[Emphere]

Others have already raised this point... but let me try to reiterate. The problem of getting priors is not just one of "acquiring more information". In many cases it's not even clear what such a probability means. For example, you believe that Trump is the 45th POTUS...and you assign it a prior of 0.8...what does the probability mean in this case? In the case of rolling dice it's clear what each probabilities mean, but not in this case. And in any case, how much should you update your probabilities for any given piece of prior? All of these questions (how to assign priors, how much to update them etc.) are the _crux_ of Bayesianism and Bayesianism itself has little to say about it. The founders of Bayesianism itself were aware of these issues. For a more substantive critique read the following.

https://www.jstor.org/stable/20079192?seq=1

Bayes is a good _tool_ but to me it's a very small one and it doesn't and _cannot_ do most of the heavy lifting of how to live my life. Suppose I want to decide what I should do next week, Bayes is close to useless for that. And that is certainly something "critical thinking" should help me with.

Keywords to search for "small worlds vs large worlds Bayes"

[john-shaffer]

The book "Probability Theory" by E.T. Jaynes is absolutely amazing at explaining how to actually think about Bayesian probabilities, how to assign priors, and the implications of different priors and posteriors. He explains it far better than I can, but one of the major points is that the prior becomes increasingly irrelevant with less data. If we have a lot of data, then we can assign just about any reasonable prior and still get accurate results. If we don't have much data, then the prior has a large influence. In this case we can't be as confident in the accuracy of our posterior probability, but we can calculate just how much confidence we can have in it.

Your example is hard to get into because it just doesn't make sense. The probability of Trump being the 45th President is 1. Garbage in, garbage out applies here.

If instead we step back to 2016 and say that we know the next president will be blue or red, then it's reasonable to assign the maximally uninformative prior of 0.5 to each outcome. Each additional piece of information modifies the probability. If we learn that the red candidate has been caught on tape speaking about groping women, we would calculate a new, lower posterior probability. How much lower depends on what we can determine about how much this hurts his election chances. We will get better results if we can estimate this factor accurately. This won't be much of an issue if we have many other pieces of data so that this one piece doesn't have a huge influence, but if this is our only piece of data then its accuracy will have a huge effect on the accuracy of our conclusion. For the sake of example, let's say that we our best information provides us with a posterior probability of 0.25. This posterior will be our prior when the next piece of information comes in (i.e., it's no longer an uninformative prior because it now contains past information).

Now suppose we see on TV that Trump has won the election. We don't have any money riding on the exact percentage, so let's just estimate the probability of the TV report being correct at 0.99. If we plug this in to Bayes' Theorem with our prior of 0.25, we get a new posterior of 0.97. We're not very confident in this posterior because we used an uninformative prior with 2 very weak updates, but if we really care about accuracy then we can seek out more and better data and get a far more accurate posterior. Moreover, we can calculate the confidence we have in the posterior based on the prior and the data.

Now in 2020 when the election can no longer be challenged, we update our posterior again, but it's not very interesting. Bayes' Theorem still applies, but since we know for a certainty who POTUS 45 is, the prior (0.97) factors out. We get a posterior probability of 1. Yet even though we've only used 3 pieces of data, we can calculate our confidence in our posterior as being extremely high since one our data points was so strong.

I don't know how much it can help you decide what to do next week, but if part of that decision involves calculating or reasoning about probabilities, than you absolutely should understand Bayes' Theorem. It's helpful to know how to think about it even if you can't assign exact numbers in the same way that understanding geometric or logarithmic curves is useful.