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Joseph Born Kadane is a statistician whose work connects Bayesian statistics, decision theory and applications of probability to practical evidence. In his own research statement, he describes work on prior elicitation: constructing probability distributions from an investigator's background knowledge, as well as on the foundations and computation of Bayesian inference.^[ref]

His institutional biography records that he joined Carnegie Mellon in 1971 and headed its statistics department from 1972 to 1981.^[ref] In computing, his name is also attached to Kadane's algorithm for the maximum subarray problem: the best segment ending at a new position either extends the preceding best-ending segment or starts at that position. This gives a linear-time dynamic programming solution.

For a segment, store its total sum $S$, maximum possibly-empty prefix sum $P$, maximum possibly-empty suffix sum $Q$ and maximum possibly-empty internal contiguous sum $B$. Adjacent segment summaries combine byA prefix either stays on the left or includes the whole left segment and a right prefix. A suffix has the symmetric decomposition. An internal segment either stays in one child or crosses the boundary as a left suffix followed by a right prefix. These cases prove the formulas, including empty candidates. Store the corresponding endpoints to recover a winning segment. This constant-size summary gives a linear time divide-and-conquer algorithm for the empty-allowed maximum subarray problem.

Let $e_i$ be the best sum of a segment ending at position $i$, allowing the empty segment with sum zero. The maximum subarray problem recurrence isAn optimal nonempty segment ending at $i$ extends the best suffix ending at $i-1$, or starts fresh; a negative suffix cannot help. Keep its start index and record endpoints whenever a new overall best is found:

suffix = 0; best = 0
start = 0; bestLeft = 0; bestRight = 0
for i = 0,...,n-1:
    suffix = suffix + A[i]
    if suffix < 0:
        suffix = 0; start = i+1
    if suffix > best:
        best = suffix
        bestLeft = start; bestRight = i+1
return A[bestLeft:bestRight]

The half-open endpoint interval is empty initially, so empty input and all-negative input correctly return an empty segment. The invariant identifies the best nonnegative suffix and the best segment in the scanned prefix, proving correctness. This is Kadane's algorithm, using $O(n)$ time and $O(1)$ auxiliary space, excluding the returned slice. The displayed example yields sum $55$, from the segment $10,-5,40,10$.