design of experiments

Bounds on Bounds

This is a TODO that reminds me to eventually write a post on limitations of MI estimators and alternative solutions.

Upper and Lower Mutual Information Bounds

Recently Foster et al (2020) introduced a couple of tractable bounds that can be used to estimate the expected information gain (EIG) of a design policy (a mapping from past designs and observations to the next design). These include the sequential Prior Contrastive Estimate (sPCE) lower bound: $$ \mathcal{L}_T(\pi) = \mathbb{E} \left[ \log \frac{p(h_T | \theta_0, \pi)}{\frac{1}{L+1} \sum_0^L p(h_T | \theta_l, \pi)} \right] $$ where $h_T = (d_0, y_0,\dots,d_T,y_T)$ is the experimental history at time $T$, $\pi$ is the design policy, $L$ is the number of contrastiv samples and $\theta$ parameterises the experimental model $p(y|\theta, d)$ that maps designs to a distribution over outcomes.