log concave confidence interval

Python implementation of the paper "Walther G, Ali A, Shen X, Boyd S. Confidence bands for a log-concave density".

Please first install CVXPY 1.1.

To test the method in one example, please run the jupyter notebook example.ipynb.

To run the experiments in the paper, please run experiment_platform.ipynb to compute and save the confidence intervals. The distribution, the number of design points, the coverage rate, and the number of threads should be chosen and set.

After running the experiments, to generate a row of Table 1 in the paper, please run compute_coverage_rate_and_width.ipynb. To generate the figures, please run plot_figure.ipynb.

Usage

Construct object. Under the directory, import confint. Given data X, to run the method with confidence level specified by 1 - alpha and ratio of optimized design points specified by opt_pts_ratio, first construct an object of the confint class. For example

from confint import confint
import numpy as np

X = np.random.randn(n)
X = np.sort(X)
alpha = 0.1
opt_pts_ratio = 0.5

conf_int = confint(X, alpha, opt_pts_ratio=opt_pts_ratio)

Solve. Then call the method compute_pw_conf_ints to compute piecewise confidence intervals. Following the above example, the code is as follows.

conf_int.compute_pw_conf_ints()

The following optional arguments can be passed into the solve method.

• thread_num gives the number of threads used to run the algorithm in parallel accross optimized design points.
• algorithm parameters
  • tau_max specifies the maximum value of penalty parameter tau. Default value is 1e3.
  • tau_init specifies the initial value of tau. Default value is 1e-5.
  • kappa specifies the penalty increase factor. Default value is 8.
  • max_iters specifies the maximum number of iterations. Default value is 50.
  • min_iters specifies the minimum number of iterations. Default value is 15.
  • M gives a numeric lower bound exp(-M) on the density, and a typical value is around 10.
• verbose is a boolean giving the choice of printing information during the iterations. Default value is False.

Retrieve result. The result is stored in the following attributes of the constructed confint object.

• opt_pts stores the optmized design points, at which the confidence intervals are optimized over.
• low_opt_pts stores the lower confidence bound at opt_pts.
• high_opt_pts stores the higher confidence bound at opt_pts.
• opt_int_pts stores the union of optmized design points and interpolation points.
• low_opt_int_pts stores the lower confidence bound at opt_int_pts.
• high_opt_int_pts stores the higher confidence bound at opt_int_pts.