Can someone give a good rundown of the differences between the Bayesian and the frequentist approach to probability?
From what I understand:
The frequentists view is that the data is a repeatable random sample (random variable) with a specific frequency/probability (which is defined as the relative frequency of an event as the number of trials approaches infinity). The underlying parameters and probabilities remain constant during this repeatable process and that the variation is due to variability in $X_n$ and not the probability distribution (which is fixed for a certain event/process).
The bayesian view is that the data is fixed while the frequency/probability for a certain event can change meaning that the parameters of the distribution changes. In effect, the data that you get changes the prior distribution of a parameter which gets updated for each set of data.
To me it seems that the frequentist approach is more practical/logical since it seems reasonable that events have a specific probability and that the variation is in our sampling.
Furthermore, most data analysis from studies is usually done using the frequentist approach (i.e. confidence intervals, hypothesis testing with p-values etc) since it is easily understandable.
I was just wondering whether anyone could give me a quick summary of their interpretation of bayesian vs frequentist approach including bayesian statistical equivalents of the frequentist p-value and confidence interval. In addition, specific examples of where 1 method would be preferable to the other is appreciated.