Grant application result from incomplete data

Results for financing applications often arrive in two steps:

1. One's personal score (confidential) and some overall statistical information (public).
2. Final decision (accepted/rejected).

For projects close to the (unknown) cutoff value, the delay between 1. and 2. can be quite long; it is imposed by the negotiation process with those higher up on the list.

A real-life example I was confronted to are the Marie Curie Fellowships: There were 3300 valid applications for the 2011 Intra-European Fellowships (IEF), out of which 800 fell below a threshold T=70%. Knowing that about 600 projects will be financed and that your score is S (say, 95%), will you make the cut ?

Let us first search for the most probable distribution $P(x)$ obeying the constraints:

• $P(x)$ is defined on (0,1) : $\int _0 ^{1} P(x) \, \text{d} x = 1$
• $\int _0 ^{T} P(x) \, \text{d} x = F_1 = 800/3300 = 0.2424$

The problem is easily solved yielding a piecewise constant $P(x)$, with $P(x\leq T) = F_1/T$ and $P(x \geq T) = (1-F_1)/(1-T)$ . Defining the success rate above the threshold $M = 600/(3300 - 800)$, we conclude:

• Your application is successful if $\frac{1-S}{1-T} \leq M$. For the numerical values above, this corresponds to $S \geq S_{\text{min}} = 94.5$.
• The number of applications below the threshold is in fact irrelevant. 

We limited the reasoning to one distribution (the most probable), for which there is a unique answer. However, we understand intuitively that the uncertainty is very large; it would be much more useful  to determine completely the success probability $g(S)$. Only for $g(S)$ significantly different from $1/2$ can one draw any conclusion from such limited information.