Is it possible to have too many priors in Bayes Theorem?

Question

Each prior restricts the sample space, so in theory, your probability becomes more and more accurate. However, if our sample space becomes too small, the tradeoff between the accurate sample and the confidence we gain from having more data points becomes apparent.

Example: Assume I want to know the odds that I will contract COVID. I ask myself what is $P(\text{getting covid} | \text{20 years old})$. Let's say I get $10\%$. Then I stack another prior to get even more accurate data. I ask myself what is $P(\text{getting covid} | \text{20 years old} \land \text{is not obese})$. Great. Even more accurate probability.

However, what if I ask myself $P(\text{getting COVID} | \text{I have a tattoo of my name on my arm})$. Let's say there are $2$ people in the world, me and my friend, who have a tattoo of my name on his/her arm. Let's say my friend got COVID. This gives me a $1$ in $2$ odds of getting COVID which dosen't make sense.

A couple of ideas - maybe Bayes Theorem should be only used as a tool, maybe the priors have to be relevant to the outcome, maybe there's a threshold below which our sample is not significant. Not sure.

Answer 1

The first thing to realise here is that "probability" does not have the same meaning for everybody. P(getting Covid|20 years old)=0.1 may mean that you personally, based on all you know, would pay up to 10 Euros for the chance of winning 100 Euros in case a randomly selected 20 years old person catches Covid (obviously assuming that all is well defined, so for example this may mean "in the next year" or "before the 21st birthday" or whatever; same is assumed for the other possibilities below). This can be referred to as subjective epistemic probability.

It may also mean that in the long run, observing all 20 years olds, the relative frequency of those who catch Covid will approach 10%. Note that this is somewhat idealised, because we can only ever observe finitely many people. That would be a frequentist probability.

There is also objective epistemic probability, which wouldn't refer to individual persons but rather involve only secured knowledge and otherwise "informationless" priors, and which are not motivated via betting rates but rather via axioms for rational reasoning.

Bayes theorem holds for all those versions of probabilities (even though some use terminology so that it implies that "Bayesian probabilities" are distinctively different from frequentist ones), but in a given situation you need to be clear about which one you mean.

Talking about P(getting COVID|I have a tattoo of my name on my arm), in a situation where you know that currently there are only two people and one of them has caught Covid whereas the other one has reached the 21st year without catching it (assuming this is how you define the event), this will not tell you the probability, but rather the relative frequency of that (conditional) event. Assume that the probability refers to a potential future situation in which somebody else will have that tatoo (and no further information given).

Thinking about subjective probability, you are still free to choose your probability according to your beliefs (as Dave Harris has explained), take into account all your need and also the relative frequency from the sample you know, but 2 is of course not a big sample size, so you may not give that result a large influence on your subjective probability choice. For sure you can play the betting game, and for sure you are not constrained to choose your betting rate as 0.5.

Thinking about frequentist probability, the probability refers to the hypothetical event that infinitely more persons occurs with your tatoo. The relative frequency $\frac{1}{2}$ then is an estimate of the assumed true probability, but every good frequentist would tell you that it can't be a very good estimate with a sample size of two (as can be reflected in wide confidence intervals). Furthermore, if in fact you believe that in reality there will be no further people with that same tatoo, the frequentist probability can be seen as a useless fiction, because it refers to the repetition of an "experiment" that in reality will never be repeated.

Note however that we can think of indirect repetition in the sense that we may have a regression model explaining some $y$ from many $x$-variables; then a specific vector of values of all the $x$-variables may never repeat precisely, but we may think of observations with other $x$-values as repetitions of the same process, the same mechanism how the $y$ depends on the $x$; formally, probability refers to the "error term" implied in the model, which is repeated even if the precise $x$ is not.

Regarding the objectivist epistemic probability, given that there is no other information, you could start from an informationless prior about the probability and then update it based on your observed sample of size 2. The resulting posterior can be used as the prior for the next observation. This seems pretty nice; trouble is that people in the literature have come up with more than one way of specifying the "informationless prior"; however you choose it, it will have some implications that some can be seen as representing some kind of "information", which we'd want to avoid. Also if there is in fact more information that does not come in a nice formal form, it is often not clear how to "objectively" involve it.

What you are right about is that the more precisely you specify the conditioning event, the smaller is the number of available cases (even potentially in the future as long as we think about real observations rather than an idealised fantasy world). This is in fact something of a dilemma for all probability interpretations, unless you feel confident to just make up a subjective value out of thin air, because all further considerations require assumptions (regression models, independence, absence of interactions etc.) that cannot be checked at any precision using available observations for the exact event you are interested in.

Answer 2

You are not asking about having too many priors, but about conditioning. The prior in Bayes theorem is this part

$$ p(A|B) = \frac{p(B|A) ~\overbrace{p(A)}^\text{prior}}{p(N)} $$

while you seem to be asking about conditioning $p(B|\cdot)$ on many different variables. First, notice that conditional probabilities $p(B|A)$, $p(B|C)$, $p(B|C,A)$, $p(B|D,A)$, etc tell you about different scenarios. The probability that a CrossValidated.com user named Neel Sandell gets COVID is a different thing than the probability that any random person gets it, or that an MD working on a COVID ward gets it. Each time you condition, you restrict the space, that is correct. When you ask about probability given that someone has blond hair, the answer would be relevant only to the blond-haired people, not people in general. So you asked about a specific scenario, your question was restricted, not the answer.

This has nothing to do with the priors, it is just the bare fact that if you ask different questions, you get different answers, and if you ask specific questions, you get specific answers that may not be relevant to the general problem.

Answer 3

There are many different ideas that you seem to be mixing and matching together. It appears to be part of your confusion.

First, let us dispense with the idea that a prior restricts your sample space. It does not. It does restrict your personal beliefs about how likely it is to see any particular element of the sample space. Suppose you believe the mean to be seven and the variance one on normally distributed data. In that case, you should be amazed to see an observation of thirty-two.

Just because you are amazed does not imply that others would be amazed, and, certainly, Nature is unsurprised. Bayesian probability is subjective.

The second element seems to be about the relative value of trying out different models of the world. If your data set is large enough, the good news is that Bayesian probability is generative. Bayesian methods favor models that are closer to the method that Nature chooses to generate data. If the COVID virus favors tattoos, then it will impact the probabilities through the likelihood function. As long as you compare multiple models and create probabilities that those models are the one and only valid model, you will eventually pare away the models less reflective of reality.

To work, even the unlikely cases need to have a prior distribution.

Finally, significance is not a Bayesian idea. Bayesian probabilities are meaningful if they are meaningful to you. It is a subjective system. Thankfully, Bayesian probabilities do not reflect long-run frequencies. There is a totally other set of rules if that is what you need. Student’s t-test is a wonderful idea.

Your prior should reflect all prior knowledge about the models and parameters that you have in your possession that comes from any other source than the data itself. You can use all kinds of other prior distributions to make other people happy, but those are not your probabilities. If an editor wants you to use a diffuse prior distribution, you are not solving your probabilities. You are solving his or her probabilities. That is okay too. Likewise, if your boss wants you to use naive Bayes, then you are solving your boss's probabilities.

However, it is true that if you are highly bigotted, then your priors will interfere with learning. Conversely, if you are highly gullible, your prior distributions will interfere with learning if your sample is not representative of the population. It will take much data to learn just a little bit in the former case. It will take much data to overcome an initial unfortunate set of observations in the latter case. It is not a trivial task to determine if your prior knowledge and beliefs reflect reality. Of course, it is the job of the data to teach you what you thought you knew was correct or incorrect.

If your prior is too concentrated, it will slow learning if you are wrong. If your prior is too diffuse, then it will slow learning if you get an unfortunate set of initial observations as they will concentrate around the likelihood. Sometimes tossing ten heads in a row happens with a fair coin. If you are naive enough, then you will believe it is a two-headed coin. Both concentration and diffusion are problems unless they reflect your real beliefs.

Answer 4

Let's say there are 2 people in the world, me and my friend, who have a tattoo of my name on his/her arm. Let's say my friend got COVID. This gives me a 1 in 2 odds of getting COVID which dosen't make sense.

This is not a problem with too many priors/features/parameters, but with too little observations. (Although having too many features can result in having few observations for a particular class)

Say you would assume a Bernoulli distribution for an observation, sick or not sick with probability $p$ and $1-p$, where our prior estimate of the parameter $p$ is a flat beta(1,1) distribution, then after one single observation of another person that happens to get sick then the posterior for $p$ would be a beta(2,1) distribution and the estimate for another person to get sick would be 66.6% (the expectation value of a beta-binomial distribution with parameters $\alpha=2$, $\beta=1$ and $n=1$).

So that 66.6% is in this application of Bayes theorem even higher than your guess of 50%.

If that 66.6% is too high, then the reasons for the error of getting that high number are the choice of prior and the low number of observations.

In this example it doesn't matter whether we are dealing with tattoos or not.

When this type of error occurs, of having too little observations giving bad results, because we are dealing with too many variables, then we call this overfitting.

Related: Is it true that Bayesian methods don't overfit?