2.1. Biased initial samples#
This example walks through correcting a biased set of initial samples to recover the true posterior using Aspire’s flow-based samplers.
Set up the environment#
Import the core scientific stack along with Aspire utilities for logging, plotting, and handling sample collections.
import matplotlib.pyplot as plt
import numpy as np
from pathlib import Path
from scipy.stats import norm, uniform, expon
from aspire import Aspire
from aspire.samples import Samples
from aspire.utils import configure_logger, AspireFile
from aspire.plot import plot_comparison
# Configure the logger
configure_logger("INFO")
<Logger aspire (INFO)>
Configure output paths#
Create a reproducible random number generator and ensure the directory used to store figures and results exists.
outdir = Path("outdir") / "biased_example"
outdir.mkdir(parents=True, exist_ok=True)
rng = np.random.default_rng(42)
Describe the target distribution#
Specify the one-dimensional marginals that define our four-dimensional target and wrap them in likelihood and prior helper functions consumed by Aspire.
dists = [
norm(loc=6, scale=0.2),
expon(scale=1),
norm(loc=5, scale=1),
uniform(loc=-10, scale=20),
]
prior_bounds = [
(-10, 10),
(0, 10),
(-10, 10),
(-10, 10),
]
dims = len(dists)
parameters = [f"x_{i}" for i in range(dims)]
def log_likelihood(samples: Samples):
x = samples.x
log_prob = np.zeros(x.shape[0])
for i, dist in enumerate(dists):
log_prob += dist.logpdf(x[:, i])
return log_prob
def log_prior(samples: Samples):
log_prob = np.zeros(samples.x.shape[0])
for i, bounds in enumerate(prior_bounds):
log_prob += uniform(bounds[0], bounds[1] - bounds[0]).logpdf(samples.x[:, i])
return log_prob
Draw biased initial samples#
Generate intentionally biased proposal samples alongside reference draws from
the true distribution, then instantiate the Aspire object that will learn a
correcting flow.
n_init = 1_000
x_initial = np.concatenate([
rng.normal(loc=2, scale=1, size=(n_init, 1)),
rng.exponential(scale=1, size=(n_init, 1)),
rng.normal(loc=0, scale=1, size=(n_init, 1)),
# rng.uniform(low=-10, high=10, size=(n_init, 1)),
rng.uniform(low=-10, high=10, size=(n_init, 1)),
], axis=1)
x_true = np.concatenate([
dist.rvs(size=(n_init, 1), random_state=rng) for dist in dists
], axis=1)
dims = x_initial.shape[1]
parameters = [f"x_{i}" for i in range(dims)]
initial_samples = Samples(
x=x_initial,
parameters=parameters,
)
true_samples = Samples(
x=x_true,
parameters=parameters,
)
# Define the parameters and prior bounds
prior_bounds_dict = {p: b for p, b in zip(parameters, prior_bounds)}
# Define the aspire object
poppy = Aspire(
log_likelihood=log_likelihood,
log_prior=log_prior,
dims=dims,
parameters=parameters,
prior_bounds=prior_bounds_dict,
flow_matching=False,
flow_backend="zuko",
bounded_to_unbounded=True,
hidden_features=[32, 32],
device="cpu",
)
Train the flow model#
Fit the normalizing flow to the initial samples and monitor optimisation via the loss curve.
# Fit the flow to the initial samples
history = poppy.fit(
initial_samples,
n_epochs=100,
lr_annealing=True,
batch_size=256,
)
# Plot the loss
fig = history.plot_loss()
fig.savefig(outdir / "loss.png")
plt.show()
2025-12-15 08:47:13,709 - aspire - INFO - Prior bounds: {'x_0': (-10, 10), 'x_1': (0, 10), 'x_2': (-10, 10), 'x_3': (-10, 10)}
2025-12-15 08:47:13,711 - aspire - INFO - Bounded parameters: ['x_0', 'x_1', 'x_2', 'x_3']
2025-12-15 08:47:13,712 - aspire - INFO - Affine transform applied to: ['x_0', 'x_1', 'x_2', 'x_3']
2025-12-15 08:47:13,713 - aspire - INFO - Configuring <class 'aspire.flows.torch.flows.ZukoFlow'> with kwargs: {'hidden_features': [32, 32]}
2025-12-15 08:47:15,504 - aspire - INFO - Initialized normalizing flow:
MAF(
(transform): LazyComposedTransform(
(0): MaskedAutoregressiveTransform(
(base): MonotonicAffineTransform()
(order): [0, 1, 2, 3]
(hyper): MaskedMLP(
(0): MaskedLinear(in_features=4, out_features=32, bias=True)
(1): ReLU()
(2): MaskedLinear(in_features=32, out_features=32, bias=True)
(3): ReLU()
(4): MaskedLinear(in_features=32, out_features=8, bias=True)
)
)
(1): MaskedAutoregressiveTransform(
(base): MonotonicAffineTransform()
(order): [3, 2, 1, 0]
(hyper): MaskedMLP(
(0): MaskedLinear(in_features=4, out_features=32, bias=True)
(1): ReLU()
(2): MaskedLinear(in_features=32, out_features=32, bias=True)
(3): ReLU()
(4): MaskedLinear(in_features=32, out_features=8, bias=True)
)
)
(2): MaskedAutoregressiveTransform(
(base): MonotonicAffineTransform()
(order): [0, 1, 2, 3]
(hyper): MaskedMLP(
(0): MaskedLinear(in_features=4, out_features=32, bias=True)
(1): ReLU()
(2): MaskedLinear(in_features=32, out_features=32, bias=True)
(3): ReLU()
(4): MaskedLinear(in_features=32, out_features=8, bias=True)
)
)
)
(base): UnconditionalDistribution(DiagNormal(loc: torch.Size([4]), scale: torch.Size([4])))
)
2025-12-15 08:47:15,504 - aspire - INFO - Training with 1000 samples
2025-12-15 08:47:15,506 - aspire - INFO - Training on 800 samples, validating on 200 samples.
Epochs: 0%| | 0/100 [00:00<?, ?it/s]
Epochs: 0%| | 0/100 [00:00<?, ?it/s, train_loss=5.7682, val_loss=5.8224]
Epochs: 1%| | 1/100 [00:00<01:20, 1.24it/s, train_loss=5.7682, val_loss=5.8224]
Epochs: 1%| | 1/100 [00:00<01:20, 1.24it/s, train_loss=5.7515, val_loss=5.7682]
Epochs: 1%| | 1/100 [00:00<01:20, 1.24it/s, train_loss=5.7143, val_loss=5.7317]
Epochs: 1%| | 1/100 [00:00<01:20, 1.24it/s, train_loss=5.7225, val_loss=5.7095]
Epochs: 1%| | 1/100 [00:00<01:20, 1.24it/s, train_loss=5.7302, val_loss=5.6977]
Epochs: 1%| | 1/100 [00:00<01:20, 1.24it/s, train_loss=5.6941, val_loss=5.6948]
Epochs: 1%| | 1/100 [00:00<01:20, 1.24it/s, train_loss=5.6558, val_loss=5.6969]
Epochs: 1%| | 1/100 [00:00<01:20, 1.24it/s, train_loss=5.7342, val_loss=5.7007]
Epochs: 8%|▊ | 8/100 [00:00<00:08, 11.45it/s, train_loss=5.7342, val_loss=5.7007]
Epochs: 8%|▊ | 8/100 [00:00<00:08, 11.45it/s, train_loss=5.7078, val_loss=5.7022]
Epochs: 8%|▊ | 8/100 [00:00<00:08, 11.45it/s, train_loss=5.7519, val_loss=5.7030]
Epochs: 8%|▊ | 8/100 [00:00<00:08, 11.45it/s, train_loss=5.6347, val_loss=5.7020]
Epochs: 8%|▊ | 8/100 [00:00<00:08, 11.45it/s, train_loss=5.7032, val_loss=5.7021]
Epochs: 8%|▊ | 8/100 [00:00<00:08, 11.45it/s, train_loss=5.5172, val_loss=5.7019]
Epochs: 8%|▊ | 8/100 [00:00<00:08, 11.45it/s, train_loss=5.6615, val_loss=5.7026]
Epochs: 8%|▊ | 8/100 [00:01<00:08, 11.45it/s, train_loss=5.6178, val_loss=5.7065]
Epochs: 8%|▊ | 8/100 [00:01<00:08, 11.45it/s, train_loss=5.8328, val_loss=5.7099]
Epochs: 16%|█▌ | 16/100 [00:01<00:03, 22.92it/s, train_loss=5.8328, val_loss=5.7099]
Epochs: 16%|█▌ | 16/100 [00:01<00:03, 22.92it/s, train_loss=5.6534, val_loss=5.7151]
Epochs: 16%|█▌ | 16/100 [00:01<00:03, 22.92it/s, train_loss=5.6888, val_loss=5.7189]
Epochs: 16%|█▌ | 16/100 [00:01<00:03, 22.92it/s, train_loss=5.6999, val_loss=5.7215]
Epochs: 16%|█▌ | 16/100 [00:01<00:03, 22.92it/s, train_loss=5.7227, val_loss=5.7238]
Epochs: 16%|█▌ | 16/100 [00:01<00:03, 22.92it/s, train_loss=5.6617, val_loss=5.7258]
Epochs: 16%|█▌ | 16/100 [00:01<00:03, 22.92it/s, train_loss=5.6256, val_loss=5.7259]
Epochs: 16%|█▌ | 16/100 [00:01<00:03, 22.92it/s, train_loss=5.5353, val_loss=5.7276]
Epochs: 16%|█▌ | 16/100 [00:01<00:03, 22.92it/s, train_loss=5.5656, val_loss=5.7282]
Epochs: 24%|██▍ | 24/100 [00:01<00:02, 33.30it/s, train_loss=5.5656, val_loss=5.7282]
Epochs: 24%|██▍ | 24/100 [00:01<00:02, 33.30it/s, train_loss=5.6148, val_loss=5.7273]
Epochs: 24%|██▍ | 24/100 [00:01<00:02, 33.30it/s, train_loss=5.6193, val_loss=5.7261]
Epochs: 24%|██▍ | 24/100 [00:01<00:02, 33.30it/s, train_loss=5.5485, val_loss=5.7265]
Epochs: 24%|██▍ | 24/100 [00:01<00:02, 33.30it/s, train_loss=5.6399, val_loss=5.7260]
Epochs: 24%|██▍ | 24/100 [00:01<00:02, 33.30it/s, train_loss=5.6048, val_loss=5.7257]
Epochs: 24%|██▍ | 24/100 [00:01<00:02, 33.30it/s, train_loss=5.6074, val_loss=5.7256]
Epochs: 24%|██▍ | 24/100 [00:01<00:02, 33.30it/s, train_loss=5.7287, val_loss=5.7247]
Epochs: 24%|██▍ | 24/100 [00:01<00:02, 33.30it/s, train_loss=5.5905, val_loss=5.7272]
Epochs: 32%|███▏ | 32/100 [00:01<00:01, 42.11it/s, train_loss=5.5905, val_loss=5.7272]
Epochs: 32%|███▏ | 32/100 [00:01<00:01, 42.11it/s, train_loss=5.6615, val_loss=5.7289]
Epochs: 32%|███▏ | 32/100 [00:01<00:01, 42.11it/s, train_loss=5.6233, val_loss=5.7312]
Epochs: 32%|███▏ | 32/100 [00:01<00:01, 42.11it/s, train_loss=5.6513, val_loss=5.7336]
Epochs: 32%|███▏ | 32/100 [00:01<00:01, 42.11it/s, train_loss=5.6302, val_loss=5.7363]
Epochs: 32%|███▏ | 32/100 [00:01<00:01, 42.11it/s, train_loss=5.6900, val_loss=5.7370]
Epochs: 32%|███▏ | 32/100 [00:01<00:01, 42.11it/s, train_loss=5.6180, val_loss=5.7363]
Epochs: 32%|███▏ | 32/100 [00:01<00:01, 42.11it/s, train_loss=5.8270, val_loss=5.7345]
Epochs: 32%|███▏ | 32/100 [00:01<00:01, 42.11it/s, train_loss=5.6799, val_loss=5.7267]
Epochs: 40%|████ | 40/100 [00:01<00:01, 49.31it/s, train_loss=5.6799, val_loss=5.7267]
Epochs: 40%|████ | 40/100 [00:01<00:01, 49.31it/s, train_loss=5.7152, val_loss=5.7193]
Epochs: 40%|████ | 40/100 [00:01<00:01, 49.31it/s, train_loss=5.6854, val_loss=5.7122]
Epochs: 40%|████ | 40/100 [00:01<00:01, 49.31it/s, train_loss=5.6053, val_loss=5.7093]
Epochs: 40%|████ | 40/100 [00:01<00:01, 49.31it/s, train_loss=5.7100, val_loss=5.7096]
Epochs: 40%|████ | 40/100 [00:01<00:01, 49.31it/s, train_loss=5.6404, val_loss=5.7118]
Epochs: 40%|████ | 40/100 [00:01<00:01, 49.31it/s, train_loss=5.5505, val_loss=5.7148]
Epochs: 40%|████ | 40/100 [00:01<00:01, 49.31it/s, train_loss=5.6185, val_loss=5.7163]
Epochs: 40%|████ | 40/100 [00:01<00:01, 49.31it/s, train_loss=5.6490, val_loss=5.7185]
Epochs: 48%|████▊ | 48/100 [00:01<00:00, 54.91it/s, train_loss=5.6490, val_loss=5.7185]
Epochs: 48%|████▊ | 48/100 [00:01<00:00, 54.91it/s, train_loss=5.6641, val_loss=5.7202]
Epochs: 48%|████▊ | 48/100 [00:01<00:00, 54.91it/s, train_loss=5.5823, val_loss=5.7219]
Epochs: 48%|████▊ | 48/100 [00:01<00:00, 54.91it/s, train_loss=5.6293, val_loss=5.7229]
Epochs: 48%|████▊ | 48/100 [00:01<00:00, 54.91it/s, train_loss=5.6556, val_loss=5.7242]
Epochs: 48%|████▊ | 48/100 [00:01<00:00, 54.91it/s, train_loss=5.6549, val_loss=5.7252]
Epochs: 48%|████▊ | 48/100 [00:01<00:00, 54.91it/s, train_loss=5.7120, val_loss=5.7252]
Epochs: 48%|████▊ | 48/100 [00:01<00:00, 54.91it/s, train_loss=5.7036, val_loss=5.7245]
Epochs: 55%|█████▌ | 55/100 [00:01<00:00, 56.87it/s, train_loss=5.7036, val_loss=5.7245]
Epochs: 55%|█████▌ | 55/100 [00:01<00:00, 56.87it/s, train_loss=5.6222, val_loss=5.7237]
Epochs: 55%|█████▌ | 55/100 [00:01<00:00, 56.87it/s, train_loss=5.6231, val_loss=5.7244]
Epochs: 55%|█████▌ | 55/100 [00:01<00:00, 56.87it/s, train_loss=5.6054, val_loss=5.7250]
Epochs: 55%|█████▌ | 55/100 [00:01<00:00, 56.87it/s, train_loss=5.6501, val_loss=5.7252]
Epochs: 55%|█████▌ | 55/100 [00:01<00:00, 56.87it/s, train_loss=5.6928, val_loss=5.7264]
Epochs: 55%|█████▌ | 55/100 [00:01<00:00, 56.87it/s, train_loss=5.6276, val_loss=5.7284]
Epochs: 55%|█████▌ | 55/100 [00:01<00:00, 56.87it/s, train_loss=5.7687, val_loss=5.7308]
Epochs: 62%|██████▏ | 62/100 [00:01<00:00, 59.53it/s, train_loss=5.7687, val_loss=5.7308]
Epochs: 62%|██████▏ | 62/100 [00:01<00:00, 59.53it/s, train_loss=5.6259, val_loss=5.7336]
Epochs: 62%|██████▏ | 62/100 [00:01<00:00, 59.53it/s, train_loss=5.6128, val_loss=5.7348]
Epochs: 62%|██████▏ | 62/100 [00:01<00:00, 59.53it/s, train_loss=5.5893, val_loss=5.7355]
Epochs: 62%|██████▏ | 62/100 [00:01<00:00, 59.53it/s, train_loss=5.5853, val_loss=5.7359]
Epochs: 62%|██████▏ | 62/100 [00:01<00:00, 59.53it/s, train_loss=5.5594, val_loss=5.7359]
Epochs: 62%|██████▏ | 62/100 [00:01<00:00, 59.53it/s, train_loss=5.5864, val_loss=5.7356]
Epochs: 62%|██████▏ | 62/100 [00:01<00:00, 59.53it/s, train_loss=5.6392, val_loss=5.7356]
Epochs: 69%|██████▉ | 69/100 [00:01<00:00, 60.77it/s, train_loss=5.6392, val_loss=5.7356]
Epochs: 69%|██████▉ | 69/100 [00:01<00:00, 60.77it/s, train_loss=5.6543, val_loss=5.7354]
Epochs: 69%|██████▉ | 69/100 [00:01<00:00, 60.77it/s, train_loss=5.6040, val_loss=5.7351]
Epochs: 69%|██████▉ | 69/100 [00:01<00:00, 60.77it/s, train_loss=5.6251, val_loss=5.7350]
Epochs: 69%|██████▉ | 69/100 [00:01<00:00, 60.77it/s, train_loss=5.6127, val_loss=5.7347]
Epochs: 69%|██████▉ | 69/100 [00:01<00:00, 60.77it/s, train_loss=5.5837, val_loss=5.7347]
Epochs: 69%|██████▉ | 69/100 [00:01<00:00, 60.77it/s, train_loss=5.6374, val_loss=5.7347]
Epochs: 69%|██████▉ | 69/100 [00:01<00:00, 60.77it/s, train_loss=5.6903, val_loss=5.7344]
Epochs: 69%|██████▉ | 69/100 [00:01<00:00, 60.77it/s, train_loss=5.6071, val_loss=5.7339]
Epochs: 77%|███████▋ | 77/100 [00:01<00:00, 63.66it/s, train_loss=5.6071, val_loss=5.7339]
Epochs: 77%|███████▋ | 77/100 [00:01<00:00, 63.66it/s, train_loss=5.6556, val_loss=5.7336]
Epochs: 77%|███████▋ | 77/100 [00:01<00:00, 63.66it/s, train_loss=5.6180, val_loss=5.7336]
Epochs: 77%|███████▋ | 77/100 [00:01<00:00, 63.66it/s, train_loss=5.6250, val_loss=5.7338]
Epochs: 77%|███████▋ | 77/100 [00:01<00:00, 63.66it/s, train_loss=5.7097, val_loss=5.7339]
Epochs: 77%|███████▋ | 77/100 [00:01<00:00, 63.66it/s, train_loss=5.6440, val_loss=5.7338]
Epochs: 77%|███████▋ | 77/100 [00:02<00:00, 63.66it/s, train_loss=5.5217, val_loss=5.7338]
Epochs: 77%|███████▋ | 77/100 [00:02<00:00, 63.66it/s, train_loss=5.6750, val_loss=5.7337]
Epochs: 77%|███████▋ | 77/100 [00:02<00:00, 63.66it/s, train_loss=5.6321, val_loss=5.7334]
Epochs: 85%|████████▌ | 85/100 [00:02<00:00, 65.74it/s, train_loss=5.6321, val_loss=5.7334]
Epochs: 85%|████████▌ | 85/100 [00:02<00:00, 65.74it/s, train_loss=5.6664, val_loss=5.7331]
Epochs: 85%|████████▌ | 85/100 [00:02<00:00, 65.74it/s, train_loss=5.6000, val_loss=5.7328]
Epochs: 85%|████████▌ | 85/100 [00:02<00:00, 65.74it/s, train_loss=5.5550, val_loss=5.7325]
Epochs: 85%|████████▌ | 85/100 [00:02<00:00, 65.74it/s, train_loss=5.5503, val_loss=5.7324]
Epochs: 85%|████████▌ | 85/100 [00:02<00:00, 65.74it/s, train_loss=5.6301, val_loss=5.7323]
Epochs: 85%|████████▌ | 85/100 [00:02<00:00, 65.74it/s, train_loss=5.5384, val_loss=5.7322]
Epochs: 85%|████████▌ | 85/100 [00:02<00:00, 65.74it/s, train_loss=5.6442, val_loss=5.7321]
Epochs: 92%|█████████▏| 92/100 [00:02<00:00, 66.90it/s, train_loss=5.6442, val_loss=5.7321]
Epochs: 92%|█████████▏| 92/100 [00:02<00:00, 66.90it/s, train_loss=5.5608, val_loss=5.7321]
Epochs: 92%|█████████▏| 92/100 [00:02<00:00, 66.90it/s, train_loss=5.6623, val_loss=5.7321]
Epochs: 92%|█████████▏| 92/100 [00:02<00:00, 66.90it/s, train_loss=5.6358, val_loss=5.7321]
Epochs: 92%|█████████▏| 92/100 [00:02<00:00, 66.90it/s, train_loss=5.6161, val_loss=5.7321]
Epochs: 92%|█████████▏| 92/100 [00:02<00:00, 66.90it/s, train_loss=5.5040, val_loss=5.7321]
Epochs: 92%|█████████▏| 92/100 [00:02<00:00, 66.90it/s, train_loss=5.6070, val_loss=5.7321]
Epochs: 92%|█████████▏| 92/100 [00:02<00:00, 66.90it/s, train_loss=5.6648, val_loss=5.7321]
Epochs: 92%|█████████▏| 92/100 [00:02<00:00, 66.90it/s, train_loss=5.5725, val_loss=5.7321]
Epochs: 100%|██████████| 100/100 [00:02<00:00, 67.98it/s, train_loss=5.5725, val_loss=5.7321]
Epochs: 100%|██████████| 100/100 [00:02<00:00, 44.45it/s, train_loss=5.5725, val_loss=5.7321]
2025-12-15 08:47:17,761 - aspire - INFO - Loaded best model with val loss 5.6948
Compare flow-based importance samples#
Use the trained flow to generate new samples, and visualise how they compare to the original biased set and the ground truth.
is_samples = poppy.sample_posterior(10_000)
fig = plot_comparison(
initial_samples,
true_samples,
is_samples,
is_samples,
per_samples_kwargs=[
dict(include_weights=True, color="C0"),
dict(include_weights=False, color="k"),
dict(include_weights=False, color="lightgrey"),
dict(include_weights=True, color="C1"),
],
labels=["Initial samples", "True samples", "Flow samples", "IS samples"],
)
fig.savefig(outdir / "initial_samples.png")
plt.show()
2025-12-15 08:47:17,915 - aspire - INFO - Sampled 10000 samples from the posterior
2025-12-15 08:47:17,916 - aspire - INFO - Number of likelihood evaluations: 10000
2025-12-15 08:47:17,916 - aspire - INFO - Sample summary:
2025-12-15 08:47:17,917 - aspire - INFO - No. samples: 10000
No. parameters: 4
Log evidence: -27.71 +/- 0.86
Effective sample size: 1.3
Efficiency: 0.00
WARNING:root:Too few points to create valid contours
Refine with sequential Monte Carlo#
Run Aspire’s SMC sampler to further correct the proposal and inspect the particle evolution.
# Produce samples from the posterior
samples, smc_history = poppy.sample_posterior(
1000,
sampler="smc",
return_history=True,
n_final_samples=5000,
sampler_kwargs=dict(
n_steps=32,
),
)
fig = smc_history.plot()
fig.savefig(outdir / "history.png")
2025-12-15 08:47:20,164 - aspire - INFO - Prior bounds: {'x_0': (-10, 10), 'x_1': (0, 10), 'x_2': (-10, 10), 'x_3': (-10, 10)}
INFO:aspire.transforms:Prior bounds: {'x_0': (-10, 10), 'x_1': (0, 10), 'x_2': (-10, 10), 'x_3': (-10, 10)}
2025-12-15 08:47:20,166 - aspire - INFO - Prior bounds: {'x_0': array([-10, 10]), 'x_1': array([ 0, 10]), 'x_2': array([-10, 10]), 'x_3': array([-10, 10])}
INFO:aspire.transforms:Prior bounds: {'x_0': array([-10, 10]), 'x_1': array([ 0, 10]), 'x_2': array([-10, 10]), 'x_3': array([-10, 10])}
2025-12-15 08:47:20,166 - aspire - INFO - Affine transform applied to: ['x_0', 'x_1', 'x_2', 'x_3']
INFO:aspire.transforms:Affine transform applied to: ['x_0', 'x_1', 'x_2', 'x_3']
2025-12-15 08:47:20,177 - aspire - INFO - it 1 - beta: 0.0118408203125
INFO:aspire.samplers.smc.base:it 1 - beta: 0.0118408203125
2025-12-15 08:47:20,178 - aspire - INFO - it 1 - ESS: 500.3 (0.50 efficiency)
INFO:aspire.samplers.smc.base:it 1 - ESS: 500.3 (0.50 efficiency)
2025-12-15 08:47:20,180 - aspire - INFO - it 1 - Log evidence ratio: -2.27 +/- 0.03
INFO:aspire.samplers.smc.base:it 1 - Log evidence ratio: -2.27 +/- 0.03
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2025-12-15 08:47:21,078 - aspire - INFO - it 2 - beta: 0.030424593947827816
INFO:aspire.samplers.smc.base:it 2 - beta: 0.030424593947827816
2025-12-15 08:47:21,079 - aspire - INFO - it 2 - ESS: 500.2 (0.50 efficiency)
INFO:aspire.samplers.smc.base:it 2 - ESS: 500.2 (0.50 efficiency)
2025-12-15 08:47:21,080 - aspire - INFO - it 2 - Log evidence ratio: -2.26 +/- 0.03
INFO:aspire.samplers.smc.base:it 2 - Log evidence ratio: -2.26 +/- 0.03
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2025-12-15 08:47:22,303 - aspire - INFO - it 3 - beta: 0.05938491891955522
INFO:aspire.samplers.smc.base:it 3 - beta: 0.05938491891955522
2025-12-15 08:47:22,304 - aspire - INFO - it 3 - ESS: 500.0 (0.50 efficiency)
INFO:aspire.samplers.smc.base:it 3 - ESS: 500.0 (0.50 efficiency)
2025-12-15 08:47:22,305 - aspire - INFO - it 3 - Log evidence ratio: -2.13 +/- 0.03
INFO:aspire.samplers.smc.base:it 3 - Log evidence ratio: -2.13 +/- 0.03
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2025-12-15 08:47:22,949 - aspire - INFO - it 4 - beta: 0.10445940717231915
INFO:aspire.samplers.smc.base:it 4 - beta: 0.10445940717231915
2025-12-15 08:47:22,950 - aspire - INFO - it 4 - ESS: 500.1 (0.50 efficiency)
INFO:aspire.samplers.smc.base:it 4 - ESS: 500.1 (0.50 efficiency)
2025-12-15 08:47:22,951 - aspire - INFO - it 4 - Log evidence ratio: -2.09 +/- 0.03
INFO:aspire.samplers.smc.base:it 4 - Log evidence ratio: -2.09 +/- 0.03
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2025-12-15 08:47:23,925 - aspire - INFO - it 5 - beta: 0.1719501754214634
INFO:aspire.samplers.smc.base:it 5 - beta: 0.1719501754214634
2025-12-15 08:47:23,926 - aspire - INFO - it 5 - ESS: 500.0 (0.50 efficiency)
INFO:aspire.samplers.smc.base:it 5 - ESS: 500.0 (0.50 efficiency)
2025-12-15 08:47:23,928 - aspire - INFO - it 5 - Log evidence ratio: -1.90 +/- 0.03
INFO:aspire.samplers.smc.base:it 5 - Log evidence ratio: -1.90 +/- 0.03
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Sampling: 62%|██████▎ | 20/32 [00:00<00:00, 197.18step/s, acceptance_rate=0.454, rho=0.99]
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Sampling: 100%|██████████| 32/32 [00:00<00:00, 191.96step/s, acceptance_rate=0.442, rho=0.99]
2025-12-15 08:47:24,970 - aspire - INFO - it 6 - beta: 0.27306206386735177
INFO:aspire.samplers.smc.base:it 6 - beta: 0.27306206386735177
2025-12-15 08:47:24,971 - aspire - INFO - it 6 - ESS: 500.0 (0.50 efficiency)
INFO:aspire.samplers.smc.base:it 6 - ESS: 500.0 (0.50 efficiency)
2025-12-15 08:47:24,972 - aspire - INFO - it 6 - Log evidence ratio: -1.74 +/- 0.03
INFO:aspire.samplers.smc.base:it 6 - Log evidence ratio: -1.74 +/- 0.03
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Sampling: 59%|█████▉ | 19/32 [00:00<00:00, 188.18step/s, acceptance_rate=0.461, rho=0.99]
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Sampling: 100%|██████████| 32/32 [00:00<00:00, 181.80step/s, acceptance_rate=0.462, rho=0.99]
2025-12-15 08:47:26,126 - aspire - INFO - it 7 - beta: 0.39977927246078704
INFO:aspire.samplers.smc.base:it 7 - beta: 0.39977927246078704
2025-12-15 08:47:26,127 - aspire - INFO - it 7 - ESS: 500.0 (0.50 efficiency)
INFO:aspire.samplers.smc.base:it 7 - ESS: 500.0 (0.50 efficiency)
2025-12-15 08:47:26,129 - aspire - INFO - it 7 - Log evidence ratio: -1.22 +/- 0.03
INFO:aspire.samplers.smc.base:it 7 - Log evidence ratio: -1.22 +/- 0.03
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Sampling: 53%|█████▎ | 17/32 [00:00<00:00, 160.66step/s, acceptance_rate=0.485, rho=0.99]
Sampling: 53%|█████▎ | 17/32 [00:00<00:00, 160.66step/s, acceptance_rate=0.478, rho=0.99]
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Sampling: 53%|█████▎ | 17/32 [00:00<00:00, 160.66step/s, acceptance_rate=0.506, rho=0.99]
Sampling: 53%|█████▎ | 17/32 [00:00<00:00, 160.66step/s, acceptance_rate=0.479, rho=0.99]
Sampling: 53%|█████▎ | 17/32 [00:00<00:00, 160.66step/s, acceptance_rate=0.47, rho=0.99]
Sampling: 53%|█████▎ | 17/32 [00:00<00:00, 160.66step/s, acceptance_rate=0.456, rho=0.99]
Sampling: 53%|█████▎ | 17/32 [00:00<00:00, 160.66step/s, acceptance_rate=0.437, rho=0.99]
Sampling: 53%|█████▎ | 17/32 [00:00<00:00, 160.66step/s, acceptance_rate=0.486, rho=0.99]
Sampling: 53%|█████▎ | 17/32 [00:00<00:00, 160.66step/s, acceptance_rate=0.488, rho=0.99]
Sampling: 100%|██████████| 32/32 [00:00<00:00, 173.05step/s, acceptance_rate=0.488, rho=0.99]
2025-12-15 08:47:28,972 - aspire - INFO - it 8 - beta: 0.5425716516579285
INFO:aspire.samplers.smc.base:it 8 - beta: 0.5425716516579285
2025-12-15 08:47:28,973 - aspire - INFO - it 8 - ESS: 500.0 (0.50 efficiency)
INFO:aspire.samplers.smc.base:it 8 - ESS: 500.0 (0.50 efficiency)
2025-12-15 08:47:28,975 - aspire - INFO - it 8 - Log evidence ratio: -0.70 +/- 0.03
INFO:aspire.samplers.smc.base:it 8 - Log evidence ratio: -0.70 +/- 0.03
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Sampling: 62%|██████▎ | 20/32 [00:00<00:00, 194.21step/s, acceptance_rate=0.495, rho=0.99]
Sampling: 62%|██████▎ | 20/32 [00:00<00:00, 194.21step/s, acceptance_rate=0.513, rho=0.99]
Sampling: 62%|██████▎ | 20/32 [00:00<00:00, 194.21step/s, acceptance_rate=0.489, rho=0.99]
Sampling: 62%|██████▎ | 20/32 [00:00<00:00, 194.21step/s, acceptance_rate=0.488, rho=0.99]
Sampling: 62%|██████▎ | 20/32 [00:00<00:00, 194.21step/s, acceptance_rate=0.513, rho=0.99]
Sampling: 62%|██████▎ | 20/32 [00:00<00:00, 194.21step/s, acceptance_rate=0.514, rho=0.99]
Sampling: 62%|██████▎ | 20/32 [00:00<00:00, 194.21step/s, acceptance_rate=0.514, rho=0.99]
Sampling: 62%|██████▎ | 20/32 [00:00<00:00, 194.21step/s, acceptance_rate=0.498, rho=0.99]
Sampling: 62%|██████▎ | 20/32 [00:00<00:00, 194.21step/s, acceptance_rate=0.488, rho=0.99]
Sampling: 62%|██████▎ | 20/32 [00:00<00:00, 194.21step/s, acceptance_rate=0.5, rho=0.99]
Sampling: 62%|██████▎ | 20/32 [00:00<00:00, 194.21step/s, acceptance_rate=0.49, rho=0.99]
Sampling: 62%|██████▎ | 20/32 [00:00<00:00, 194.21step/s, acceptance_rate=0.499, rho=0.99]
Sampling: 62%|██████▎ | 20/32 [00:00<00:00, 194.21step/s, acceptance_rate=0.533, rho=0.99]
Sampling: 100%|██████████| 32/32 [00:00<00:00, 189.59step/s, acceptance_rate=0.533, rho=0.99]
2025-12-15 08:47:30,092 - aspire - INFO - it 9 - beta: 0.678531228033588
INFO:aspire.samplers.smc.base:it 9 - beta: 0.678531228033588
2025-12-15 08:47:30,093 - aspire - INFO - it 9 - ESS: 500.0 (0.50 efficiency)
INFO:aspire.samplers.smc.base:it 9 - ESS: 500.0 (0.50 efficiency)
2025-12-15 08:47:30,094 - aspire - INFO - it 9 - Log evidence ratio: 0.03 +/- 0.03
INFO:aspire.samplers.smc.base:it 9 - Log evidence ratio: 0.03 +/- 0.03
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Sampling: 62%|██████▎ | 20/32 [00:00<00:00, 194.34step/s, acceptance_rate=0.513, rho=0.99]
Sampling: 62%|██████▎ | 20/32 [00:00<00:00, 194.34step/s, acceptance_rate=0.517, rho=0.99]
Sampling: 62%|██████▎ | 20/32 [00:00<00:00, 194.34step/s, acceptance_rate=0.489, rho=0.99]
Sampling: 62%|██████▎ | 20/32 [00:00<00:00, 194.34step/s, acceptance_rate=0.502, rho=0.99]
Sampling: 62%|██████▎ | 20/32 [00:00<00:00, 194.34step/s, acceptance_rate=0.515, rho=0.99]
Sampling: 62%|██████▎ | 20/32 [00:00<00:00, 194.34step/s, acceptance_rate=0.499, rho=0.99]
Sampling: 62%|██████▎ | 20/32 [00:00<00:00, 194.34step/s, acceptance_rate=0.499, rho=0.99]
Sampling: 62%|██████▎ | 20/32 [00:00<00:00, 194.34step/s, acceptance_rate=0.515, rho=0.99]
Sampling: 62%|██████▎ | 20/32 [00:00<00:00, 194.34step/s, acceptance_rate=0.514, rho=0.99]
Sampling: 62%|██████▎ | 20/32 [00:00<00:00, 194.34step/s, acceptance_rate=0.515, rho=0.99]
Sampling: 62%|██████▎ | 20/32 [00:00<00:00, 194.34step/s, acceptance_rate=0.538, rho=0.99]
Sampling: 62%|██████▎ | 20/32 [00:00<00:00, 194.34step/s, acceptance_rate=0.5, rho=0.99]
Sampling: 62%|██████▎ | 20/32 [00:00<00:00, 194.34step/s, acceptance_rate=0.523, rho=0.99]
Sampling: 100%|██████████| 32/32 [00:00<00:00, 193.50step/s, acceptance_rate=0.523, rho=0.99]
2025-12-15 08:47:31,743 - aspire - INFO - it 10 - beta: 0.8206846248168311
INFO:aspire.samplers.smc.base:it 10 - beta: 0.8206846248168311
2025-12-15 08:47:31,745 - aspire - INFO - it 10 - ESS: 500.0 (0.50 efficiency)
INFO:aspire.samplers.smc.base:it 10 - ESS: 500.0 (0.50 efficiency)
2025-12-15 08:47:31,746 - aspire - INFO - it 10 - Log evidence ratio: 0.65 +/- 0.03
INFO:aspire.samplers.smc.base:it 10 - Log evidence ratio: 0.65 +/- 0.03
Sampling: 0%| | 0/32 [00:00<?, ?step/s]
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2025-12-15 08:47:34,941 - aspire - INFO - it 11 - beta: 0.9262939822893645
INFO:aspire.samplers.smc.base:it 11 - beta: 0.9262939822893645
2025-12-15 08:47:34,942 - aspire - INFO - it 11 - ESS: 500.0 (0.50 efficiency)
INFO:aspire.samplers.smc.base:it 11 - ESS: 500.0 (0.50 efficiency)
2025-12-15 08:47:34,944 - aspire - INFO - it 11 - Log evidence ratio: 1.12 +/- 0.03
INFO:aspire.samplers.smc.base:it 11 - Log evidence ratio: 1.12 +/- 0.03
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2025-12-15 08:47:42,335 - aspire - INFO - it 12 - beta: 1.0
INFO:aspire.samplers.smc.base:it 12 - beta: 1.0
2025-12-15 08:47:42,336 - aspire - INFO - it 12 - ESS: 534.9 (0.53 efficiency)
INFO:aspire.samplers.smc.base:it 12 - ESS: 534.9 (0.53 efficiency)
2025-12-15 08:47:42,337 - aspire - INFO - it 12 - Log evidence ratio: 1.30 +/- 0.03
INFO:aspire.samplers.smc.base:it 12 - Log evidence ratio: 1.30 +/- 0.03
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2025-12-15 08:47:49,740 - aspire - INFO - Generating 5000 final samples
INFO:aspire.samplers.smc.base:Generating 5000 final samples
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2025-12-15 08:47:55,650 - aspire - INFO - Log evidence: -11.21 +/- 0.11
INFO:aspire.samplers.smc.base:Log evidence: -11.21 +/- 0.11
2025-12-15 08:47:55,651 - aspire - INFO - Sampled 5000 samples from the posterior
INFO:aspire.aspire:Sampled 5000 samples from the posterior
2025-12-15 08:47:55,652 - aspire - INFO - Number of likelihood evaluations: 579000
INFO:aspire.aspire:Number of likelihood evaluations: 579000
2025-12-15 08:47:55,653 - aspire - INFO - Sample summary:
INFO:aspire.aspire:Sample summary:
2025-12-15 08:47:55,653 - aspire - INFO - No. samples: 5000
No. parameters: 4
Log evidence: -11.21 +/- 0.11
INFO:aspire.aspire:No. samples: 5000
No. parameters: 4
Log evidence: -11.21 +/- 0.11
Persist results#
Write the flow history, SMC diagnostics, and posterior samples to disk so the analysis can be reproduced or extended.
# Save the the results to a file
# The AspireFile is a small wrapper around h5py.File that automatically
# includes additional metadata
with AspireFile(outdir / "poppy_result.h5", "w") as f:
# poppy.save_config(f, "poppy_config")
samples.save(f, "smc/posterior_samples")
history.save(f, "flow/history")
smc_history.save(f, "smc/history")
Final posterior comparison#
Contrast the SMC posterior samples against the earlier flow draws to check that corrections removed the bias.
fig = plot_comparison(
initial_samples,
true_samples,
is_samples,
samples,
per_samples_kwargs=[
dict(include_weights=True, color="C0"),
dict(include_weights=False, color="k"),
dict(include_weights=False, color="lightgrey"),
dict(include_weights=False, color="C1"),
],
labels=["Initial samples", "True samples", "Flow samples", "SMC samples"],
)
fig.savefig(outdir / "posterior_samples.png")
plt.show()
