beta distribution
conjugate
prior
binomial distribution
bernoulli distribution
posterior
Because I could easily grasp that it is a "trust formula" in the way mg described it. But this way to "view" the formula is a mistery to me.
Certainly! Let's try to express the formula in simpler terms without using the technical terms you mentioned:
The formula f(N)=N/(N+X) can be viewed as follows:
Imagine you have a scenario where you want to estimate the probability of a certain event happening, but you don't have much information about it. The value of N represents the number of times the event has been observed to happen, and X represents the number of times the event has not been observed (or happened negatively).
In this context, you can consider X as a measure of your initial belief or skepticism about the event happening. If X is small, you are more open to the possibility of the event occurring, and if X is large, you are more doubtful or skeptical about the event.
As you observe more instances of the event happening (N increasing), the formula allows you to update your belief about the event's probability. The larger the N value, the more you rely on the actual observed data to determine the probability of the event, rather than your initial skepticism represented by X.
In simpler terms, the formula takes into account the number of positive occurrences (N) and a measure of initial skepticism (X), and it gives you a revised probability of the event happening based on the available data.
So, X can be interpreted as the "strength of your initial belief" in the event happening, and as you observe more positive occurrences (N), your estimation becomes more influenced by the actual data rather than your initial belief.
You're asking "Can this way of viewing the formula in terms of Bayesian probability be expressed without any of the machinery of Bayesian probability?".
The probability machinery (Bayes rule) is a principled way to do this, and in the case of count data (number of positive reviews for the cafe) works out to give be a simple fraction n/(n+x).
Define: x = parameter of how skeptical you are in general about the quality of cafes (large x very sceptical), m = number of positive reviews for the cafe,
p = m+1 / (m+1+x) your belief (expressed as a probability) that the cafe is good after hearing m positive reviews about it.
Learning about the binomial and the beta distribution would help you see where the formula comes from. People really like Bayesian machinery, because it has a logical/consistent feel: i.e. rather than coming up with some formula out of thin air, you derive the formula based on general rules about reasoning under uncertainty + updating beliefs.
Essentially if you have a Beta(a, b) prior then your prior mean is a/(a+b) and after observing n samples from a Bernoulli distribution that are all positive, your posterior is Beta(a+n, b) with posterior mean (a+n)/(a+n+b). So in your example you effectively have a Beta(0, x) prior and x (“suspicious”/“gullible”) is directly interpreted as the strength of your prior!