Frequentists and Bayesians both agree on all of the math. The difference lies in the interpretation of what the math means, and how useful different types of ma

Frequentists and Bayesians both agree on all of the math. The difference lies in the interpretation of what the math means, and how useful different types of math are. A rough summary: Frequentist methods tend to try to characterize the question of interest in terms of a single measurable statistic. This is particularly useful in circumstances where memory/cpu is tight (i.e. every single circumstance before the last few decades.) This necessarily means though that these techniques lose much of the information in the data. They also try to make very few a priori assumptions about the underlying data. The bayesian critique of this is that the assumptions are still there, just hidden, and as a result are much more likely to be wrong. Bayesian methods tend to try to characterize the question of interest in terms of the entire distribution of outcomes. This causes there to be little information lost between the data and the conclusions, but is often computationally intractable in common circumstances. Bayesians tend to make strong a priori assumptions about the form the data will take, and to ameliorate this they make their assumptions very explicit in the model. The Frequentist critique of this is that every time you read a bayesian piece of analysis, you have to be very careful to determine whether the outcome described is merely the assumed prior + noise. Both cultures provide essential mathematical tools for any statistician's toolbox, and it's often just a matter of taste which get used. Hence the bike-shedding.