>regardless of what the result of the first throw is, the chance that it is fair is necessarily lowered after the first throw.
This isn't true. You probably have a higher prior for it being unfair towards 6 or 1 (and how much depends on the scenario).
However, if for example, you have priors for it being potentially unfair towards every number equally, 1 throw only changes the posterior probability of whether it is biased towards that number but not whether it is unfair. In case of unequal priors for which numbers it might be biased (e.g. 0.01 chance of it being biased towards 6/1 and 0.001 for 2, 3, 4, 5 and 0.976 for unbiased) getting a number changes all those probabilities yes but for example if you get a 3 the chance of bias there gets higher but mostly at the expense of the probabilities for bias for the other numbers - which get lower (I am too lazy to calculate the posteriors).
All in all, you should know beforehand what your posteriors are going to be for any case based on any outcome and if you know that any one probability (like that of it being fair) is definitely going to lower no matter the outcome then that should be your current prior for that case.
This isn't true. You probably have a higher prior for it being unfair towards 6 or 1 (and how much depends on the scenario).
However, if for example, you have priors for it being potentially unfair towards every number equally, 1 throw only changes the posterior probability of whether it is biased towards that number but not whether it is unfair. In case of unequal priors for which numbers it might be biased (e.g. 0.01 chance of it being biased towards 6/1 and 0.001 for 2, 3, 4, 5 and 0.976 for unbiased) getting a number changes all those probabilities yes but for example if you get a 3 the chance of bias there gets higher but mostly at the expense of the probabilities for bias for the other numbers - which get lower (I am too lazy to calculate the posteriors).
All in all, you should know beforehand what your posteriors are going to be for any case based on any outcome and if you know that any one probability (like that of it being fair) is definitely going to lower no matter the outcome then that should be your current prior for that case.