Any case studies comparing HMC, VI, and Marginal Maximum Likelihood?


I’m looking for some case studies comparing accuracy of HMC, variational inference, and marginal maximum likelihood for hierarchical regression models.

I found this interesting paper which compares stan vs lme4 for mixed effects models. If anyone has found something out there that also looks at variational inference please let me know, thanks!


I tweeted something on this some time ago:

Codes below for the lazy:

import pymc3 as pm
import numpy as np
import matplotlib.pylab as plt

L = np.array([[2, 1]]).T
Sigma = + np.diag([1e-2, 1e-2])
L_chol = np.linalg.cholesky(Sigma)

with pm.Model() as model:
    y = pm.MvNormal('y', mu=np.zeros(2), chol=L_chol, shape=2)
    tr0 = pm.sample(500, chains=1)
    tr1 ='advi').sample(500)
    tr2 ='fullrank_advi').sample(500)
    tr3 ='svgd').sample(500)

plt.plot(tr0['y'][:,0], tr0['y'][:,1], 'o', alpha=.1, label='NUTS')
plt.plot(tr1['y'][:,0], tr1['y'][:,1], 'o', alpha=.1, label='ADVI')
plt.plot(tr2['y'][:,0], tr2['y'][:,1], 'o', alpha=.1, label='FullRank')
plt.plot(tr3['y'][:,0], tr3['y'][:,1], 'o', alpha=.1, label='SVGD')

import matplotlib.pylab as plt
from mpl_toolkits import mplot3d
import numpy as np
import pymc3 as pm

def cust_logp(z):
    return -(1.-z[0])**2 - 100.*(z[1] - z[0]**2)**2

grid = np.mgrid[-2:2:100j,-1:3:100j]
Z = -np.asarray([cust_logp(g) for g in grid.reshape(2, -1).T])
fig = plt.figure()
ax = fig.gca(projection='3d')
surf = ax.plot_surface(grid[0], grid[1], Z.reshape(100,100), cmap='viridis',
                       linewidth=0, antialiased=False)

with pm.Model():
    pm.DensityDist('pot1', logp=cust_logp, shape=(2,))
    tr1 = pm.sample(500, step=pm.NUTS())['pot1']
    tr2 = pm.sample(500, step=pm.Metropolis())['pot1']
    tr3 = (n=50000, method='fullrank_advi').sample(500)['pot1'] #VI, cause whynot

import matplotlib.pylab as plt
_, ax = plt.subplots(1,3,figsize=(15,5), sharex=True, sharey=True)
ax[0].imshow(Z.reshape(100,100), extent=[-1,3,-2,2,]);
ax[0].plot(tr1[:,1], tr1[:,0], 'ro-',alpha=.1)
ax[1].imshow(Z.reshape(100,100), extent=[-1,3,-2,2,]);
ax[1].plot(tr2[:,1], tr2[:,0], 'ro-',alpha=.1)
ax[2].imshow(Z.reshape(100,100), extent=[-1,3,-2,2,]);
ax[2].plot(tr3[:,1], tr3[:,0], 'ro', alpha=.1)

with pm.Model():
    pm.DensityDist('pot1', logp=cust_logp, shape=(2,))

You can also find a case study in for hierarchical linear models, especially comparing different fitting in PyMC3.

Something I would like to point out is that, even though sometimes the estimation of regression coefficients seems to differ among different samplers, the fitting and prediction are actually very close.


Cool! I’ll check it out thanks.