There are a number of philosophical issues raised in any
discussion on prior probabilities. For further information on such
discussion see [38], [27], [5].
It is essential, when considering 
as a random
variable, to assign prior probabilities, simply because such must
exist. In the case where prior knowledge shows that no particular
value or values of are more likely than any others,
then will be uniformly distributed. That is to say,
. It is important to note that such a
statement of initial belief is saying that at the outset, it is
believed that for example 100% of cars being rusty is as equally
likely to be prevalent as 0%, or indeed any other intermediate
value.
A more reasonable situation would be one where cars are being surveyed in the light of previous work and with some knowledge of the materials involved. Then the prior might take the form of a normal distribution with some mean and (perhaps large) variance.
For notational simplicity 
is written and taken to
mean 
from here onwards.