Linear Bivariate BMM with SAMBA toy models : switchcos mixing

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 (Avilable 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. (uncoment 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 0x7f408d13adb0>
../../_images/40a8347d1e1505256b0234ff2df2de783cf82bb7c99b1db81893e082e65e8e76.png

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') 

switchcos mixing function has 3                   free parameter(s)
Warning : Default prior is set to {'switchcos_0': Uniform(minimum=0, maximum=1, name='switchcos_0', latex_label='switchcos_0', unit=None, boundary=None), 'switchcos_1': Uniform(minimum=0, maximum=1, name='switchcos_1', latex_label='switchcos_1', unit=None, boundary=None), 'switchcos_2': Uniform(minimum=0, maximum=1, name='switchcos_2', latex_label='switchcos_2', unit=None, boundary=None)}
To change the prior use `set_prior` method

Need to speceify a constained prior for switchcos. g1 < g3 < g2

We follow Bilby documentation on how to specefy 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)

14:49 bilby INFO    : Running for label 'switchcos_mix', output will be saved to 'outdir/samba_bivariate_1'

The following settings were                     provided for sampler 
{'sampler': 'ptemcee', 'ntemps': 5, 'nwalkers': 50, 'Tmax': 100, 'burn_in_fixed_discard': 50, 'nsamples': 2000, 'npool': 1}

14:49 bilby INFO    : Analysis priors:

14:49 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)

14:49 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)

14:49 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)

14:49 bilby INFO    : Analysis likelihood class: <class 'Taweret.sampler.likelihood_wrappers.likelihood_wrapper_for_bilby'>

14:49 bilby INFO    : Analysis likelihood noise evidence: nan

14:49 bilby INFO    : Single likelihood evaluation took nan s

14:49 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}

14:49 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)

14:49 bilby INFO    : Generating pos0 samples

14:49 bilby INFO    : Starting to sample

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14:55 bilby INFO    : Finished sampling

14:55 bilby INFO    : Writing checkpoint and diagnostics

14:55 bilby INFO    : Finished writing checkpoint

14:55 bilby INFO    : Sampling time: 0:06:05.268091

14:55 bilby ERROR   : 

Saving the data has failed with the following message:
Object of type custom_prior_switchcos is not JSON serializable
Data has been dumped to outdir/samba_bivariate_1/switchcos_mix_result.pkl.

14:55 bilby ERROR   : 

Saving the data has failed with the following message:
Object of type custom_prior_switchcos is not JSON serializable
Data has been dumped to outdir/samba_bivariate_1/switchcos_mix_result.pkl.

14:55 bilby INFO    : Summary of results:
nsamples: 4550
ln_noise_evidence:    nan
ln_evidence:  4.467 +/-  0.003
ln_bayes_factor:    nan +/-  0.003

# Posterior of the mixing parameters.
result.plot_corner()
../../_images/4ed1c3e2513744de40c1fbea00461035716a0295af804b15f17b35069850753f.png 
#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()
../../_images/c2e4af5104811c4c5e70d917f19fe340fd348fa55eca446939ce9b583244e8f9.png

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])

(10000, 3)
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)

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

(4550, 100)

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

using provided samples instead of posterior

(5000, 100)

%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')
../../_images/24ee95f5a30c7c22ded9a70c64bc5fc52f1ea9c5a4d7e32f42d321ac3a540074.png 
%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')
../../_images/d79b75c2290553f506d0d1a0dac2c52fd948d8def82bf1b5a3103eebc15f04ca.png
Contents