How do Frequentist vs. Bayesian views of probability lead to their different treatments of data/hypotheses? I have read that a key difference between Bayesians and Frequentists is their treatment of probability. Frequentists treat probability as the frequency with which something will happen over the long run. Bayesians treat probability as a measure of their confidence in the outcome of a single event.
I've also read that Frequentists consider models to be fixed, and data to vary; while Bayesians consider models to vary and data to be fixed. 
How exactly do the different treatments of probability lead to these different views of data/models?
And then, the crux of my question is, why do these different views allow Bayesians to talk about the probability of a hypothesis being true given some data, while Frequentists are restricted to talking about the probability of data being true given some hypothesis?
 A: To perhaps overly simplify, a key distinction is that Bayesian statisticians are willing to use tools of probability in areas the frequentist wouldn't. Imagine we have a thumbtack that could land heads or tails. Let the probability of landing heads be $\theta$.


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*In frequentist statistics, $\theta$ is a real number.

*In Bayesian statistics, $\theta$ is a random variable.


Further explanation.


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*To the frequentist, the probability $\theta$ a thumbtack will land heads is a real number. We don't know what that number is $\theta$, but the parameter is a single real number in $[0, 1]$. Talking about the probability that $\theta > .5$ is bizarre because $\theta$ is not a random variable! It's a number. Asking for $P(\theta > .5)$ is like asking $P(1>2)$ or $P(2<3)$. $\theta > .5$ has no uncertainty. 

*In contrast, the subjective Bayesian is willing to model her uncertainty (in her own head) over $\theta$'s value using the tools of probability. To the subjective Bayesian, the probability $\theta$ a thumbtack will land heads is a random variable: it's a measurable function from some sample space to the space of real numbers. The expression $\theta > .5$ is another random variable: a measurable function from the sample space $\Omega$ to the space of binary values $\{0, 1\}$.
If you have access, David Kreps's Notes on the Theory of Choice has a beautiful dialogue that brings out these concepts (and from which I'm drawing the thumbtack example).
Once you treat $\theta$ as a random variable, you can talk about $P(\theta \mid X)$, the probability of parameter values given the data (using Bayes rule). It also raises the question of what's the unconditional distribution of $\theta$ (i.e. the prior).
