 bilby INFO    : Analysis likelihood noise evidence: nan

19:38 bilby INFO    : Single likelihood evaluation took 1.889e-04 s

19:38 bilby INFO    : Using sampler Ptemcee with kwargs {'ntemps': 5, 'nwalkers': 40, 'Tmax': 100, 'betas': None, 'a': 2.0, 'adaptation_lag': 10000, 'adaptation_time': 100, 'random': None, 'adapt': False, 'swap_ratios': False}

19:38 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

19:38 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=500, mean_logl_frac=0.01, thin_by_nact=0.5, nsamples=3000, ignore_keys_for_tau=None, min_tau=1, niterations_per_check=5)

19:38 bilby INFO    : Generating pos0 samples

19:38 bilby INFO    : Starting to sample

19:40 bilby INFO    : Finished sampling

19:40 bilby INFO    : Writing checkpoint and diagnostics

19:40 bilby INFO    : Finished writing checkpoint

19:40 bilby INFO    : Sampling time: 0:01:27.096973

19:40 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. 

19:40 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. 

19:40 bilby INFO    : Summary of results:
nsamples: 3040
ln_noise_evidence:    nan
ln_evidence: -1.214 +/-  3.631
ln_bayes_factor:    nan +/-  3.631

posteriors = [0,0,0]
for i in range(0,3):
    result = results[i]
    label = labels[i]
    result = result.posterior.iloc[:,0:-2]
    result['model'] = label
    posteriors[i]=result

import pandas as pd

df = pd.concat(posteriors, ignore_index=True, sort=False)

df.head(-10)

	addstepasym_0	addstepasym_1	addstepasym_2	model1_0	model2_0	model
0	2.486527	7.941589	0.662796	5.542920	0.976820	BMMC
1	3.097445	6.956633	0.789695	4.455517	0.881384	BMMC
2	3.430944	7.229666	0.863489	4.690683	1.299283	BMMC
3	3.461479	7.118759	0.855797	4.728741	1.325310	BMMC
4	3.912860	6.651543	0.870824	4.571047	1.443501	BMMC
...	...	...	...	...	...	...
9985	3.848702	0.460433	0.992055	5.103604	1.501484	BMMmean
9986	3.884630	5.423056	0.990370	4.689796	1.283540	BMMmean
9987	3.363562	7.162613	0.996488	5.167660	1.053737	BMMmean
9988	3.394550	6.904732	0.991850	5.186315	1.081165	BMMmean
9989	3.580620	4.098574	0.985367	4.716994	1.617868	BMMmean

9990 rows × 6 columns

df_renamed=df.rename(columns={'addstepasym_0':r'$\beta_0$', 'addstepasym_1':r'$\beta_1$', 
                              'addstepasym_2':r'$\alpha$', 'model1_0':r'$\theta_1$', 
                              'model2_0':r'$\theta_2$', 'model':'method'})

#g.savefig('temp_save')

import seaborn as sns
sns.set_context('paper', font_scale=1.5)
gg = sns.PairGrid(df_renamed, hue="method", diag_sharey=False, hue_kws={'alpha':0.5}, corner=True,
                palette={'BMMC':sns.color_palette()[2],'BMMcor':sns.color_palette()[3], 'BMMmean':sns.color_palette()[-1]})
gg.map_lower(sns.kdeplot, fill=True)
gg.map_diag(sns.kdeplot, linewidth=2, fill=True)
gg.add_legend(loc='upper center')
plt.tight_layout()
plt.savefig('comparative_posterior', dpi=100)

2.5.3.1. 4. Predictions

sns.set_context('paper', font_scale=1.9)
fig, axs = plt.subplots(1,2,figsize=(20,10))
ax, ax2 = axs.flatten()
#fig2, ax2 = plt.subplots(figsize=(10,10))
colors = {'BMMC':sns.color_palette()[2],'BMMcor':sns.color_palette()[3], 'BMMmean':sns.color_palette()[-1]}
for i, mix_model in enumerate(mix_models):
    _,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

    model_params = [np.array(mix_model.map[3]), np.array(mix_model.map[4])]
    map_prediction = mix_model.evaluate(mix_model.map[0:3], plot_g, model_params=model_params)
    print(mix_model.map)
    
    _,_,CI_weights,_=mix_model.predict_weights(plot_g, CI=[5,20, 80, 95])
    perw_5, perw_20, perw_80, perw_95 = CI_weights
    
    
    #ax.fill_between(plot_g,perw_5,perw_95,color=colors[labels[i]], alpha=0.2, label='90% C.I.')
    ax.fill_between(plot_g,perw_20,perw_80, color=colors[labels[i]], alpha=0.3, label=labels[i])
    
    if i==0:
        ax2.fill_between(plot_g,prior20.flatten(),prior80.flatten(),color=sns.color_palette()[7], alpha=0.2, label='60% C.I. Prior')
        ax2.errorbar(g,exp_data[0],yerr=exp_data[1], marker='x', label='experimental data', color='red', fmt='.')
        ax2.plot(plot_g, mean_prior.flatten(), label='prior mean')
    #ax2.plot(plot_g, mean.flatten(), label=labels[i])
    #ax2.fill_between(plot_g,per5.flatten(),per95.flatten(),color=sns.color_palette()[4], alpha=0.2, label='90% C.I.')
    ax2.fill_between(plot_g,per20.flatten(),per80.flatten(), color=colors[labels[i]], alpha=0.3, label=labels[i])

    ax2.plot(plot_g, map_prediction.flatten(), color=colors[labels[i]], linestyle='dashed')

    
ax.legend()
ax.set_xlabel('x')
ax.set_ylabel('Model weight (w)')


ax2.set_ybound(-1,4)
ax2.legend(loc='upper center')
ax2.set_xlabel('x')
ax2.set_ylabel('Model output')

ax.set_title('(a)')
ax2.set_title('(b)')

plt.tight_layout()
fig.savefig('comparative_posterior_prditcions', dpi=100)
#fig2.savefig('comparative_posterior_predict', dpi=100)

[3.11361453 7.44964417 0.99807884 5.00308475 1.30365511]

/tmp/ipykernel_4738/1972745161.py:26: UserWarning: marker is redundantly defined by the 'marker' keyword argument and the fmt string "." (-> marker='.'). The keyword argument will take precedence.
  ax2.errorbar(g,exp_data[0],yerr=exp_data[1], marker='x', label='experimental data', color='red', fmt='.')

[3.99225858 0.51047784 0.98947215 5.00438811 1.26912869]

[3.83041157 4.92808133 0.99695917 4.99453917 1.24114518]