Linear Bivariate BMM with calibration of Coleman toy models.

The best way to learn Taweret is to use it. You can run, modify and experiment with this notebook on 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 package Taweret for a toy problem.

Author : Dan Liyanage

Date : 11/13/2022

[2]:

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

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
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

[2]:

# 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()

[3]:

g = np.linspace(-1,9,10)
plot_g = np.linspace(-1,9,100)
true_output = truth.evaluate(plot_g)
exp_data = truth.evaluate(g)

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

[4]:

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)
            ax.plot(plot_g, predict_1[0])
            ax.set_ylabel(r'$f_1(x)$')
        if i==1:
            predict_2 = m2.evaluate(plot_g, value)
            ax.plot(plot_g, predict_2[0])
            ax.set_ylabel(r'$f_2(x)$')
    ax.set_xlabel('x')

../_images/notebooks_Linear_BMM_with_cdf_function_for_coleman_models_7_0.png

2. Choose a Mixing method

[7]:

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

models= {'model1':m1,'model2':m2}
mix_model = BL(models_dic=models, method='cdf', nargs_model_dic={'model1':1, 'model2':1})

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
Using default priors for model 1
{'model1_0': Uniform(minimum=1, maximum=6, name='model1_0', latex_label='model1_0', unit=None, boundary=None)}
Using default priors for model 2
{'model2_0': Uniform(minimum=-2, maximum=3, name='model2_0', latex_label='model2_0', unit=None, boundary=None)}

[8]:

## uncoment to change the prior from the default
import bilby

priors = bilby.core.prior.PriorDict()
priors['cdf_0'] = bilby.core.prior.Normal(0, 1, name="cdf_0")
priors['cdf_1'] = bilby.core.prior.Normal(0, 1, name="cdf_1")
mix_model.set_prior(priors)

[8]:

{'cdf_0': Normal(mu=0, sigma=1, name='cdf_0', latex_label='cdf_0', unit=None, boundary=None),
 'cdf_1': Normal(mu=0, sigma=1, name='cdf_1', latex_label='cdf_1', 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)}

[9]:

mix_model.prior

[9]:

{'cdf_0': Normal(mu=0, sigma=1, name='cdf_0', latex_label='cdf_0', unit=None, boundary=None),
 'cdf_1': Normal(mu=0, sigma=1, name='cdf_1', latex_label='cdf_1', 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)}

3. Train to find posterior

[10]:

kwargs_for_sampler = {'sampler': 'ptemcee',
                    'ntemps': 5,
                    'nwalkers': 50,
                    'Tmax': 100,
                    'burn_in_fixed_discard': 50,
                    'nsamples': 2000,
                    'threads': 6,
                    'printdt': 60}

result = mix_model.train(x_exp=g, y_exp=exp_data[0], y_err=exp_data[1],
                        label='cdf_mix',
                        outdir='outdir/coleman',
                        kwargs_for_sampler=kwargs_for_sampler,
                        load_previous=True)

/Users/alexandra/Documents/Taweret/Taweret/mix/bivariate_linear.py:639: UserWarning: 'threads' dectected in 'kwargs_for_sampler' on Darwin. Setting `start_method` to `fork`
  warnings.warn("'threads' dectected in 'kwargs_for_sampler'" +
10:03 bilby INFO    : Running for label 'cdf_mix', output will be saved to 'outdir/coleman'

Saved results for cdf_mix do not exist in : outdir/coleman
The following settings were                     provided for sampler
{'sampler': 'ptemcee', 'ntemps': 5, 'nwalkers': 50, 'Tmax': 100, 'burn_in_fixed_discard': 50, 'nsamples': 2000, 'threads': 6, 'printdt': 60}

10:03 bilby INFO    : Analysis priors:
10:03 bilby INFO    : cdf_0=Normal(mu=0, sigma=1, name='cdf_0', latex_label='cdf_0', unit=None, boundary=None)
10:03 bilby INFO    : cdf_1=Normal(mu=0, sigma=1, name='cdf_1', latex_label='cdf_1', unit=None, boundary=None)
10:03 bilby INFO    : model1_0=Uniform(minimum=1, maximum=6, name='model1_0', latex_label='model1_0', unit=None, boundary=None)
10:03 bilby INFO    : model2_0=Uniform(minimum=-2, maximum=3, name='model2_0', latex_label='model2_0', unit=None, boundary=None)
10:03 bilby INFO    : Analysis likelihood class: <class 'Taweret.sampler.likelihood_wrappers.likelihood_wrapper_for_bilby'>
10:03 bilby INFO    : Analysis likelihood noise evidence: nan
10:03 bilby INFO    : Single likelihood evaluation took 1.116e-04 s
10:03 bilby WARNING : Supplied argument 'printdt' not an argument of 'Ptemcee', removing.
10:03 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}
10:03 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)
10:03 bilby INFO    : Generating pos0 samples
10:03 bilby INFO    : Starting to sample

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431|0:00:15|nc:5.4e+05|a0:0.56-0.60|swp:0.27-0.67|n:750<2000|t!2(+0.02,+0.17)|q:1.01|0.03ms/ev
432|0:00:15|nc:5.4e+05|a0:0.56-0.60|swp:0.27-0.67|n:800<2000|t!2(+0.03,+0.70)|q:1.01|0.03ms/ev
433|0:00:15|nc:5.4e+05|a0:0.56-0.60|swp:0.27-0.67|n:850<2000|t!2(+0.02,+0.38)|q:1.01|0.03ms/ev
434|0:00:15|nc:5.4e+05|a0:0.56-0.60|swp:0.27-0.67|n:900<2000|t!2(+0.02,+0.43)|q:1.01|0.03ms/ev
435|0:00:15|nc:5.4e+05|a0:0.56-0.60|swp:0.27-0.67|n:950<2000|t!2(+0.01,+0.25)|q:1.01|0.03ms/ev
436|0:00:16|nc:5.4e+05|a0:0.56-0.60|swp:0.27-0.67|n:1000<2000|t!2(+0.04,+0.27)|q:1.01|0.03ms/ev
437|0:00:16|nc:5.5e+05|a0:0.56-0.60|swp:0.27-0.67|n:1050<2000|t!2(+0.02,+0.17)|q:1.01|0.03ms/ev
438|0:00:16|nc:5.5e+05|a0:0.56-0.60|swp:0.27-0.67|n:1100<2000|t!2(+0.02,+0.19)|q:1.01|0.03ms/ev
439|0:00:16|nc:5.5e+05|a0:0.56-0.60|swp:0.27-0.67|n:1150<2000|t!2(+0.02,+0.27)|q:1.01|0.03ms/ev
440|0:00:16|nc:5.5e+05|a0:0.56-0.60|swp:0.27-0.67|n:1200<2000|t!2(+0.04,+0.26)|q:1.01|0.03ms/ev
441|0:00:16|nc:5.5e+05|a0:0.56-0.60|swp:0.27-0.67|n:1250<2000|t!2(+0.01,+0.31)|q:1.01|0.03ms/ev
442|0:00:16|nc:5.5e+05|a0:0.56-0.60|swp:0.27-0.67|n:1300<2000|t!2(+0.01,+0.33)|q:1.00|0.03ms/ev
443|0:00:16|nc:5.5e+05|a0:0.56-0.60|swp:0.27-0.67|n:1350<2000|t!2(+0.02,+0.35)|q:1.00|0.03ms/ev
444|0:00:16|nc:5.6e+05|a0:0.56-0.60|swp:0.27-0.67|n:1400<2000|t!2(+0.02,+0.28)|q:1.00|0.03ms/ev
445|0:00:16|nc:5.6e+05|a0:0.56-0.60|swp:0.27-0.67|n:1450<2000|t!2(+0.02,+0.12)|q:1.00|0.03ms/ev
446|0:00:16|nc:5.6e+05|a0:0.56-0.60|swp:0.27-0.67|n:1500<2000|t!2(+0.03,+0.19)|q:1.00|0.03ms/ev
447|0:00:16|nc:5.6e+05|a0:0.56-0.60|swp:0.27-0.67|n:1550<2000|t!2(+0.02,+0.43)|q:1.00|0.03ms/ev
448|0:00:16|nc:5.6e+05|a0:0.56-0.60|swp:0.27-0.67|n:1600<2000|t!2(+0.02,+0.30)|q:1.00|0.03ms/ev
449|0:00:16|nc:5.6e+05|a0:0.56-0.60|swp:0.27-0.67|n:1650<2000|t!2(+0.01,+0.28)|q:1.00|0.03ms/ev
450|0:00:16|nc:5.6e+05|a0:0.56-0.60|swp:0.27-0.67|n:1700<2000|t!2(+0.02,+0.31)|q:1.00|0.03ms/ev
451|0:00:16|nc:5.6e+05|a0:0.56-0.60|swp:0.27-0.67|n:1750<2000|t!2(+0.02,+0.29)|q:1.00|0.03ms/ev
452|0:00:16|nc:5.6e+05|a0:0.56-0.60|swp:0.27-0.67|n:1800<2000|t!2(+0.02,+0.12)|q:1.00|0.03ms/ev
453|0:00:16|nc:5.7e+05|a0:0.56-0.60|swp:0.27-0.67|n:1850<2000|t!2(+0.02,+0.24)|q:1.00|0.03ms/ev
454|0:00:16|nc:5.7e+05|a0:0.56-0.60|swp:0.27-0.67|n:1900<2000|t!2(+0.01,+0.29)|q:1.00|0.03ms/ev
455|0:00:16|nc:5.7e+05|a0:0.56-0.60|swp:0.27-0.67|n:1950<2000|t!2(+0.01,+0.39)|q:1.00|0.03ms/ev
456|0:00:16|nc:5.7e+05|a0:0.56-0.60|swp:0.27-0.67|n:2000<2000|t!2(+0.01,+0.21)|q:1.00|0.03ms/ev
457|0:00:16|nc:5.7e+05|a0:0.56-0.60|swp:0.27-0.67|n:2050<2000|t!2(+0.01,+0.29)|q:1.00|0.03ms/ev
458|0:00:16|nc:5.7e+05|a0:0.56-0.60|swp:0.27-0.67|n:2100<2000|t!2(+0.01,+0.46)|q:1.00|0.03ms/ev
459|0:00:16|nc:5.7e+05|a0:0.56-0.60|swp:0.27-0.67|n:2150<2000|t!2(+0.03,+0.41)|q:1.00|0.03ms/ev
460|0:00:16|nc:5.8e+05|a0:0.56-0.60|swp:0.27-0.67|n:2200<2000|t!2(+0.03,+0.40)|q:1.00|0.03ms/ev
461|0:00:16|nc:5.8e+05|a0:0.56-0.60|swp:0.27-0.67|n:2250<2000|t!2(+0.01,+0.32)|q:1.00|0.03ms/ev
462|0:00:16|nc:5.8e+05|a0:0.56-0.60|swp:0.27-0.67|n:2300<2000|t!2(+0.02,+0.24)|q:1.00|0.03ms/ev
463|0:00:17|nc:5.8e+05|a0:0.56-0.60|swp:0.27-0.67|n:2350<2000|t!2(+0.02,+0.43)|q:1.00|0.03ms/ev
464|0:00:17|nc:5.8e+05|a0:0.56-0.60|swp:0.27-0.67|n:2400<2000|t!2(+0.03,+0.16)|q:1.00|0.03ms/ev
465|0:00:17|nc:5.8e+05|a0:0.56-0.60|swp:0.27-0.67|n:2450<2000|t!2(+0.01,+0.55)|q:1.00|0.03ms/ev
466|0:00:17|nc:5.8e+05|a0:0.56-0.60|swp:0.27-0.67|n:2500<2000|t!2(+0.02,+0.33)|q:1.00|0.03ms/ev
467|0:00:17|nc:5.8e+05|a0:0.56-0.60|swp:0.27-0.67|n:2550<2000|t!2(+0.02,+0.23)|q:1.00|0.03ms/ev
468|0:00:17|nc:5.8e+05|a0:0.56-0.60|swp:0.27-0.67|n:2600<2000|t!2(+0.01,+0.31)|q:1.00|0.03ms/ev
469|0:00:17|nc:5.9e+05|a0:0.56-0.60|swp:0.27-0.67|n:2650<2000|t!2(+0.01,+0.41)|q:1.00|0.03ms/ev
470|0:00:17|nc:5.9e+05|a0:0.56-0.60|swp:0.27-0.67|n:2700<2000|t!2(+0.01,+0.28)|q:1.00|0.03ms/ev
471|0:00:17|nc:5.9e+05|a0:0.56-0.60|swp:0.27-0.67|n:2750<2000|t!2(+0.03,+0.21)|q:1.00|0.03ms/ev
472|0:00:17|nc:5.9e+05|a0:0.56-0.60|swp:0.27-0.67|n:2800<2000|t!2(+0.02,+0.34)|q:1.00|0.03ms/ev
473|0:00:17|nc:5.9e+05|a0:0.56-0.60|swp:0.27-0.67|n:2850<2000|t!2(+0.02,+0.28)|q:1.00|0.03ms/ev
474|0:00:17|nc:5.9e+05|a0:0.56-0.60|swp:0.27-0.67|n:2900<2000|t!2(+0.03,+0.36)|q:1.00|0.03ms/ev
475|0:00:17|nc:5.9e+05|a0:0.56-0.60|swp:0.27-0.67|n:2950<2000|t!2(+0.01,+0.33)|q:1.00|0.03ms/ev
476|0:00:17|nc:6.0e+05|a0:0.56-0.60|swp:0.27-0.67|n:3000<2000|t!2(+0.02,+0.31)|q:1.00|0.03ms/ev
477|0:00:17|nc:6.0e+05|a0:0.56-0.60|swp:0.27-0.67|n:3050<2000|t!2(+0.01,+0.23)|q:1.00|0.03ms/ev
478|0:00:17|nc:6.0e+05|a0:0.56-0.60|swp:0.27-0.67|n:3100<2000|t!2(+0.01,+0.13)|q:1.00|0.03ms/ev
479|0:00:17|nc:6.0e+05|a0:0.56-0.60|swp:0.27-0.67|n:3150<2000|t!2(+0.02,+0.32)|q:1.00|0.03ms/ev
480|0:00:17|nc:6.0e+05|a0:0.56-0.60|swp:0.27-0.67|n:3200<2000|t!2(+0.02,+0.28)|q:1.00|0.03ms/ev
481|0:00:17|nc:6.0e+05|a0:0.56-0.60|swp:0.27-0.67|n:3250<2000|t!2(+0.02,+0.35)|q:1.00|0.03ms/ev
482|0:00:17|nc:6.0e+05|a0:0.56-0.60|swp:0.27-0.67|n:3300<2000|t!2(+0.01,+0.23)|q:1.00|0.03ms/ev
483|0:00:17|nc:6.0e+05|a0:0.56-0.60|swp:0.27-0.67|n:3350<2000|t!2(+0.01,+0.36)|q:1.00|0.03ms/ev
484|0:00:17|nc:6.0e+05|a0:0.56-0.60|swp:0.27-0.67|n:3400<2000|t!2(+0.01,+0.28)|q:1.00|0.03ms/ev
485|0:00:17|nc:6.1e+05|a0:0.56-0.60|swp:0.27-0.67|n:3450<2000|t!2(+0.02,+0.30)|q:1.00|0.03ms/ev
486|0:00:17|nc:6.1e+05|a0:0.56-0.60|swp:0.27-0.67|n:3500<2000|t!2(+0.03,+0.39)|q:1.00|0.03ms/ev
487|0:00:17|nc:6.1e+05|a0:0.56-0.60|swp:0.27-0.67|n:3550<2000|t!2(+0.02,+0.19)|q:1.00|0.03ms/ev
488|0:00:17|nc:6.1e+05|a0:0.56-0.60|swp:0.27-0.67|n:3600<2000|t!2(+0.02,+0.23)|q:1.00|0.03ms/ev
489|0:00:17|nc:6.1e+05|a0:0.56-0.60|swp:0.27-0.67|n:3650<2000|t=2(+0.01,+0.10)|q:1.00|0.03ms/ev

10:03 bilby INFO    : Finished sampling
10:03 bilby INFO    : Writing checkpoint and diagnostics
10:03 bilby INFO    : Finished writing checkpoint
10:03 bilby INFO    : Sampling time: 0:00:17.999632
10:03 bilby INFO    : Summary of results:
nsamples: 3650
ln_noise_evidence:    nan
ln_evidence: -9.482 +/-  3.012
ln_bayes_factor:    nan +/-  3.012

[11]:

result.plot_corner()

[11]:

../_images/notebooks_Linear_BMM_with_cdf_function_for_coleman_models_14_0.png

4. Predictions

[12]:

_,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, 4)
using provided samples instead of posterior

[13]:

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

[14]:

mix_model.map

[14]:

array([ 2.96825776, -0.80797699,  4.7239345 ,  1.44454939])

[15]:

# Map value prediction for the step mixing function parameter
model_params = [np.array(mix_model.map[2]), np.array(mix_model.map[3])]
map_prediction = mix_model.evaluate(mix_model.map[0:2], plot_g, model_params=model_params)

[16]:

_,_,CI_weights,_=mix_model.predict_weights(plot_g, CI=[5,20, 80, 95])
perw_5, perw_20, perw_80, perw_95 = CI_weights

(3650, 100)

[17]:

%matplotlib inline
fig, ax = plt.subplots()
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.legend()
#ax.plot(plot_g, true_output[0], label='truth')
#ax.set_ylim(1.2,3.2)

[17]:

<matplotlib.legend.Legend at 0x167bd9410>

../_images/notebooks_Linear_BMM_with_cdf_function_for_coleman_models_21_1.png

[18]:

%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.errorbar(g,exp_data[0],yerr=exp_data[1], 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(-2,4)
ax.legend()

[18]:

<matplotlib.legend.Legend at 0x167b70bd0>

../_images/notebooks_Linear_BMM_with_cdf_function_for_coleman_models_22_1.png

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