Decision Trees, Protocols, and the Fourier Entropy-Influence Conjecture

Given $f:\{-1, 1\}^n \rightarrow \{-1, 1\}$, define the \emph{spectral distribution} of $f$ to be the distribution on subsets of $[n]$ in which the set $S$ is sampled with probability $\widehat{f}(S)^2$. Then the Fourier Entropy-Influence (FEI) conjecture of Friedgut and Kalai (1996) states that there is some absolute constant $C$ such that $\operatorname{H}[\widehat{f}^2] \leq C\cdot\operatorname{Inf}[f]$. Here, $\operatorname{H}[\widehat{f}^2]$ denotes the Shannon entropy of $f$'s spectral distribution, and $\operatorname{Inf}[f]$ is the total influence of $f$. This conjecture is one of the major open problems in the analysis of Boolean functions, and settling it would have several interesting consequences. Previous results on the FEI conjecture have been largely through direct calculation. In this paper we study a natural interpretation of the conjecture, which states that there exists a communication protocol which, given subset $S$ of $[n]$ distributed as $\widehat{f}^2$, can communicate the value of $S$ using at most $C\cdot\operatorname{Inf}[f]$ bits in expectation. Using this interpretation, we are able show the following results: 1. First, if $f$ is computable by a read-$k$ decision tree, then $\operatorname{H}[\widehat{f}^2] \leq 9k\cdot \operatorname{Inf}[f]$. 2. Next, if $f$ has $\operatorname{Inf}[f] \geq 1$ and is computable by a decision tree with expected depth $d$, then $\operatorname{H}[\widehat{f}^2] \leq 12d\cdot \operatorname{Inf}[f]$. 3. Finally, we give a new proof of the main theorem of O'Donnell and Tan (ICALP 2013), i.e. that their FEI$^+$ conjecture composes. In addition, we show that natural improvements to our decision tree results would be sufficient to prove the FEI conjecture in its entirety. We believe that our methods give more illuminating proofs than previous results about the FEI conjecture

Well-being across America

This paper uses new Behavioral Risk Factor Surveillance System data to provide the first estimates of well-being across the states of America. From this sample of 1.3 million US citizens, we analyze measures of life satisfaction and mental health. Adjusting for people's characteristics, states such as Louisiana and DC have high psychological well-being levels while California and West Virginia have low well-being; there is no correlation between states' well-being and their GDP per capita. Correcting for people's incomes, satisfaction with life is lowest in the rich states. We discuss implications for the arbitrage theory that regions provide equal utility and compensating differentials.compensating differentials, BRFSS, happiness, geography, GHQ, Easterlin Paradox, mental health, depression, life course

Objective Confirmation of Subjective Measures of Human Well-being: Evidence from the USA

A huge research literature, across the behavioral and social sciences, uses information on individuals' subjective well-being. These are responses to questions – asked by survey interviewers or medical personnel – such as "how happy do you feel on a scale from 1 to 4?" Yet there is little scientific evidence that such data are meaningful. This study examines a 2005-2008 Behavioral Risk Factor Surveillance System random sample of 1.3 million United States citizens. Life-satisfaction in each U.S. state is measured. Across America, people's answers trace out the same pattern of quality of life as previously estimated, using solely non-subjective data, in a literature from economics (so-called 'compensating differentials' neoclassical theory due originally to Adam Smith). There is a state-by-state match (r = 0.6, pcompensating differentials, well-being, happiness, spatial equilibrium