def ci_cmle_is(X, v, theta_grid, alpha_level, T = 100, verbose = False):
cmle_is = np.empty_like(theta_grid)
r = X.sum(1)
c = X.sum(0)
for l, theta_l in enumerate(theta_grid):
logit_P_l = theta_l * v
w_l = np.exp(logit_P_l)
z = cond_a_sample(r, c, w_l, T)
logf = np.empty(T)
for t in range(T):
logQ, logP = z[t][1], z[t][2]
logf[t] = logP - logQ
logkappa = -np.log(T) + logsumexp(logf)
if verbose:
logcvsq = -np.log(T - 1) - 2 * logkappa + \
logsumexp(2 * logabsdiffexp(logf, logkappa))
print 'est. cv^2 = %.2f (T = %d)' % (np.exp(logcvsq), T)
cmle_is[l] = np.sum(np.log(w_l[X])) - logkappa
crit = -0.5 * chi2.ppf(1 - alpha_level, 1)
ci = invert_test(theta_grid, cmle_is - cmle_is.max(), crit)
if params['plot']:
plot_statistics(ax_cmle_is, theta_grid, cmle_is - cmle_is.max(), crit)
cmle_is_coverage_data['cis'].append(ci)
cmle_is_coverage_data['theta_grid'] = theta_grid
cmle_is_coverage_data['crit'] = crit
return ci
def log_partition_is(z, cvsq = False):
"""From importance-weighted sampled, estimate log-partition function."""
T = len(z)
logf = np.empty(T)
for t in range(T):
logf[t] = z[t][2] - z[t][1]
logkappa = -np.log(T) + logsumexp(logf)
if not cvsq:
return logkappa
else:
logcvsq = -np.log(T - 1) - 2 * logkappa + \
logsumexp(2 * logabsdiffexp(logf, logkappa))
return logkappa, logcvsq
def log_partition_is(z, cvsq=False):
"""From importance-weighted sampled, estimate log-partition function."""
T = len(z)
logf = np.empty(T)
for t in range(T):
logf[t] = z[t][2] - z[t][1]
logkappa = -np.log(T) + logsumexp(logf)
if not cvsq:
return logkappa
else:
logcvsq = -np.log(T - 1) - 2 * logkappa + \
logsumexp(2 * logabsdiffexp(logf, logkappa))
return logkappa, logcvsq
def ci_cmle_is(X, v, theta_grid, alpha_level, T = 100, verbose = False):
cmle_is = np.empty_like(theta_grid)
r = X.sum(1)
c = X.sum(0)
for l, theta_l in enumerate(theta_grid):
logit_P_l = theta_l * v
w_l = np.exp(logit_P_l)
z = cond_a_sample(r, c, w_l, T)
logf = np.empty(T)
for t in range(T):
logQ, logP = z[t][1], z[t][2]
logf[t] = logP - logQ
logkappa = -np.log(T) + logsumexp(logf)
if verbose:
logcvsq = -np.log(T - 1) - 2 * logkappa + \
logsumexp(2 * logabsdiffexp(logf, logkappa))
print 'est. cv^2 = %.2f (T = %d)' % (np.exp(logcvsq), T)
cmle_is[l] = np.sum(np.log(w_l[X])) - logkappa
return invert_test(theta_grid, cmle_is - cmle_is.max(),
-0.5 * chi2.ppf(1 - alpha_level, 1))
def ci_cmle_is(X, v, theta_grid, alpha_level, T=100, verbose=False):
cmle_is = np.empty_like(theta_grid)
r = X.sum(1)
c = X.sum(0)
for l, theta_l in enumerate(theta_grid):
logit_P_l = theta_l * v
w_l = np.exp(logit_P_l)
z = cond_a_sample(r, c, w_l, T)
logf = np.empty(T)
for t in range(T):
logQ, logP = z[t][1], z[t][2]
logf[t] = logP - logQ
logkappa = -np.log(T) + logsumexp(logf)
if verbose:
logcvsq = -np.log(T - 1) - 2 * logkappa + \
logsumexp(2 * logabsdiffexp(logf, logkappa))
print 'est. cv^2 = %.2f (T = %d)' % (np.exp(logcvsq), T)
cmle_is[l] = np.sum(np.log(w_l[X])) - logkappa
return invert_test(theta_grid, cmle_is - cmle_is.max(),
-0.5 * chi2.ppf(1 - alpha_level, 1))
def ci_conservative_generic(X,
K,
theta_grid,
alpha_level,
suff,
log_likelihood,
sample,
t,
corrected=True,
two_sided=True,
verbose=False):
L = len(theta_grid)
# Generate samples from the mixture proposal distribution
Y = [sample(theta_grid[np.random.randint(L)]) for k in range(K)]
# Test statistic at observation, for each grid point
t_X = t(X).reshape((L, 1))
if verbose:
print 'X: t_min = %.2f, t_max = %.2f' % (t_X.min(), t_X.max())
# Statistics for the samples from the proposal distribution only
# need to be calculated once...
t_Y = np.empty((L, K + 1))
t_Y[:, K] = 0.0
for k in range(K):
t_Y[:, k] = t(Y[k])
if verbose:
print 'Y_%d: t_min = %.2f, t_max = %.2f' % \
(k, t_Y[:,k].min(), t_Y[:,k].max())
if two_sided:
I_t_Y_plus = t_Y >= t_X
I_t_Y_plus[:, K] = True
I_t_Y_minus = -t_Y >= -t_X
I_t_Y_minus[:, K] = True
# Probabilities under each component of the proposal distribution
# only need to be calculated once...
log_Q_X = np.empty(L)
log_Q_Y = np.empty((L, K))
for l in range(L):
theta_l = theta_grid[l]
log_Q_X[l] = log_likelihood(X, theta_l)
for k in range(K):
log_Q_Y[l, k] = log_likelihood(Y[k], theta_l)
if verbose:
print '%.2f: %.2g, %.2g' % \
(theta_l, np.exp(log_Q_X[l]), np.exp(log_Q_Y[l].max()))
log_Q_sum_X = logsumexp(log_Q_X)
log_Q_sum_Y = np.empty(K)
for k in range(K):
log_Q_sum_Y[k] = logsumexp(log_Q_Y[:, k])
# Step over the grid, calculating approximate p-values
if two_sided:
log_p_plus = np.empty(L)
log_p_minus = np.empty(L)
for l in range(L):
theta_l = theta_grid[l]
log_w_l = np.empty(K + 1)
# X contribution
if corrected:
log_w_l[K] = theta_l * (suff * X).sum() - log_Q_sum_X
else:
log_w_l[K] = -np.inf
# Y contribution
for k in range(K):
log_w_l[k] = theta_l * (suff * Y[k]).sum() - log_Q_sum_Y[k]
if two_sided:
log_p_num_plus = logsumexp(log_w_l[I_t_Y_plus[l]])
log_p_num_minus = logsumexp(log_w_l[I_t_Y_minus[l]])
log_p_denom = logsumexp(log_w_l)
if verbose:
print '%.2f: %.2g (%.2g, %.2g)' % \
(theta_l, log_w_l[K], log_w_l[0:K].min(), log_w_l[0:K].max())
if two_sided:
log_p_plus[l] = log_p_num_plus - log_p_denom
log_p_minus[l] = log_p_num_minus - log_p_denom
if two_sided:
# p_pm = min(1, 2 * min(p_plus, p_minus))
log_p_vals = np.fmin(0, np.log(2) + np.fmin(log_p_plus, log_p_minus))
else:
log_p_vals = np.fmin(0, log_p_minus)
return invert_test(theta_grid, log_p_vals, np.log(alpha_level))
def ci_conservative_generic(X, K, theta_grid, alpha_level,
log_likelihood, sample, t,
verbose = False):
L = len(theta_grid)
# Generate samples from the mixture proposal distribution
Y = []
for k in range(K):
l_k = np.random.randint(L)
theta_k = theta_grid[l_k]
Y.append(sample(theta_k))
# Test statistic at observation
t_X = t(X)
# Statistics for the samples from the proposal distribution only
# need to be calculated once...
t_Y = np.zeros(K + 1)
for k in range(K):
t_Y[k] = t(Y[k])
I_t_Y_plus = t_Y >= t_X
I_t_Y_plus[K] = True
I_t_Y_minus = -t_Y >= -t_X
I_t_Y_minus[K] = True
# Probabilities under each component of the proposal distribution
# only need to be calculated once...
log_Q_X = np.empty(L)
log_Q_Y = np.empty((L,K))
for l in range(L):
theta_l = theta_grid[l]
log_Q_X[l] = log_likelihood(X, theta_l)
for k in range(K):
log_Q_Y[l,k] = log_likelihood(Y[k], theta_l)
if verbose:
print '%.2f: %.2g, %.2g' % \
(theta_l, np.exp(log_Q_X[l]), np.exp(log_Q_Y[l].max()))
log_Q_sum_X = logsumexp(log_Q_X)
log_Q_sum_Y = np.empty(K)
for k in range(K):
log_Q_sum_Y[k] = logsumexp(log_Q_Y[:,k])
# Step over the grid, calculating approximate p-values
log_p_plus = np.empty(L)
log_p_minus = np.empty(L)
for l in range(L):
theta_l = theta_grid[l]
log_w_l = np.empty(K + 1)
# X contribution
log_w_l[K] = (theta_l * t_X) - log_Q_sum_X
# Y contribution
for k in range(K):
log_w_l[k] = (theta_l * t_Y[k]) - log_Q_sum_Y[k]
log_p_num_plus = logsumexp(log_w_l[I_t_Y_plus])
log_p_num_minus = logsumexp(log_w_l[I_t_Y_minus])
log_p_denom = logsumexp(log_w_l)
if verbose:
print '%.2f: %.2g (%.2g, %.2g)' % \
(theta_l, log_w_l[K], log_w_l[0:K].min(), log_w_l[0:K].max())
log_p_plus[l] = log_p_num_plus - log_p_denom
log_p_minus[l] = log_p_num_minus - log_p_denom
# p_pm = min(1, 2 * min(p_plus, p_minus))
log_p_pm = np.fmin(0, np.log(2) + np.fmin(log_p_plus, log_p_minus))
return invert_test(theta_grid, log_p_pm, np.log(alpha_level))
def ci_conservative_generic(X, K, theta_grid, alpha_level,
suff, log_likelihood, sample, t,
corrected = True, two_sided = True,
verbose = False):
L = len(theta_grid)
# Generate samples from the mixture proposal distribution
Y = [sample(theta_grid[np.random.randint(L)]) for k in range(K)]
# Test statistic at observation, for each grid point
t_X = t(X).reshape((L, 1))
if verbose:
print 'X: t_min = %.2f, t_max = %.2f' % (t_X.min(), t_X.max())
# Statistics for the samples from the proposal distribution only
# need to be calculated once...
t_Y = np.empty((L, K+1))
t_Y[:,K] = 0.0
for k in range(K):
t_Y[:,k] = t(Y[k])
if verbose:
print 'Y_%d: t_min = %.2f, t_max = %.2f' % \
(k, t_Y[:,k].min(), t_Y[:,k].max())
if two_sided:
I_t_Y_plus = t_Y >= t_X
I_t_Y_plus[:,K] = True
I_t_Y_minus = -t_Y >= -t_X
I_t_Y_minus[:,K] = True
# Probabilities under each component of the proposal distribution
# only need to be calculated once...
log_Q_X = np.empty(L)
log_Q_Y = np.empty((L,K))
for l in range(L):
theta_l = theta_grid[l]
log_Q_X[l] = log_likelihood(X, theta_l)
for k in range(K):
log_Q_Y[l,k] = log_likelihood(Y[k], theta_l)
if verbose:
print '%.2f: %.2g, %.2g' % \
(theta_l, np.exp(log_Q_X[l]), np.exp(log_Q_Y[l].max()))
log_Q_sum_X = logsumexp(log_Q_X)
log_Q_sum_Y = np.empty(K)
for k in range(K):
log_Q_sum_Y[k] = logsumexp(log_Q_Y[:,k])
# Step over the grid, calculating approximate p-values
if two_sided:
log_p_plus = np.empty(L)
log_p_minus = np.empty(L)
for l in range(L):
theta_l = theta_grid[l]
log_w_l = np.empty(K + 1)
# X contribution
if corrected:
log_w_l[K] = theta_l * (suff * X).sum() - log_Q_sum_X
else:
log_w_l[K] = -np.inf
# Y contribution
for k in range(K):
log_w_l[k] = theta_l * (suff * Y[k]).sum() - log_Q_sum_Y[k]
if two_sided:
log_p_num_plus = logsumexp(log_w_l[I_t_Y_plus[l]])
log_p_num_minus = logsumexp(log_w_l[I_t_Y_minus[l]])
log_p_denom = logsumexp(log_w_l)
if verbose:
print '%.2f: %.2g (%.2g, %.2g)' % \
(theta_l, log_w_l[K], log_w_l[0:K].min(), log_w_l[0:K].max())
if two_sided:
log_p_plus[l] = log_p_num_plus - log_p_denom
log_p_minus[l] = log_p_num_minus - log_p_denom
if two_sided:
# p_pm = min(1, 2 * min(p_plus, p_minus))
log_p_vals = np.fmin(0, np.log(2) + np.fmin(log_p_plus, log_p_minus))
else:
log_p_vals = np.fmin(0, log_p_minus)
return invert_test(theta_grid, log_p_vals, np.log(alpha_level))
def ci_conservative_generic(X,
K,
theta_grid,
alpha_level,
log_likelihood,
sample,
t,
verbose=False):
L = len(theta_grid)
# Generate samples from the mixture proposal distribution
Y = []
for k in range(K):
l_k = np.random.randint(L)
theta_k = theta_grid[l_k]
Y.append(sample(theta_k))
# Test statistic at observation
t_X = t(X)
# Statistics for the samples from the proposal distribution only
# need to be calculated once...
t_Y = np.zeros(K + 1)
for k in range(K):
t_Y[k] = t(Y[k])
I_t_Y_plus = t_Y >= t_X
I_t_Y_plus[K] = True
I_t_Y_minus = -t_Y >= -t_X
I_t_Y_minus[K] = True
# Probabilities under each component of the proposal distribution
# only need to be calculated once...
log_Q_X = np.empty(L)
log_Q_Y = np.empty((L, K))
for l in range(L):
theta_l = theta_grid[l]
log_Q_X[l] = log_likelihood(X, theta_l)
for k in range(K):
log_Q_Y[l, k] = log_likelihood(Y[k], theta_l)
if verbose:
print '%.2f: %.2g, %.2g' % \
(theta_l, np.exp(log_Q_X[l]), np.exp(log_Q_Y[l].max()))
log_Q_sum_X = logsumexp(log_Q_X)
log_Q_sum_Y = np.empty(K)
for k in range(K):
log_Q_sum_Y[k] = logsumexp(log_Q_Y[:, k])
# Step over the grid, calculating approximate p-values
log_p_plus = np.empty(L)
log_p_minus = np.empty(L)
for l in range(L):
theta_l = theta_grid[l]
log_w_l = np.empty(K + 1)
# X contribution
log_w_l[K] = (theta_l * t_X) - log_Q_sum_X
# Y contribution
for k in range(K):
log_w_l[k] = (theta_l * t_Y[k]) - log_Q_sum_Y[k]
log_p_num_plus = logsumexp(log_w_l[I_t_Y_plus])
log_p_num_minus = logsumexp(log_w_l[I_t_Y_minus])
log_p_denom = logsumexp(log_w_l)
if verbose:
print '%.2f: %.2g (%.2g, %.2g)' % \
(theta_l, log_w_l[K], log_w_l[0:K].min(), log_w_l[0:K].max())
log_p_plus[l] = log_p_num_plus - log_p_denom
log_p_minus[l] = log_p_num_minus - log_p_denom
# p_pm = min(1, 2 * min(p_plus, p_minus))
log_p_pm = np.fmin(0, np.log(2) + np.fmin(log_p_plus, log_p_minus))
return invert_test(theta_grid, log_p_pm, np.log(alpha_level))
