Log-Partition Convexity Check

Problem

Check numerically that the log-partition function A(eta) = ln(sum_t exp(eta * t)) is convex. Return True when the etas are equally spaced within 1e-9 and every second difference A[i+1] - 2 A[i] + A[i-1] is at least -1e-12. Otherwise return False.

Hint

Convexity of A(eta) is equivalent to the variance of the statistic being nonnegative.

Test cases

case 1

in: [[-1,0,1],[0,1]]

expected: true

case 2

in: [[0,1,2],[0]]

expected: true

case 3

in: [[0,1,2],[1]]

expected: true

case 4

in: [[0,1,3],[0,1]]

expected: false

Starter codePython

import math
def log_partition_convexity_check(etas, t_values):
    # Your code here
    pass

4 test cases · run your code against them in the editor