This paper studies the implications of a simple theorem, which states that for arbitrary underlying dynamics and diffusive information flows, the cumulants of Bayesian beliefs have a recursive structure: the sensitivity of the mean to news is proportional to the variance; the sensitivity of the $n$th cumulant to news is proportional to the $n+1$th. The specific application is the US aggregate stock market, because it has a long time series of high-frequency data along with option-implied higher moments. The model qualitatively and quantitatively generates a range of observed features of the data: negative skewness and positive excess kurtosis in stock returns, positive skewness and kurtosis and long memory in volatility, a negative relationship between returns and volatility changes, and predictable variation in the strength of that relationship. Those results have a simple necessary and sufficient condition, which is model-free: beliefs must be negatively skewed in all states of the world.