There is a layer of probability in the question which I think obscures the issue without adding anything. Suppose $X_1,...,X_{10000}$ are IID random variables (whose values are numbers which are not necessarily probabilities of anything). Suppose $Y_1,...,Y_{10000}$ are IID standard Gaussians. Let $X_i + Y_i=Z_i$ be observable, and let $a_1,...,a_{100}$ be the indices with the highest values among the $Z_i$. The question is how much larger $(Z_{a_1} + ... + Z_{a_{100}})/100$ is over $(X_{a_1} + ... + X_{a_{100}})/100$ (perhaps conditioned on the values observed). This problem seems fundamental, and I hope a statistician can say how this is typically addressed.

Even if the $X_i$ are constant, this is still interesting, and it shows that you need to know the values $100$ and $10000$ (with a good approximation coming from the ratio). If you simply know that you have noisy versions of the top $100$, you don't know how strong the bias is. (In concrete terms, you need to know how many experiments were not published to estimate the bias toward type I statistical errors in what is reported.)

The bias in the case that the $X_i$ are constant is an upper bound for the bias in general. The proof is simply that $\sum Y_{a_i}$ is at most the sum of the greatest $100$ values among the $Y_i$.