I have a fundamental question about "Bayes Factors". Specifically, I understand if we want to compute the posterior probability of a hypothesis (e.g., $p(H_1 |Data)$, we need a prior probability (i.e., $p(H_1)$) for the hypothesis in question as per Bayes' rule.
BUT I'm wondering when computing only a "Bayes Factor" which is a factor by which we update our prior belief about the hypothesis, why we need a prior for the determination of $H_1$. In other words, when we do NOT want to compute any posterior probability, why we talk about a prior when computing a Bayes Factor?
In fact, I even want to know what does the mathematical integration for the following 2 $H_1$s (one with a prior and the other without a prior) that I have used below (R code) exactly do?:
(For simplicity's sake, below, I compute a Bayes Factor for a binomial experiment that has resulted in 35 successes out of 100 trials.)
## With a Beta prior: H1 = integrate(function(p) dbeta(p, 1, 10)*dbinom(35, 100, p), 0, 1)[] # > 0.00235671 ## Without any kind of prior: H11 = integrate(function(p) dbinom(35, 100, p), 0, 1)[] # > 0.00990099 ## H0 H0 = dbinom(35, 100, .5) ## BF10: BF10 = H1 / H0 BF10 = H11 / H0