How to sample from a log-uniform distribution.

log_unif_example.py

"""
How we might sample from a log-uniform distribution
https://stats.stackexchange.com/questions/155552/what-does-log-uniformly-distribution-mean

Only run one of these three cases at a time, otherwise the plots update each
other. Run with these versions:

matplotlib      3.2.1
numpy           1.18.3
"""
import numpy as np
import matplotlib.pyplot as plt
np.set_printoptions(suppress=True, linewidth=200, edgeitems=10)
low = 0.01
high = 0.7
size = 10000
nb_bins = 50

if False:
    data = np.random.uniform(low=low, high=high, size=size)
    count, bins, ignored = plt.hist(data, bins=nb_bins, align='mid')
    plt.title('Uniform({}, {})'.format(low, high))
    plt.xlabel('Epsilon')
    plt.savefig('distr_uniform.png')

if False:
    data = np.random.uniform(low=np.log(low), high=np.log(high), size=size)
    count, bins, ignored = plt.hist(data, bins=nb_bins, align='mid')
    plt.title('Uniform(log({}), log({})'.format(low, high))
    plt.xlabel('Epsilon')
    plt.savefig('distr_uniform_log.png')

# Number of classes are the number of intervals.
nb_classes = 5 + 1

if True:
    data = np.random.uniform(low=np.log(low), high=np.log(high), size=size)
    discretized = np.linspace(np.log(low), np.log(high), num=nb_classes)
    data = np.exp(data)
    count, bins, ignored = plt.hist(data, bins=nb_bins, align='mid')
    plt.title('exp( Uniform(log({}), log({}) )'.format(low, high))
    plt.xlabel('Epsilon')
    plt.savefig('distr_uniform_log_true.png')

    # Now let's add dicretized ranges.
    print('Discretized bounds (len {}) for epsilons:\nLog: {}\nNormal: {}'.format(
            len(discretized), discretized, np.exp(discretized)))
    for idx,item in enumerate(discretized):
        plt.axvline(x=np.exp(item), color='black')
        if idx < len(discretized) - 1:
            start = np.exp(discretized[idx])
            end = np.exp(discretized[idx+1])
            count = np.sum( (start <= data) & (data < end) )
            print('{:.3f} <= x < {:.3f} count: {}'.format(start, end, count))
    plt.savefig('distr_uniform_log_true_bounds.png')

0.01 to 0.2

The standard uniform distribution.

A visualization of what the log looks like:

And the exponentiated version which is what we will actually be using.

Note that this is similar but not quite the same as calling something like torch.logspace(low, high). What the torch logspace does is get a set of values within that range which are "spaced logarithmically." E.g.:

For 5 and 10 numbered spacings, between 0.1 and 0.01, we call torch.logspace( torch.log10(0.01), torch.log10(0.1) ):

tensor([0.1000, 0.0562, 0.0316, 0.0178, 0.0100])
tensor([0.1000, 0.0774, 0.0599, 0.0464, 0.0359, 0.0278, 0.0215, 0.0167, 0.0129, 0.0100])

0.01 to 0.5 now (same three plots in order)

(July 26) Now log(0.01) to log(0.7) with discretized bins.

Here is a plot which also has nb_classes=5+1 (because nb_classes is really the number of vertical ticks).

For classes, I get:

0.010 <= x < 0.023 count: 1998
0.023 <= x < 0.055 count: 1968
0.055 <= x < 0.128 count: 2020
0.128 <= x < 0.299 count: 2021
0.299 <= x < 0.700 count: 1993

If it's nb_classes=10+1) then we get:

0.010 <= x < 0.015 count: 991
0.015 <= x < 0.023 count: 1034
0.023 <= x < 0.036 count: 1043
0.036 <= x < 0.055 count: 968
0.055 <= x < 0.084 count: 983
0.084 <= x < 0.128 count: 1020
0.128 <= x < 0.196 count: 1051
0.196 <= x < 0.299 count: 1022
0.299 <= x < 0.458 count: 961
0.458 <= x < 0.700 count: 927