Frequentist approach to prior information From the frequentist point of view, is it possible to incorporate prior information about a parameter into a probability model? Let me illustrate with an example: suppose you have a die with black and white faces but you do not know how many of each; you are then told that you can flip a coin for each face of the die and if it lands heads, then the face will be black, otherwise it will be white. Then, we are interested in finding out the number of times $X$ among $n$ die throws that the upper face will be black. 
From the Bayesian point of view, it is clear that we can consider a binomial distribution for the prior for the probability of success $\theta$ (black face being upper). We then compute the likelihood of $\theta$ for some observed data $x$, then the posterior, and we are done. But from a frequentist point of view, how can we incorporate the information that we have about $\theta$? Do we just consider the fact that $\theta = 3/6$ is the most probable value and thus just use the probability model for $X$ evaluated at that given value of $\theta$; or is the incorporation of prior knowledge incompatible with frequentist assumptions? Does this mean that in cases like this, a frequentist is limited to using a point estimate for $\theta$? Do prior information/posterior distribution have a frequentist equivalent?
edit: I had already seen from the suggested duplicate post that you can incorporate prior information with Bayes rule, but it is still not entirely clear to me what this means: is this just a way of saying that you need to use a point estimate for the parameter according to what you learnt from the prior information?
 A: Your question can be disentangled in a few distinct ones:
" Do prior information/posterior distribution have a frequentist equivalent?"
No, the frequentist approach does assume that the parameters of interest are unknown constant, which need to be estimated, but does not assign any prior distribution to them. 
"Is  the incorporation of prior knowledge incompatible with frequentist assumptions?"
Strictly speaking, incorporating prior information in a frequentist setting is not impossible, but more difficult. For example, if you knew that your parameter were dependent on the values of other covariates, you could in principle incorporate that information in your likelihood.
As far as your examples, they are a bit difficult to follow: in the frequentist setting, the estimation of the parameters of interest may be done by maximum likelihood, i.e. by taking into consideration the data in your likelihood. All the "biases" you mention in your examples would be easily captured by the available data, hence by MLE and I guess that the Bayesian and frequentist answers would be very similar. To clarify, there's no real prior information that you can use in your example, which will not be captured through the data in order to help you with your inference problem. In your case, a prior information may be that white faces are only contiguous (if such scheme fits the die). That is information which you would not be able to capture by the simple rolling of the dice, but can help your estimation (since the number of configurations is limited).
