Bayesian methodologies are oftentimes referred to as being subjective in nature, given the involvement of prior specification, which is usually dependent on researchers' preferences. On the other hand, the frequentists do not require any specification of prior and are said to be objective. However, in the Bayesians paradigm, the process of prior specification has been formalized such that the most appropriate priors suitable for given cases are elicited. Thus, bayesian methods are totally transparent and very much objective as the frequentist methods, if not more.
Where is objectivity in the frequentists paradigm?
The only interpretation of probability I know that aims to give an objective foundation is that the probability of an event is the limiting relative frequency of the event occuring in an infinitively long series of trials/observations.* However, this interpretation lacks a physical (i.e. "objective") basis (as infinite series can in principle not be observed, and any result from a finite series is arbitrarily unrepresentative as an estimate, because it resembles only an infinitively small part of that infinite series the probability should refer to), and is mathematically (i.e. logically) impracticable. There need only two cases be considered: in this infinite series, the event will either happen an infinite number of times, or it will happen only finitly often. In the second case, the limit is 0, and in the first case, the limit is either undefined or 1 (depending if you want to define Inf/Inf=1). Apart from that, there are many other points that demonstrate that the limiting frequency definition is actually not a definition, it is neither objective nor logically sound. See http://www.joelvelasco.net/teaching/3865/hajek%20-%20mises%20redux%20redux.pdf and http://www.oxfordhandbooks.com/view/10.1093/oxfordhb/9780199607617.001.0001/oxfordhb-9780199607617-e-17.
After all, any frequentist method starts with assuming a particular probability model. Then, the probability of observed data can be evaluated w.r.t. that given model. That's "objective", as anyone following this method arrives at the same conclusions from the same data. However, the conclusions refer to a "subjectively chosen" model - and it gives just the probability (whatever it is!) of the data, where we are typically interested in statements about the likely cause of these observations (i.e. the probabilities of hypotheses - a term that is not even thinkable in a frequentist world, because hypotheses are no events that would occur in series of trials that could be repeated an infinite number of times!).
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*another candidate might be the disputed propensity interpretation, but as being a metaphysical concept I wouldn't connect it to "objectivity".
Thank you very much for that detailed contribution. The debate on objectivity of the frequentist approach call for further exposition, as it seem to more subjective than the bayesian equivalent.
For me there is not one Bayesian paradigm, but two: its subjectivist interpretation and its logicist interpretation. Its subjectivist interpretation follows de Finetti asumption of coherence, which leads to a psychological approach for the concept of probability. Its logicist interpretation follows the Jeffreys and later Cox assumpton of consistency, which leads to a logical approach.
For more details on these two approaches you can read the two chapters I wrote for the book Probability and social science published by Springer in the Methodos Series.