from collections.abc import Sequence, Mapping
import matplotlib.pyplot as plt
import numpy as np
from bayesflow.utils import prepare_plot_data, add_titles_and_labels, prettify_subplots
[docs]
def z_score_contraction(
estimates: Mapping[str, np.ndarray] | np.ndarray,
targets: Mapping[str, np.ndarray] | np.ndarray,
variable_keys: Sequence[str] = None,
variable_names: Sequence[str] = None,
figsize: Sequence[int] = None,
label_fontsize: int = 16,
title_fontsize: int = 18,
tick_fontsize: int = 12,
color: str = "#132a70",
num_col: int = None,
num_row: int = None,
) -> plt.Figure:
"""
Implements a graphical check for global model sensitivity by plotting the
posterior z-score over the posterior contraction for each set of posterior
samples in ``estimates`` according to [1].
- The definition of the posterior z-score is:
post_z_score = (posterior_mean - true_parameters) / posterior_std
And the score is adequate if it centers around zero and spreads roughly
in the interval [-3, 3]
- The definition of posterior contraction is:
post_contraction = 1 - (posterior_variance / prior_variance)
In other words, the posterior contraction is a proxy for the reduction in
uncertainty gained by replacing the prior with the posterior.
The ideal posterior contraction tends to 1.
Contraction near zero indicates that the posterior variance is almost
identical to the prior variance for the particular marginal parameter
distribution.
Note:
Means and variances will be estimated via their sample-based estimators.
[1] Schad, D. J., Betancourt, M., & Vasishth, S. (2021).
Toward a principled Bayesian workflow in cognitive science.
Psychological methods, 26(1), 103.
Paper also available at https://arxiv.org/abs/1904.12765
Parameters
----------
estimates : np.ndarray of shape (num_datasets, num_post_draws, num_params)
The posterior draws obtained from num_datasets
targets : np.ndarray of shape (num_datasets, num_params)
The prior draws (true parameters) used for generating the num_datasets
variable_keys : list or None, optional, default: None
Select keys from the dictionaries provided in estimates and targets.
By default, select all keys.
variable_names : list or None, optional, default: None
The parameter names for nice plot titles. Inferred if None
figsize : tuple or None, optional, default : None
The figure size passed to the matplotlib constructor. Inferred if None.
label_fontsize : int, optional, default: 16
The font size of the y-label text
title_fontsize : int, optional, default: 18
The font size of the title text
tick_fontsize : int, optional, default: 12
The font size of the axis ticklabels
color : str, optional, default: '#8f2727'
The color for the true vs. estimated scatter points and error bars
num_row : int, optional, default: None
The number of rows for the subplots. Dynamically determined if None.
num_col : int, optional, default: None
The number of columns for the subplots. Dynamically determined if None.
Returns
-------
f : plt.Figure - the figure instance for optional saving
Raises
------
ShapeError
If there is a deviation from the expected shapes of ``estimates`` and ``targets``.
"""
# Gather plot data and metadata into a dictionary
plot_data = prepare_plot_data(
estimates=estimates,
targets=targets,
variable_keys=variable_keys,
variable_names=variable_names,
num_col=num_col,
num_row=num_row,
figsize=figsize,
)
estimates = plot_data.pop("estimates")
targets = plot_data.pop("targets")
# Estimate posterior means and stds
post_means = estimates.mean(axis=1)
post_vars = estimates.var(axis=1, ddof=1)
post_stds = np.sqrt(post_vars)
# Estimate prior variance
prior_vars = targets.var(axis=0, keepdims=True, ddof=1)
# Compute contraction and z-score
contraction = np.clip(1 - (post_vars / prior_vars), 0, 1)
z_score = (post_means - targets) / post_stds
# Loop and plot
for i, ax in enumerate(plot_data["axes"].flat):
if i >= plot_data["num_variables"]:
break
ax.scatter(contraction[:, i], z_score[:, i], color=color, alpha=0.5)
ax.set_xlim([-0.05, 1.05])
prettify_subplots(plot_data["axes"], num_subplots=plot_data["num_variables"], tick_fontsize=tick_fontsize)
# Add labels, titles, and set font sizes
add_titles_and_labels(
axes=plot_data["axes"],
num_row=plot_data["num_row"],
num_col=plot_data["num_col"],
title=plot_data["variable_names"],
xlabel="Posterior contraction",
ylabel="Posterior z-score",
title_fontsize=title_fontsize,
label_fontsize=label_fontsize,
)
plot_data["fig"].tight_layout()
return plot_data["fig"]
