Suppose there is a particle in a box. We are interested in identifying what type of particle it is, but are not allowed look inside the box. All we can do is observe the particles that are entering the box and how their behaviour changes as they exit. For example if a proton enters the box and leaves with it's trajectory bent towards the centre of the box we get some evidence that the unknown particle is negatively charged. Assume the order of incoming particles is deterministic but unknown in advance.
All exit behaviour is subject to random fluctuation. So it's a possible a proton will be bent away from the centre of the box by a negatively-charged particle. However we know the parameters of the random fluctuations for each type of entering particle. For example the means and subgaussian proxies of all possible interactions. So for example if a series of protons enters and is bent towards the centre, we can compute a confidence with which the unknown particle is negative.
Mathematically we can reduce this to a seres of random varaiables $X_1,X_2,\ldots, X_N$ with known proxies $\sigma_1^2,\sigma_2^2,\ldots, \sigma_N^2$ and mean $1$ in case the unknown particle is positive charge, and mean $0$ otherwise.
The optimal way to combine all the data after $N$ observations is using the linear combination $Y_N = \sum_i \alpha_i X_i$ for $\alpha_i = \frac{1/\sigma_i^2 }{\sum_j 1/\sigma_j^2}$. This gives $Y_N$ with subgaussian proxy $\Sigma_N = \sum_i \sigma_i^2$. Then we can guess positive charge if $Y_N > 1/2$ and negative if $Y_N \le 1/2$. The subgaussian proxy says the guess is correct with probability $O(e^{-K \Sigma_n})$ for some fixed $K >0$.
So far so good. But what if we are instead interested in tail bounds for the guesses. In other words we want to bound the probability all the guesses on turns $m,m+1,\ldots, N$ are correct?
This is easy if for example all $\sigma_i^2 = c$. In that case the linear combination is just $Y_n = \frac{X_1 + \ldots + X_n}{n}$ and the guess has accuracy $O( e^{-K n})$ which we can integrate to get $O(\frac{e^{-K m}}{K})$ for the tail.
Following this approach we could try to sum up $e^{-K \Sigma_n}$ for $n= m,m+1,\ldots, N$. But this raises at least one silly problem. Namely if some $\sigma_n^2 = \infty$ say (or in other words particle $n$ provides no information) then it makes sense to assign zero coefficient to that random variable, and get $Y_{n} = Y_{n-1}$, but no sense to include both terms $e^{-K \Sigma_{n-1}}$ and $e^{-K \Sigma_n}$ in the sum, since guesses $n-1$ and $n$ are the same.
This particular silly example can be remedied by just ignoring all bad particles, for example ignore all neutrino-type particles. The interesting case is when there are particles that provide a small yet nonzero amount of information, and we want to include them in our guesses in a way that improves the theoretical bounds rather than damages them.
Informally I would suspect that if sample $n$ provides very little information we will give it a very small coefficient and a good guess on turn $n-1$ increases the likelihood of a good guess on turn $n$. I wonder is there a way to make this formal?
