Suppose that I have a coin whose output set is $\{+,-\}$. Suppose that the output distribution, $p:(1-p)$, is fixed. Our goal is to guess what $p$ is based on real-world experiments.
Suppose that I have tossed it for $100$ times, and exactly $70$ times were positive $+$ and exactly $30$ times were negative $-$. How is this experiment supposed to affect my guess about $p$? This is not a math question, but rather a philosophical one.
Of course, we cannot conclude that $p = 0.7$. All we can say is that under certain assumptions/requirements it seems that $ p = 0.7$ is the most reasonable choice, so far.
Questions
What really are assumptions/requirements in the field of statistics? This philosophical choices are often, if not always, omitted or obscured in textbooks, wikipedia, or blog posts.
Based on the assumptions/requirements given in 1., how does one compare the ($p=0.7$)-model with any other model, say the ($p=0.6$)-model?
Maybe one does not want to compare a single model to another, but rather a chunk of models to another chunk. For example, one might wish to compare that the model has parameter $p \in [0.6,0.8]$ with that the model has parameter $p \in [0.3,0.5]$. To do this, it seems to me that we need to integrate some "confidence" function over the intervals $[0.6,0.8]$ and $[0.3,0.5]$. How to define this "confidence" function (essentially reduced to 2.)? And how does one choose the measure on $[0,1]$ for the integrations?