我已经实施了贝叶斯概率矩阵分解 https://www.cs.toronto.edu/~amnih/papers/bpmf.pdf算法使用pymc3在Python中。我还实现了它的前身，概率矩阵分解（PMF）。看我之前的问题 https://stats.stackexchange.com/questions/146547/pymc3-implementation-of-probabilistic-matrix-factorization-pmf-map-produces-a以供参考此处使用的数据。
我在使用 NUTS 采样器绘制 MCMC 样本时遇到问题。我使用 PMF 中的 MAP 初始化模型参数，并使用高斯随机绘制的超参数在 0 附近进行初始化。但是，我得到了PositiveDefiniteError为采样器设置步骤对象时。我已经验证 PMF 的 MAP 估计是合理的，因此我预计它与超参数的初始化方式有关。 PMF 模型如下：
import pymc3 as pm
import numpy as np
import pandas as pd
import theano
import scipy as sp
data = pd.read_csv('jester-dense-subset-100x20.csv')
n, m = data.shape
test_size = m / 10
train_size = m - test_size
train = data.copy()
train.ix[:,train_size:] = np.nan # remove test set data
train[train.isnull()] = train.mean().mean() # mean value imputation
train = train.values
test = data.copy()
test.ix[:,:train_size] = np.nan # remove train set data
test = test.values
# Low precision reflects uncertainty; prevents overfitting
alpha_u = alpha_v = 1/np.var(train)
alpha = np.ones((n,m)) * 2 # fixed precision for likelihood function
dim = 10 # dimensionality
# Specify the model.
with pm.Model() as pmf:
pmf_U = pm.MvNormal('U', mu=0, tau=alpha_u * np.eye(dim),
shape=(n, dim), testval=np.random.randn(n, dim)*.01)
pmf_V = pm.MvNormal('V', mu=0, tau=alpha_v * np.eye(dim),
shape=(m, dim), testval=np.random.randn(m, dim)*.01)
pmf_R = pm.Normal('R', mu=theano.tensor.dot(pmf_U, pmf_V.T),
tau=alpha, observed=train)
# Find mode of posterior using optimization
start = pm.find_MAP(fmin=sp.optimize.fmin_powell)
这是 BPMF：
n, m = data.shape
dim = 10 # dimensionality
beta_0 = 1 # scaling factor for lambdas; unclear on its use
alpha = np.ones((n,m)) * 2 # fixed precision for likelihood function
logging.info('building the BPMF model')
std = .05 # how much noise to use for model initialization
with pm.Model() as bpmf:
# Specify user feature matrix
lambda_u = pm.Wishart(
'lambda_u', n=dim, V=np.eye(dim), shape=(dim, dim),
testval=np.random.randn(dim, dim) * std)
mu_u = pm.Normal(
'mu_u', mu=0, tau=beta_0 * lambda_u, shape=dim,
testval=np.random.randn(dim) * std)
U = pm.MvNormal(
'U', mu=mu_u, tau=lambda_u, shape=(n, dim),
testval=np.random.randn(n, dim) * std)
# Specify item feature matrix
lambda_v = pm.Wishart(
'lambda_v', n=dim, V=np.eye(dim), shape=(dim, dim),
testval=np.random.randn(dim, dim) * std)
mu_v = pm.Normal(
'mu_v', mu=0, tau=beta_0 * lambda_v, shape=dim,
testval=np.random.randn(dim) * std)
V = pm.MvNormal(
'V', mu=mu_v, tau=lambda_v, shape=(m, dim),
testval=np.random.randn(m, dim) * std)
# Specify rating likelihood function
R = pm.Normal(
'R', mu=theano.tensor.dot(U, V.T), tau=alpha,
observed=train)
# `start` is the start dictionary obtained from running find_MAP for PMF.
for key in bpmf.test_point:
if key not in start:
start[key] = bpmf.test_point[key]
with bpmf:
step = pm.NUTS(scaling=start)
在最后一行，我收到以下错误：
PositiveDefiniteError: Scaling is not positive definite. Simple check failed. Diagonal contains negatives. Check indexes [ 0 2 ... 2206 2207 ]
据我了解，我不能使用find_MAP使用具有超先验（如 BPMF）的模型。这就是为什么我尝试使用 PMF 中的 MAP 值进行初始化，PMF 使用 U 和 V 上参数的点估计，而不是参数化的超先验。
