Allowing No

One of the more applicable insights to fall out of a generally Bayesian way of thinking is in the practice of soliciting opinions from others. If you want to know what someone thinks about a proposition, there needs to be some connection between how they respond and their internally held beliefs. The weaker this connection, the less certain you can be of their stated beliefs.

As an example, take this conversation at the end of a dinner party:

Alice : "Well. Thank you for the lovely meal. I should be heading off about now, see you all at the next one!"
Bob : "Yeah I think I should head home as well. You live north of here right Alice?"
Alice : "Yeah, about 10 minutes away"
Bob : "Oh sweet, any chance I could get a lift?"
Alice : "Yeah. No worries."

On one interpretation, this is a regular, polite conversation: Alice was driving and Bob needed a lift. What would we expect to be different, though, if Alice in fact did not want to give Bob a lift? Sketching out this interaction in more Bayesian terms: suppose that Alice is truly indifferent to giving Bob a lift 70% of the time, but on 30% of occasions she is tired or doesn't want to listen to Bob re-tell the story she already heard three times tonight. Suppose further that in cases where she doesn't want to drive Bob, in order to save face and avoid accidentally reusing an excuse, she grits her teeth and responds positively about 90% of the time. Its only a short car ride home she reminds herself.

Doing the simple math, we see that Bob will only hear a concrete no about 3% of the time, compared to Alice's perfect world figure of ~30%. This is clearly less than ideal for Alice but may also be bad by Bob's own lights. Sure, he may occasionally indulge his choice of dinnertime conversation topic for slightly longer than his fellow interlocutor prefers, but he does not want to be disrespecting Alice's preferences a whole 27% of the time! Here the social norms have had a net negative effect: making Bob feel guilty and frustrating Alice's true preferences.

What then, can Bob do? Clearly, if it is knowledge of Alice's true preferences that he wants to act on, he must change his tack. Just asking Alice what she wants in front of the whole crowd of guests does not work. He could try explicitly declaiming his willingness to catch an uber or otherwise find another way home. This goes some of the way, but if Alice is especially people-pleasing she may still feel that she doesn't really have a choice in responses. What Bob really needs to do is to give Alice an easy, socially acceptable way out . Something that shifts the expected response from a yes to a no. "You probably need to get home quick to get those leftovers in the fridge, right!". Here Alice can graciously accept Bob's remark and leave freely, or she can choose to offer a second time, making clear her original offer's legitimacy.

Put again in more Bayesian terms: we might think that Alice has some probability of answering yes or no, and model Alice's actual response as the argmax of the posterior distribution of this answer after it has been updated based on her preference. We can define some internal state that corresponds to how strong that preference is, eg. maybe tonight she has an 80-20 preference (or likelihood ratio) for not taking Bob. Assuming that Alice starts out with a 90% chance of saying yes (due to her social obligations), we can use the odds form of Bayes rule to calculate the posterior distribution:

$$ 1:9 \times 8:2 = 8:18 = \frac{8}{8+18} = 31\% $$

Importantly, the answer implied by this distribution is not the same as that implied by Alice's actual preference. In fact, Alice would need to have a preference stronger than the prior to shift the posterior meaningfully. Clearly, naively looking at the maximum a posteriori (MAP estimate) here is not going to effectively recover Alice's internal state. A strong social prior towards saying yes has drowned out the influence of Alice's preference. If Bob really wants to recover Alice's preference, he needs to strive to create an unbiased, uniform prior so that the signal isn't lost and Alice's response actually tracks her preferences.

The abstracted principle here is that: if no isn't a real option, don't expect that the stated answer reflects the truth.

All of this is quite obvious of course, and all of us have unconsciously navigated a situation like this before. Nonetheless, I find that explicitly thinking about whether the conditions at hand allow for clear signals is helpful, and allows me to better interpret other's preferences and know when to do small adjustments.