# Bayes and serial correlation in disagreements

People who have been in a long-term relationship are likely to recognise that fights between a couple are not Markovian – in that the likelihood of fighting today is not independent of the likelihood of having fought yesterday.

In fact, if you had fought in a particular time period, it increases the likelihood that you’ll fight in the next time period. As a consequence, what you are likely to find is that there are times when you go days, or weeks, or even months, together in a perennial state of disagreement, while you’ll also have long periods of peace and bliss.

While this serial correlation can be disconcerting at times, and make you wonder whether you are in a relationship with the right person, it is not hard to understand why this happens. Once again, our old friend Reverend Thomas Bayes comes to the rescue here.

This is an extremely simplified model, but will serve the purpose of this post. Each half of a couple beliefs that the other (better?) half can exist in one of two states – “nice” and “jerk”. In fact, it’s unlikely anyone will completely exist in one of these states – they’re likely to exist in a superposition of these states.

So let’s say that the probability of your partner being a jerk is $P(J)$, which makes the probability of him/her being “nice” at $P(N) = 1- P(J)$. Now when he/she does or says something (let’s call this event $E$), you implicitly do a Bayesian updation of these probabilities.

For every word/action of your partner, you can estimate the probabilities in the two cases of your partner being jerk, and nice. After every action E by the partner, you update your priors about them with the new information.

So the new probability of him being a jerk (given event E) will be given by
$P(J|E) = \frac{P(J).P(E|J)}{P(J).P(E|J) + P(N).P(E|N)}$ (the standard Bayesian  formula).

Now notice that the new probability of the partner being a jerk is dependent upon the prior probability. So when $P(J)$ is already high, it is highly likely that whatever action the partner does will not move the needle significantly. And the longer $P(J)$ stays high, the higher the probability that you’ll lapse into a fight again. Hence the prolonged periods of fighting, and serial correlation.

This equation also explains why attempts to resolve a fight quickly can backfire. When you are fighting, the normal reaction to resolve it is by committing actions that indicate that you are actually nice. The problem is that the equation above has both $P(E|N)$ and $P(E|J)$ in it.

So, in order to resolve a fight, you should not only commit actions that you would do when you are perceived nice, but also actions that you would NOT do if you are a jerk. In other words, the easiest way to pull $P(J)$ down in the above equation is to commit $E$ with high $P(E|N)$ and low $P(E|J)$.

What complicates things is that if you use one such weapon too many times, the partner will will begin to see through you, and up her $P(E|J)$ for this event. So you need to keep coming up with new tricks to defuse fights.

In short, that serial correlation exists in relationship fights is a given, and there is little you can do to prevent it. So if you go through a long period of continuous disagreement with your partner, keep in mind that such things are par for the course, and don’t do something drastic like breaking up.