Primero, algunas notas que me ayudan a dar sentido al artículo de SVI:
kμk,τkηg
μ,τ∼N(μ|γ,τ(2α−1)Ga(τ|α,β)
η0=2α−1η1=γ∗(2α−1)η2=2β+γ2(2α−1)a,b,mα,β,μ
μk,τkη˙+⟨∑Nzn,k∑Nzn,kxN∑Nzn,kx2n⟩η˙zn,kexpln(p))∏Np(xn|zn,α,β,γ)=∏N∏K(p(xn|αk,βk,γk))zn,k
Con eso, podemos completar el paso (5) del pseudocódigo SVI con:
ϕn,k∝exp(ln(π)+Eqln(p(xn|αk,βk,γk))=exp(ln(π)+Eq[⟨μkτk,−τ2⟩⋅⟨x,x2⟩−μ2τ−lnτ2)]
Actualizar los parámetros globales es más fácil, ya que cada parámetro corresponde a un recuento de los datos o una de sus estadísticas suficientes:
λ^=η˙+Nϕn⟨1,x,x2⟩
0a,b,mα,β,μ
#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
Created on Sun Aug 12 12:49:15 2018
@author: SeanEaster
"""
import numpy as np
from matplotlib import pylab as plt
from scipy.stats import t
from scipy.special import digamma
# These are priors for mu, alpha and beta
def calc_rho(t, delay=16,forgetting=1.):
return np.power(t + delay, -forgetting)
m_prior, alpha_prior, beta_prior = 0., 1., 1.
eta_0 = 2 * alpha_prior - 1
eta_1 = m_prior * (2 * alpha_prior - 1)
eta_2 = 2 * beta_prior + np.power(m_prior, 2.) * (2 * alpha_prior - 1)
k = 3
eta_shape = (k,3)
eta_prior = np.ones(eta_shape)
eta_prior[:,0] = eta_0
eta_prior[:,1] = eta_1
eta_prior[:,2] = eta_2
np.random.seed(123)
size = 1000
dummy_data = np.concatenate((
np.random.normal(-1., scale=.25, size=size),
np.random.normal(0., scale=.25,size=size),
np.random.normal(1., scale=.25, size=size)
))
N = len(dummy_data)
S = 1
# randomly init global params
alpha = np.random.gamma(3., scale=1./3., size=k)
m = np.random.normal(scale=1, size=k)
beta = np.random.gamma(3., scale=1./3., size=k)
eta = np.zeros(eta_shape)
eta[:,0] = 2 * alpha - 1
eta[:,1] = m * eta[:,0]
eta[:,2] = 2. * beta + np.power(m, 2.) * eta[:,0]
phi = np.random.dirichlet(np.ones(k) / k, size = dummy_data.shape[0])
nrows, ncols = 4, 5
total_plots = nrows * ncols
total_iters = np.power(2, total_plots - 1)
iter_idx = 0
x = np.linspace(dummy_data.min(), dummy_data.max(), num=200)
while iter_idx < total_iters:
if np.log2(iter_idx + 1) % 1 == 0:
alpha = 0.5 * (eta[:,0] + 1)
beta = 0.5 * (eta[:,2] - np.power(eta[:,1], 2.) / eta[:,0])
m = eta[:,1] / eta[:,0]
idx = int(np.log2(iter_idx + 1)) + 1
f = plt.subplot(nrows, ncols, idx)
s = np.zeros(x.shape)
for _ in range(k):
y = t.pdf(x, alpha[_], m[_], 2 * beta[_] / (2 * alpha[_] - 1))
s += y
plt.plot(x, y)
plt.plot(x, s)
f.axes.get_xaxis().set_visible(False)
f.axes.get_yaxis().set_visible(False)
# randomly sample data point, update parameters
interm_eta = np.zeros(eta_shape)
for _ in range(S):
datum = np.random.choice(dummy_data, 1)
# mean params for ease of calculating expectations
alpha = 0.5 * ( eta[:,0] + 1)
beta = 0.5 * (eta[:,2] - np.power(eta[:,1], 2) / eta[:,0])
m = eta[:,1] / eta[:,0]
exp_mu = m
exp_tau = alpha / beta
exp_tau_m_sq = 1. / (2 * alpha - 1) + np.power(m, 2.) * alpha / beta
exp_log_tau = digamma(alpha) - np.log(beta)
like_term = datum * (exp_mu * exp_tau) - np.power(datum, 2.) * exp_tau / 2 \
- (0.5 * exp_tau_m_sq - 0.5 * exp_log_tau)
log_phi = np.log(1. / k) + like_term
phi = np.exp(log_phi)
phi = phi / phi.sum()
interm_eta[:, 0] += phi
interm_eta[:, 1] += phi * datum
interm_eta[:, 2] += phi * np.power(datum, 2.)
interm_eta = interm_eta * N / S
interm_eta += eta_prior
rho = calc_rho(iter_idx + 1)
eta = (1 - rho) * eta + rho * interm_eta
iter_idx += 1