In general, in quantum mechanics (under most standard interpretations), ‘X is Y when unobserved’ doesn’t hold true – assuming that the spin of an electron has a definite value even when it isn’t being measured, together with some other, usually thought to be harmless assumptions (like locality, or noncontextuality, which roughly means that the measured value should be independent of the context of the measurement), yields a contradiction. The usual way out is to reject realism – i.e. the assumption that a quantum measurement reveals a pre-existing reality, such as the reality of the orientation of the electron’s spin. It’s also possible to reject locality and noncontextuality, which is the route taken by Bohm theory. But either locality or noncontextuality together with realism yields a contradiction – the former is exemplified in Bell’s famous theorem, the latter in the less well known Kochen-Specker theorem.
So there can’t be evidence that X is Y when unobserved because generally, X isn’t Y when unobserved.
(I see on preview that iamnotbatman has already mentioned Bell’s theorem, but I’m still gonna let this post stand, because IMO, Kochen-Specker yields yet tighter bounds on ‘hidden realities’: either they don’t exist, or reality depends on how you look at it – both of which seems to be difficult to square with the notion of something objective happening behind our backs.)
As for how this is to be squared with the possibility of making predictions about the world, well, it isn’t: quantum mechanics doesn’t make any definite predictions, just probabilistic ones, so for any given outcome, we are indeed ‘lucky’ that it obtains. It’s just that we can quantify how lucky. At any given time, anything could indeed happen – but this would only make science impossible if anything could happen with the same probability. But the set of possible outcomes is constrained, and these constraints provide the possibility of probabilistic predictions. These aren’t metaphysical, imposed-from-up-high constraints, but simply follow logically.
Perhaps an example makes this more clear. Picture a toy universe comprised, in a sense, of atomic propositions. Such a proposition can only be true or false, so any given ‘measurement’ will yield either 1 or 0; as what these propositions are about is left wholly arbitrary, either obtains with 50% probability. Now, model ‘macroscopic’ measurement as compounds of atomic propositions. Any given measurement then yields a string of binary digits. Macroscopic properties, such as ‘the moon is made of rock’ then correspond to properties of strings of bits, such as ‘there are equally many 0s as there are 1s’. Probabilities then arise from counting: there are very few strings that are comprised only of one kind of digit – two, in fact, so that the probability of a measurement having the outcome ‘all digits equal’ is very small (2/2[sup]n[/sup] if the ‘macroscopic measurement’ is composed of n atomic propositions, there being 2[sup]n[/sup] n-bit strings), while the probability of there being as many 1s as 0s is considerably higher. So, in this toy model, things like ‘the moon is made of cheese’ correspond to strings of all 1s or 0s, while ‘the moon is made of rock’ correspond to strings of equally many 1s and 0s. Clearly, finding the latter is much more probable than finding the former.
So even though there’s nothing ‘rocky’ out there in between observations, we can make predictions about the likelihood that we will see rock when looking at the moon, versus the likelihood that we will see cheese, just as we can make predictions about the likelihood of randomly generating uniform or varied strings of bits (or other properties of bit-strings).