MCMC: Hamiltonian sampling

Florent Leclercq,
Imperial Centre for Inference and Cosmology, Imperial College London,
[email protected]

In [1]:
import numpy as np
from scipy.integrate import quad
from scipy.stats import norm
from matplotlib import pyplot as plt
from matplotlib.ticker import NullFormatter
from cycler import cycler
np.random.seed(123456)
%matplotlib inline
plt.rcParams.update({'lines.linewidth': 2})

The target pdf

In [2]:
def target_joint(x,y):
    return x*x * np.exp(-x*y*y -y*y +2.*y -4.*x)
def target_marginal_x(x):
    return x*x/np.sqrt(x+1) * np.exp(-4.*x -1./(x+1.))
def target_marginal_y(y):
    return np.exp(-y*y+2.*y) / (y*y+4.)**3
def psi(x,y):
    # psi(x,y)=-ln(target_joint(x,y))
    return np.where(x>1e-10, -2.*np.log(np.fabs(x)) +x*y*y +y*y -2.*y +4.*x, +1e8)
def dpsi_dx(x,y):
    return np.where(x>1e-10, -2./x +y*y* +4., +1e8)
def dpsi_dy(x,y):
    return 2.*x*y +2.*y -2.
In [3]:
# Normalization of the marginals
Nx=quad(target_marginal_x,0.0001,100.)[0]
Ny=quad(target_marginal_y,-100.,100.)[0]
In [4]:
xmin=0.01
xmax=2.
ymin=-1.
ymax=2.5
x=np.linspace(xmin,xmax,1000)
y=np.linspace(ymin,ymax,1000)
X,Y=np.meshgrid(x,y)
Z=target_joint(X,Y)
Psi=psi(X,Y)
dPsi_dx=dpsi_dx(X,Y)
dPsi_dy=dpsi_dy(X,Y)
In [5]:
f, (ax1, ax2) = plt.subplots(1, 2, figsize=(12,12))
f.subplots_adjust(wspace=0.15)
ax1.set_xlim(xmin,xmax)
ax1.set_ylim(ymin,ymax)
ax1.set_xlabel("$x$")
ax1.set_ylabel("$y$")
ax1.imshow(Z,extent=(xmin,xmax,ymin,ymax),origin='lower',cmap='YlGnBu')
ax1.contour(X,Y,Z,extent=(xmin,xmax,ymin,ymax),origin='lower',cmap='viridis_r')
ax1.set_title("Target pdf")
ax2.set_xlim(xmin,xmax)
ax2.set_ylim(ymin,ymax)
ax2.set_xlabel("$x$")
ax2.imshow(Psi,extent=(xmin,xmax,ymin,ymax),origin='lower',cmap='RdPu')
ax2.contour(X,Y,Psi,extent=(xmin,xmax,ymin,ymax),origin='lower',cmap='inferno_r')
ax2.set_title("Potential $\psi$")
plt.show()
In [6]:
f, (ax1, ax2) = plt.subplots(1, 2, figsize=(12,12))
f.subplots_adjust(wspace=0.1)
ax1.set_xlim(xmin,xmax)
ax1.set_ylim(ymin,ymax)
ax1.set_xlabel("$x$")
ax1.set_ylabel("$y$")
ax1.imshow(dPsi_dx,extent=(xmin,xmax,ymin,ymax),origin='lower',cmap='RdPu')
ax1.contour(X,Y,dPsi_dx,extent=(xmin,xmax,ymin,ymax),origin='lower',cmap='inferno_r')
ax1.set_title("$\partial \psi / \partial x$")
ax2.set_xlim(xmin,xmax)
ax2.set_ylim(ymin,ymax)
ax2.set_xlabel("$x$")
ax2.imshow(dPsi_dy,extent=(xmin,xmax,ymin,ymax),origin='lower',cmap='RdPu')
ax2.contour(X,Y,dPsi_dy,extent=(xmin,xmax,ymin,ymax),origin='lower',cmap='inferno_r')
ax2.set_title("$\partial \psi / \partial y$")
plt.show()
In [7]:
# Plot slices of psi
f, (ax1, ax2) = plt.subplots(1, 2, figsize=(14,5))
f.subplots_adjust(wspace=0.4)
ax1.set_prop_cycle(cycler('color', [plt.cm.inferno_r(i) for i in np.linspace(0.1,0.9,11)]))
ax2.set_prop_cycle(cycler('color', [plt.cm.inferno_r(i) for i in np.linspace(0.1,0.9,11)]))
for this_y in np.array((-1.0,-0.65,-0.30,0.05,0.40,0.75,0.10,1.45,1.80,2.15,2.5)):
    ax1.plot(x,psi(x,this_y),label='y='+str(this_y))
ax1.set_xlabel("$x$")
ax1.set_title("Slices of $\psi$ at fixed $y$")
ax1.legend(frameon=False,loc='center left',bbox_to_anchor=(1, 0.5))
for this_x in np.array((0.1,0.2,0.4,0.6,0.8,1.0,1.2,1.4,1.6,1.8,2.0)):
    ax2.plot(y,psi(this_x,y),label='x='+str(this_x))
ax2.set_xlabel("$y$")
ax2.set_title("Slices of $\psi$ at fixed $x$")
ax2.legend(frameon=False,loc='center left',bbox_to_anchor=(1, 0.5))
plt.show()
In [8]:
# Plot slices of dpsi/dx
f, (ax1, ax2) = plt.subplots(1, 2, figsize=(14,5))
f.subplots_adjust(wspace=0.4)
ax1.set_prop_cycle(cycler('color', [plt.cm.inferno_r(i) for i in np.linspace(0.1,0.9,11)]))
ax2.set_prop_cycle(cycler('color', [plt.cm.inferno_r(i) for i in np.linspace(0.1,0.9,11)]))
for this_y in np.array((-1.0,-0.65,-0.30,0.05,0.40,0.75,0.10,1.45,1.80,2.15,2.5)):
    ax1.plot(x,dpsi_dx(x,this_y),label='y='+str(this_y))
ax1.set_xlabel("$x$")
ax1.set_title("$\partial \psi /\partial x$ at fixed $y$")
ax1.legend(frameon=False,loc='center left',bbox_to_anchor=(1, 0.5))
for this_x in np.array((0.1,0.2,0.4,0.6,0.8,1.0,1.2,1.4,1.6,1.8,2.0)):
    ax2.plot(y,dpsi_dx(this_x,y),label='x='+str(this_x))
ax2.set_xlabel("$y$")
ax2.set_title("$\partial \psi /\partial x$ at fixed $x$")
ax2.legend(frameon=False,loc='center left',bbox_to_anchor=(1, 0.5))
plt.show()
In [9]:
# Plot slices of dpsi/dy
f, (ax1, ax2) = plt.subplots(1, 2, figsize=(14,5))
f.subplots_adjust(wspace=0.4)
ax1.set_prop_cycle(cycler('color', [plt.cm.inferno_r(i) for i in np.linspace(0.1,0.9,11)]))
ax2.set_prop_cycle(cycler('color', [plt.cm.inferno_r(i) for i in np.linspace(0.1,0.9,11)]))
for this_y in np.array((-1.0,-0.65,-0.30,0.05,0.40,0.75,0.10,1.45,1.80,2.15,2.5)):
    ax1.plot(x,dpsi_dy(x,this_y),label='y='+str(this_y))
ax1.set_xlabel("$x$")
ax1.set_title("$\partial \psi /\partial y$ at fixed $y$")
ax1.legend(frameon=False,loc='center left',bbox_to_anchor=(1, 0.5))
for this_x in np.array((0.1,0.2,0.4,0.6,0.8,1.0,1.2,1.4,1.6,1.8,2.0)):
    ax2.plot(y,dpsi_dy(this_x,y),label='x='+str(this_x))
ax2.set_xlabel("$y$")
ax2.set_title("$\partial \psi /\partial y$ at fixed $x$")
ax2.legend(frameon=False,loc='center left',bbox_to_anchor=(1, 0.5))
plt.show()

Hamiltonian sampling

In [10]:
def Hamiltonian_sampler(psi,dpsi_dx,dpsi_dy,MassMatrix,stepstd,Ntries,x_start,y_start):
    InvMassMatrix=np.linalg.inv(MassMatrix)
    Naccepted=0
    x=x_start
    y=y_start
    samples_x=np.zeros(Ntries+1)
    samples_x[0]=x
    samples_y=np.zeros(Ntries+1)
    samples_y[0]=y
    for i in range(Ntries):
        # compute potential energy and gradient
        old_x = x
        old_y = y
        old_psi = psi(old_x,old_y)
        dpsidx = dpsi_dx(old_x,old_y)
        dpsidy = dpsi_dy(old_x,old_y)
        
        # randomly draw momenta
        p_x = norm(0.,1.).rvs()
        p_y = norm(0.,1.).rvs()
        p = np.array((p_x,p_y))
        
        # compute kinetic energy
        old_K = p.T.dot(InvMassMatrix).dot(p)/2.
        
        # compute Hamiltonian
        old_H = old_K + old_psi
        
        # do 3 leapfrog step
        for tau in range(3):
            # draw stepsize
            stepsize = norm(0.,stepstd).rvs()
            
            # Kick: make half step in p_x, p_y
            p_x -= stepsize*dpsidx/2.0
            p_y -= stepsize*dpsidy/2.0
            # compute velocities
            p = np.array((p_x,p_y))
            v_x,v_y = InvMassMatrix.dot(p)
            # Drift: make full step in (x,y)
            new_x = old_x+stepsize*v_x
            new_y = old_y+stepsize*v_y
            # compute new gradient
            dpsidx = dpsi_dx(new_x,new_y)
            dpsidy = dpsi_dy(new_x,new_y)
            # Kick: make half step in p_x, p_y
            p_x -= stepsize*dpsidx/2.0
            p_y -= stepsize*dpsidy/2.0
            p = np.array((p_x,p_y))
        
        # compute new energy and Hamiltonian
        new_psi = psi(new_x,new_y)
        new_K = p.T.dot(InvMassMatrix).dot(p)/2.
        new_H = new_K + new_psi
        dH = new_H - old_H
        
        # accept/reject new candidate x,y using the standard Monte Carlo rule
        if(x<0.):
            accept=False
        else:
            if(dH<0.0):
                accept=True
            else:
                a = np.exp(-dH)
                u = np.random.uniform()
                if(u < a):
                    accept=True
                else:
                    accept=False
                    
        if(accept):
            samples_x[i+1]=x=new_x
            samples_y[i+1]=y=new_y
            Naccepted+=1
        else:
            samples_x[i+1]=x=old_x
            samples_y[i+1]=y=old_y

    return Naccepted, samples_x, samples_y
In [11]:
Ntries=3000
x_start=1.8
y_start=-0.8
MassMatrix=np.array([[1., 0.], [0., 1.]])
stepstd=0.1
Naccepted,samples_x,samples_y=Hamiltonian_sampler(psi,dpsi_dx,dpsi_dy,MassMatrix,stepstd,Ntries,x_start,y_start)
In [12]:
fraction_accepted=float(Naccepted)/Ntries
fraction_accepted
Out[12]:
0.8406666666666667
In [13]:
f, (ax1, ax2) = plt.subplots(1, 2, figsize=(10,5))
f.subplots_adjust(wspace=0.2)
ax1.set_xlim(0,Ntries)
ax1.set_ylabel("$x$")
ax1.scatter(np.arange(Ntries+1),samples_x,color='C4',marker='.')
ax2.set_xlim(0,Ntries)
ax2.set_ylabel("$y$")
ax2.scatter(np.arange(Ntries+1),samples_y,color='C2',marker='.')
plt.show()
In [14]:
nullfmt = NullFormatter()         # no labels

# definitions for the axes
left, width = 0., xmax-xmin
bottom, height = 0., ymax-ymin
left_h = left + width + 0.1
bottom_h = bottom + height + 0.1

rect_pdf = [left, bottom, width, height]
rect_pdfx = [left, bottom_h, width, 1.]
rect_pdfy = [left_h, bottom, 1., height]

# start with a rectangular Figure
plt.figure(1, figsize=(2, 2))

ax = plt.axes(rect_pdf)
axpdfx = plt.axes(rect_pdfx)
axpdfy = plt.axes(rect_pdfy)

# no labels
axpdfx.xaxis.set_major_formatter(nullfmt)
axpdfy.yaxis.set_major_formatter(nullfmt)

# the scatter plot:
ax.set_xlim(xmin,xmax)
ax.set_ylim(ymin,ymax)
ax.set_xlabel("$x$")
ax.set_ylabel("$y$")
ax.imshow(Z,extent=(xmin,xmax,ymin,ymax),origin='lower',cmap='YlGnBu')
ax.contour(X,Y,Z,extent=(xmin,xmax,ymin,ymax),origin='lower',cmap='viridis_r')
ax.scatter(samples_x,samples_y,marker='.',color='black')

# the histograms:
axpdfx.set_xlim(xmin,xmax)
axpdfx.set_ylim([0,1.4])
axpdfx.plot(x,target_marginal_x(x)/Nx,color='C4')
axpdfx.hist(samples_x,40,density=True,histtype='step',color='C4')

axpdfy.set_xlim([0,0.8])
axpdfy.set_ylim(ymin,ymax)
axpdfy.plot(target_marginal_y(y)/Ny,y,color='C2')
axpdfy.hist(samples_y,40,density=True,histtype='step',color='C2',orientation='horizontal')

axpdfx.set_title("Hamiltonian sampling")
plt.show()

Checking trajectories

In [15]:
Ntries=1000
MassMatrix=np.array([[1., 0.], [0., 1.]])
stepstd=0.03
In [16]:
x_start_1=1.8
y_start_1=-0.8
N1,sx_1,sy_1=Hamiltonian_sampler(psi,dpsi_dx,dpsi_dy,MassMatrix,stepstd,Ntries,x_start_1,y_start_1)
In [17]:
x_start_2=1.5
y_start_2=2.0
N2,sx_2,sy_2=Hamiltonian_sampler(psi,dpsi_dx,dpsi_dy,MassMatrix,stepstd,Ntries,x_start_2,y_start_2)
In [18]:
x_start_3=0.1
y_start_3=-0.7
N3,sx_3,sy_3=Hamiltonian_sampler(psi,dpsi_dx,dpsi_dy,MassMatrix,stepstd,Ntries,x_start_3,y_start_3)
In [19]:
x_start_4=0.2
y_start_4=2.2
N4,sx_4,sy_4=Hamiltonian_sampler(psi,dpsi_dx,dpsi_dy,MassMatrix,stepstd,Ntries,x_start_4,y_start_4)
In [20]:
plt.figure(figsize=(12,12))
plt.xlim([xmin,xmax])
plt.ylim([ymin,ymax])
plt.xlabel("$x$")
plt.ylabel("$y$")
plt.imshow(Z,extent=(xmin,xmax,ymin,ymax),origin='lower',cmap='YlGnBu')
plt.contour(X,Y,Z,extent=(xmin,xmax,ymin,ymax),origin='lower',cmap='viridis_r')
plt.scatter(x_start_1,y_start_1,s=500,marker='*',color='C0')
plt.scatter(sx_1,sy_1,marker='.',color='C0')
plt.scatter(x_start_2,y_start_2,s=500,marker='*',color='C1')
plt.scatter(sx_2,sy_2,marker='.',color='C1')
plt.scatter(x_start_3,y_start_3,s=500,marker='*',color='C3')
plt.scatter(sx_3,sy_3,marker='.',color='C3')
plt.scatter(x_start_4,y_start_4,s=500,marker='*',color='C5')
plt.scatter(sx_4,sy_4,marker='.',color='C5')
plt.title("Different trajectories")
plt.show()