Total Stake Inference

Definitions

$\ell$ - enumeration of epochs **$\ell=0,1,\ldots$ corresponds to epochs 0,1,…
$\hat{D}_\ell$ - is our estimate for total stake at epoch $\ell$
$D_0$ - is our initial estimate of total stake (i.e. initial conditions)
$D^0$ - the true total stake
$s_i(t)[\hat{D_\ell}] \in 0,1 \text{ sampled from } \phi_f(w_i/\hat{D}_\ell)$ - the leader election result for node $i$ at slot $t$ in epoch $\ell$
$h > 0$ - the learning rate, i.e how quickly $\hat{D}_\ell$ adapts to changes in observations
$T$ - the length of each epoch
$N$ - the number of nodes in the system

For given initial estimate, $D_0$ we learn total stake by running the following process.

$$ \hat{D}{\ell+1}=\hat{D}\ell-h\left(\log\left(\frac{1}{1-f}\right) - \frac{\sum_t^T\sum_i^N{s_i(t)[\hat{D}_\ell]}}{T}\right) $$

In words, $\frac{\sum_t^T\sum_i^N{s_i(t)[\hat{D}_\ell]}}{T}$ gives the average number of leaders per slot observed over the course of an epoch of length $T$. We expect to see roughly $\log\left(\frac{1}{1-f}\right)$ leaders per slot., the difference $\log\left(\frac{1}{1-f}\right) - \frac{\sum_t^T\sum_i^Ns_i(t)}{T}$ gives us our “error” term in the inference algorithm. We then scale this error by $h$ to get our correction.