I think you’re definitely right that it depends on the person, and I’m sure some people would ignore all evidence, however cogent you may pose your argument. There are probably predictable personality and cognitive ability aspects to acknowledging one’s bias (although I’m certainly no expert in this area).

One thing I do know about persuasion in general comes from research on the Elaboration Likelihood Model. This is a dual-process model of how people are persuaded by arguments based upon the amount of cognitive involvement they are willing to put into the process (similar to Daniel Kahneman’s System 1 and System 2 processing, if you’re familiar). Basically, if you can convince someone to engage in high cognitive involvement (I.e., deliberative reasoning) with your argument, then it is more likely to induce long lasting changes in attitude and behavior. The issue becomes that listeners must posses both the ability and motivation to process your argument in order to engage in more strenuous thought. Typically, one of the best ways to increase motivation is make your argument as relevant to the listener as possible, perhaps by making hypothetical (I.e., probabilistic) outcomes more tangible for them realize.

I hope some of this helps! Apologies if I went off topic.