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hmc_gaussian.py
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hmc_gaussian.py
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import numpy as np
import matplotlib.pyplot as plt
from matplotlib import rc
def font_stile(a):
rc('font',**{'family':'serif','serif':[a]})
rc('text', usetex=True)
return True
activated = font_stile('Palatino')
def gaussian(mu, sigma):
data = np.random.normal(mu, sigma, 10000)
#for i in range(len(data)):
## if data[i] < 0:
# data[i] = np.random.normal(0,1,1)
return data
transition_model = lambda x: np.random.normal(x,[0.05,0.01],(2,))
def prior(w):
if(w[0]<=0 or w[1] <=0):
return 0
else:
return 1
def manual_log_like_normal(x,data):
#x[0]=mu, x[1]=sigma (new or current)
#data = the observation
val = np.sum(-np.log(x[1] * np.sqrt(2* np.pi) )-((data-x[0])**2) / (2*x[1]**2))
return val
#Defines whether to accept or reject the new sample
def acceptance(x, x_new):
accept=np.random.uniform(0,1)
return (accept < (np.exp(x_new-x)))
def metropolis_hastings(likelihood_computer,prior, transition_model, param_init,iterations,data,acceptance_rule):
x = param_init
accepted = []
rejected = []
for i in range(iterations):
x_new = transition_model(x)
x_lik = likelihood_computer(x,data)
x_new_lik = likelihood_computer(x_new,data)
if (acceptance_rule(x_lik + np.log(prior(x)),x_new_lik+np.log(prior(x_new)))):
x = x_new
accepted.append(x_new)
else:
rejected.append(x_new)
return np.array(accepted), np.array(rejected)
def grad_analitic(data, q):
grad1 = np.sum( - (data-q[0]) / (q[1]**2))
grad2 = np.sum( 1/q[1] - (data-q[0])**2/ (q[1]**3))
return np.array([grad1, grad2])
def leapfrog(q, p, data, path_len, step_size, massinverse):
for _ in range(path_len):
p = p - 0.5*step_size*grad_analitic(data, q)
q = q + step_size*np.matmul(p, massinverse)
#print(np.shape(q), np.shape(np.matmul(p, massinverse)))
p = p - 0.5*step_size*grad_analitic(data, q)
#print(step_size*np.sum( - (data-q[0]) / (q[1]**2)))
return np.array(q), np.array(p)
def kinematic_computer(p, massinverse):
k = 0.5*np.matmul(p, np.matmul(massinverse, p))
#print(k)
#k = 0.5*(p[0]**2)/mass[0] + 0.5*(p[1]**2)/mass[1]
return k
def acceptance_h(Up, Uc, Kp, Kc):
val = np.exp( Uc - Up - Kp + Kc )
accept=np.random.uniform(0,1)
return (accept < val)
def Hamiltonian_sampler(likelihood_computer,integrator, prior,
param_init_real,iterations,data,
acceptance_rule,path_len,step,mass):
x = param_init_real
accepted = []
rejected = []
acceptedp = []
rejectedp = []
massinverse = np.linalg.inv(mass)
print('inverse mass',massinverse)
for i in range(iterations):
pm = np.random.normal(0,1,(2,))
x_new, p_new = integrator(x, pm, data, path_len, step, massinverse)
u_lik = -likelihood_computer(x,data)
u_new_lik = -likelihood_computer(x_new,data)
k_lik = kinematic_computer(pm, massinverse)
k_new_lik = kinematic_computer(p_new, massinverse)
#print ('hmc',x_new, p_new)
if (acceptance_h(u_new_lik, u_lik ,k_new_lik , k_lik)):
x = x_new
accepted.append(x_new)
acceptedp.append(p_new)
else:
rejected.append(x_new)
rejectedp.append(p_new)
return np.array(accepted), np.array(rejected),np.array(acceptedp), np.array(rejectedp)
def lag_calculator(aray1, aray2, lagmax):
mean1 = np.mean(aray1)
mean2 = np.mean(aray2)
sigma1 =0.
sigma2 =0.
for m in range (0, len(aray1)):
sigma1 += (aray1[m] - mean1 )*(aray1[m] - mean1 )
sigma2 += (aray2[m] - mean2 )*(aray2[m] - mean2 )
lag1 = np.zeros(lagmax)
lag2 = np.zeros(lagmax)
for k in range(0, lagmax):
for l in range (0, len(aray1) - k):
lag1[k] += (aray1[l] - mean1 )*(aray1[l+k] - mean1 )/sigma1
lag2[k] += (aray2[l] - mean2 )*(aray2[l+k] - mean2 )/sigma2
index = np.arange(lagmax)
return index, lag1, lag2
# Mock data
data = gaussian(3, 4)
#plt.hist(data, bins = 100)
#plt.show()
# HMC parameters
path_len = 6
mass = [[2,1.5],[1.5,2]]
step = 0.003
# MCMC input
samps_num = 50000
start = [4,3]
# Metropolis sampling
samples_accepted, samples_rejected = metropolis_hastings(manual_log_like_normal,
prior,transition_model,start, samps_num,data,acceptance)
# HMC sampling
samples_accepted_h, samples_rejected_h, samples_accepted_ph, samples_rejected_ph = Hamiltonian_sampler(manual_log_like_normal,
leapfrog,prior, start, samps_num,data,
acceptance,path_len,step , mass)
# Variying paramaters sector
#fig, (ax1, ax2) = plt.subplots(1, 2,figsize=(10,7))
#fig.suptitle('Correlation')
#for i in range (0,6):
# print('iteration:', i+1)
#a = (i+1)*0.001
# a = i*0.25
# print(a)
# samples_accepted_h, samples_rejected_h, samples_accepted_ph, samples_rejected_ph = Hamiltonian_sampler(manual_log_like_normal,
# leapfrog,prior, start, samps_num,data,
# acceptance, path_len, step , [[2,a],[a,2]] )
#
#print('iteration:', i+1,(i+1)*0.001)
# plt.plot(samples_accepted_h[:,0], samples_accepted_h[:,1], label = r'off diag mass element = '+str(a))
# #plt.plot(samples_accepted_h2[:,0], samples_accepted_h2[:,1], color = 'red')
# a1,b1,c1 = lag_calculator(samples_accepted_h[:,0],samples_accepted_h[:,0], 100)
# a2,b2,c2 = lag_calculator(samples_accepted_h[:,1],samples_accepted_h[:,1], 200)
# ax1.plot(a1,c1, label = r'Integrations = '+str(i+3))
# ax2.plot(a2,c2,label = r'Integrations = '+str(i+3))
#ax1.legend(loc = 'upper right')
#ax2.legend(loc = 'upper right')
#plt.legend(loc='upper right')
#plt.savefig('off_diag_mass_effect.png')
#plt.show()
# Shapes useful for efficiency
print(np.shape(samples_accepted_h), np.shape(samples_accepted))
# Plotting
# Precontour plots (Not 68% and 95%)
plt.scatter(samples_accepted[:,0], samples_accepted[:,1], color = 'red', marker = '+', label = 'MH')
plt.plot(samples_accepted[:,0], samples_accepted[:,1] , color = 'red')
plt.scatter(samples_accepted_h[:,0], samples_accepted_h[:,1], color = 'blue', marker = '+', label = 'HMC')
plt.plot(samples_accepted_h[:,0], samples_accepted_h[:,1], color = 'blue')
# Reality check params
plt.axvline(start[0], color='black', linestyle = 'dashed')
plt.axhline(start[1], color='black', label = 'Starting', linestyle = 'dashed')
plt.axvline(3, color='red', linestyle = 'dashed')
plt.axhline(4, color='red', label = 'True value', linestyle = 'dashed')
#plt.xlim([2.5, 4.5])
#plt.ylim([2.5, 4.5])
plt.ylabel(r'$\sigma$')
plt.xlabel(r'$\mu$')
plt.legend(loc = 'upper right')
plt.savefig('HMC.png')
plt.close()
#Phase space plotting
fig, (ax1, ax2) = plt.subplots(1, 2,figsize=(10,7))
fig.suptitle('Phase space diagrams')
ax1.scatter(samples_accepted_h[:,0], samples_accepted_ph[:,0], label=r'Accepted samples HMC')
ax2.scatter(samples_accepted_h[:,1], samples_accepted_ph[:,1],label=r'Accepted samples HMC')
ax1.set(xlabel=r'$\mu$', ylabel=r'$P_{\mu}$')
ax2.set(xlabel=r'$\sigma$', ylabel=r'$P_{\sigma}$')
plt.legend(loc = 'upper right')
plt.savefig('HMC_phase.png')
plt.close()
# Correlation lengths
a1,b1,c1 = lag_calculator(samples_accepted[:,0],samples_accepted_h[:,0], 100)
a2,b2,c2 = lag_calculator(samples_accepted[:,1],samples_accepted_h[:,1], 200)
fig, (ax1, ax2) = plt.subplots(1, 2,figsize=(10,7))
fig.suptitle('Correlation')
ax1.plot(a1,b1, label=r'MH' + r'$\mu$')
ax1.plot(a1,c1, label=r'HMC'+ r'$\mu$')
ax1.legend(loc = 'upper right')
ax2.plot(a2,b2 ,label=r'MH'+ r'$\sigma$')
ax2.plot(a2,c2 ,label=r'HMC'+ r'$\sigma$')
ax2.legend(loc = 'upper right')
plt.savefig('correlation_length.png')
plt.close()
## Convergence test: Trace plots
a = 5000
fig, (ax1, ax2) = plt.subplots(1, 2,figsize=(10,7))
fig.suptitle('Traceplots')
ax1.plot(np.arange(a),samples_accepted[:a,0], label=r'MH' + r'$\mu$')
ax1.plot(np.arange(a),samples_accepted_h[:a,0], label=r'HMC'+ r'$\mu$')
ax1.legend(loc = 'upper right')
ax2.plot(np.arange(a),samples_accepted[:a,1] ,label=r'MH'+ r'$\sigma$')
ax2.plot(np.arange(a),samples_accepted_h[:a,1] ,label=r'HMC'+ r'$\sigma$')
ax2.legend(loc = 'upper right')
plt.savefig('TracePlots.png')
plt.close()