Now that I knew about the bug, the next thing to do was tell my advisor. It had been just a few weeks since he said to a luminary visiting campus, “One of my students generalized the entropy power inequality,” to which the luminary replied, “That’s impressive!” What had led me to smile then felt embarrassing now. Had the news spread elsewhere? Would this affect my advisor’s reputation in addition to my own if we ultimately had to retract the result? Why hadn’t I noticed the issue earlier?
To reach a plausible explanation for that last question, it helps to understand how my biases played a role in checking the proof. The incorrect proof had been following a technique similar to the one in Blackman’s “The convolution inequality for entropy powers”, and while Blackman confidently swapped things like the order of derivatives and expectations, I was less confident about these steps in my proof. To justify these steps required applying some measure-theoretic results, and while I’d been exposed to these in a couple probability theory courses, this was the first time I needed to apply them in my research. As a consequence of this insecurity, I focused my attention on making sure that these parts of the argument were watertight and failed to notice that I ultimately wouldn’t be able to apply these watertight arguments to prove the result.
The actual bug came from somewhere that didn’t require any measure-theoretic sophistication. Both Blackman’s proof and mine applied Gaussian perturbations to random variables, with the difference that I had introduced an auxiliary random variable and required a Markov chain to hold. The problem for my proof was that the way I was applying the Gaussian perturbations broke the Markov structure.
I swallowed my pride, found my advisor, and explained the technical issue to him. Our discussion immediately shifted to practical matters. Could we repair the bug in time for the camera-ready deadlines of the different conferences? Part of this would depend on whether we could use the same proof technique or not. We still had a couple weeks before the first camera-ready deadline (a paper that depended on the result) and three months before the camera-ready deadline for the last one (the result itself), so the decision was to work on a patch first and retract later if necessary. We started brainstorming some attacks and looking up possible references that could help.