The paper by Alexander Frankel and Emir Kamenica, featured in the American Economic Review in 2019, is exceptionally remarkable. Their work, titled Quantifying Information and Uncertainty, delves into the intricate dynamics of information and belief systems.
Reference: Quantifying Information and Uncertainty (2019) by Alexander Frankel and Emir Kamenica. Published in the American Economic Review, Volume 109, Issue 10, pages 3650–3680. DOI: 10.1257/aer.20181897.
Suppose we observe some pieces of news. How might we quantify the amount of information contained in it? Another related question, how might we quantify the uncertainty of a belief? One desideratum might be that the measure of information/uncertainty should correspond to the instrumental value/loss associated with some decision problem.
Let’s get down to business. Consider a finite state space $\Omega = {1, 2, …, n}$, with a typical state denoted as $\omega \in \Omega$. A belief $q$ is distribution on $\Omega$ that puts weight $q^\omega$ on state $\omega$. For a believe that is degenerate on $\omega$, denote $\delta_\omega$.
Information is generated by signals. Let $S$ be the set of non-empty Lebesgue-measurable subsets of $\Omega \times [0, 1]$, then, $\pi \subset S$ is a signal and some $s\in \pi$ would be the realization of the signal. Intuitively, for state $\omega$, a random variable $x$ WLOG drawn uniformly from $[0, 1]$ determines the signal realization conditional on the state, where the probability of observing realization of signal $s\in \pi$ in state $w$ is the Lebesgue measure of ${x \in [0, 1]|(\omega, x)\in s}$.
Usually, we write $\alpha$ as the realized random variable induced by signal $\pi_\alpha$. Given prior $q$, denote the posterior induced by signal realization $\alpha$ by $q(\alpha)$.
Join us tomorrow for Part II of our exploration.