[conjectureproof]

Lenny Baum, Lloyd Welch, and their colleagues at IDA were using the EM algorithm for code cracking well before they were able to prove anything about its convergence.

EM worked in practice, so they spent a long time trying to prove convergence. Modern proofs are simpler.

Could be the case that this method also works in practice. I haven't the faintest idea whether it will.