[ajtulloch]

One way to view this formula is to use the fact that the Beta distribution is a conjugate prior for the binomial distribution.

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!

[TekMol]

Can this way to view the formula be expressed without the terms

    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.

[ivansavz]

Yeah, that's a lot of jargon associated with Bayesian statistics, but at it's root the idea is simple. How to merge information you have before observing some data (a.k.a. prior) with new information you just observed, to obtain updated information (a.k.a. posterior) that includes both what you believed initially + the new evidence you observed.

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.

[bigbillheck]

> Can this way to view the formula be expressed without the terms

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?".

[checkyoursudo]

Have you ever heard of the "Up Goer 5"?

[svalorzen]

Also, in case anyone is interested, the uninformative Jeffreys prior for this in Bayesian statistics (meaning it does not assume anything and is invariant to certain transformations of the inputs) is Beta(0.5, 0.5). Thus the initial guess is 0.5, and it evolves from there from the data.

[CrazyStat]

Isn't 0.5 an absurd guess for the probability of a new restaurant being exceptionally good?