def test_frozen(self):
# Test that the frozen and non-frozen inverse Wishart gives the same
# answers
# Construct an arbitrary positive definite scale matrix
dim = 4
scale = np.diag(np.arange(dim) + 1)
scale[np.tril_indices(dim, k=-1)] = np.arange(dim * (dim - 1) / 2)
scale = np.dot(scale.T, scale)
# Construct a collection of positive definite matrices to test the PDF
X = []
for i in range(5):
x = np.diag(np.arange(dim) + (i + 1)**2)
x[np.tril_indices(dim, k=-1)] = np.arange(dim * (dim - 1) / 2)
x = np.dot(x.T, x)
X.append(x)
X = np.array(X).T
# Construct a 1D and 2D set of parameters
parameters = [
(10, 1, np.linspace(0.1, 10, 5)), # 1D case
(10, scale, X)
]
for (df, scale, x) in parameters:
iw = invwishart(df, scale)
assert_equal(iw.var(), invwishart.var(df, scale))
assert_equal(iw.mean(), invwishart.mean(df, scale))
assert_equal(iw.mode(), invwishart.mode(df, scale))
assert_allclose(iw.pdf(x), invwishart.pdf(x, df, scale))
def test_frozen(self):
# Test that the frozen and non-frozen inverse Wishart gives the same
# answers
# Construct an arbitrary positive definite scale matrix
dim = 4
scale = np.diag(np.arange(dim)+1)
scale[np.tril_indices(dim, k=-1)] = np.arange(dim*(dim-1)/2)
scale = np.dot(scale.T, scale)
# Construct a collection of positive definite matrices to test the PDF
X = []
for i in range(5):
x = np.diag(np.arange(dim)+(i+1)**2)
x[np.tril_indices(dim, k=-1)] = np.arange(dim*(dim-1)/2)
x = np.dot(x.T, x)
X.append(x)
X = np.array(X).T
# Construct a 1D and 2D set of parameters
parameters = [
(10, 1, np.linspace(0.1, 10, 5)), # 1D case
(10, scale, X)
]
for (df, scale, x) in parameters:
iw = invwishart(df, scale)
assert_equal(iw.var(), invwishart.var(df, scale))
assert_equal(iw.mean(), invwishart.mean(df, scale))
assert_equal(iw.mode(), invwishart.mode(df, scale))
assert_allclose(iw.pdf(x), invwishart.pdf(x, df, scale))
# -
# ## [Input parameters](https://www.arpm.co/lab/redirect.php?permalink=s_reg_lfm_bayes_prior_niw-parameters)
beta_pri = 0.5 # prior location parameter of the loadings
sigma2_pri = 3 # prior location parameter of the variance
sigma2_zpri = 2 # prior dispersion parameter of the loadings
t_pri = 3 # confidence on the prior loadings
v_pri = 10 # confidence on the prior variance
k_ = 500 # number of grid points
# ## [Step 1](https://www.arpm.co/lab/redirect.php?permalink=s_reg_lfm_bayes_prior_niw-implementation-step01): Compute the expectation and standard deviations of Sigma2 and B
# +
exp_sigma2 = invwishart.mean(v_pri, v_pri * sigma2_pri)
std_sigma2 = np.sqrt(invwishart.var(v_pri, v_pri * sigma2_pri))
exp_beta = beta_pri
std_beta = np.sqrt(sigma2_pri / (sigma2_zpri * t_pri) * v_pri / (v_pri - 2.))
# -
# ## [Step 2](https://www.arpm.co/lab/redirect.php?permalink=s_reg_lfm_bayes_prior_niw-implementation-step02): Compute the marginal pdf of Sigma2
s = np.linspace(0.1, exp_sigma2 + 3 * std_sigma2, k_)
f_sigma2 = invwishart.pdf(s, v_pri, v_pri * sigma2_pri)
# ## [Step 3](https://www.arpm.co/lab/redirect.php?permalink=s_reg_lfm_bayes_prior_niw-implementation-step03): Compute the marginal pdf of B
b = np.linspace(exp_beta - 3 * std_beta, exp_beta + 3 * std_beta, k_)
f_beta = t.pdf((b - beta_pri) / np.sqrt(sigma2_pri / (sigma2_zpri * t_pri)),
v_pri) / np.sqrt(sigma2_pri / (sigma2_zpri * t_pri))
# ## [Step 2](https://www.arpm.co/lab/redirect.php?permalink=s_bayes_posterior_niw-implementation-step02): Compute the parameters of the posterior distribution
mu_pos = (t_pri / (t_pri + t_)) * mu_pri + (t_ / (t_pri + t_)) * mu_hat
sigma2_pos = (v_pri / (v_pri + t_)) * sigma2_pri + \
(t_ / (v_pri + t_)) * sigma2_hat + \
(mu_pri - mu_hat) ** 2 / ((v_pri + t_) * (1 / t_ + 1 / t_pri))
t_pos = t_pri + t_
v_pos = v_pri + t_
# ## [Step 3](https://www.arpm.co/lab/redirect.php?permalink=s_bayes_posterior_niw-implementation-step03): Compute the mean and standard deviations of the sample, prior and posterior distributions of Sigma2
exp_sigma2_hat = wishart.mean(t_ - 1, sigma2_hat / t_)
std_sigma2_hat = np.sqrt(wishart.var(t_ - 1, sigma2_hat / t_))
exp_sigma2_pri = invwishart.mean(v_pri, v_pri * sigma2_pri)
std_sigma2_pri = np.sqrt(invwishart.var(v_pri, v_pri * sigma2_pri))
exp_sigma2_pos = invwishart.mean(v_pos, v_pos * sigma2_pos)
std_sigma2_pos = np.sqrt(invwishart.var(v_pos, v_pos * sigma2_pos))
# ## [Step 4](https://www.arpm.co/lab/redirect.php?permalink=s_bayes_posterior_niw-implementation-step04): Compute marginal pdfs of the sample, prior and posterior distributions of Sigma2
# +
s_max = np.max([
exp_sigma2_hat + 3. * std_sigma2_hat, exp_sigma2_pri + 3. * std_sigma2_pri,
exp_sigma2_pos + 3. * std_sigma2_pos
])
s = np.linspace(0.01, s_max, k_) # grid
f_sigma2_hat = wishart.pdf(s, t_ - 1, sigma2_hat / t_) # sample pdf
f_sigma2_pri = invwishart.pdf(s, v_pri, v_pri * sigma2_pri) # prior pdf
f_sigma2_pos = invwishart.pdf(s, v_pos, v_pos * sigma2_pos) # posterior pdf