This paper studies the joint dynamics of real time U.S. inflation and the mean inflation predictions of the Survey of Professional Forecasters (SPF) on a 1968Q4 to 2017Q2 sample. The joint data generating process (DGP) is an unobserved components (UC) model of inflation and a sticky information (SI) prediction mechanism for SPF inflation predictions. We add drifting gap inflation persistence to a UC model that already has stochastic volatility (SV) afflicting trend and gap inflation. Another innovation puts a time-varying frequency of inflation forecast updating into the SI-prediction mechanism. The joint DGP is a nonlinear state space model (SSM). We estimate the SSM using Bayesian tools grounded in a Rao-Blackwellized auxiliary particle filter, particle learning, and a particle smoother. The estimates show (i) longer horizon average SPF inflation predictions inform estimates of trend inflation, (ii) gap inflation persistence is pro-cyclical, and SI inflation updating is frequent before the Volcker disinflation, and (iii) subsequently, trend inflation and its SV fall, gap inflation persistence turns counter-cyclical, and SI inflation updating becomes infrequent.