Analyze Distribution of Daily Page Shares Per User

(2) The core concept: regression to the mean Before even thinking about regression models, ask: what happens naturally to extreme values over time? The p50 user (0 shares today) is a typical user. They'll stay roughly typical. Not much change expected. The p95 user (say, 8 shares today) is unusually active today. But are they genuinely a high-sharer, or did they just have an unusually active day? Likely some of both. Over the next 14 days, the extreme behavior will partially "revert" back toward their true average. This is called regression to the mean — extreme observations today are partly signal (their true rate) and partly noise (a lucky/unlucky day). So directionally: p50 user → stays near 0, maybe slightly above p95 user → comes down noticeably from their day-1 peak, settles at something lower Why simple linear regression isn't quite right here Linear regression would work if the outcome were continuous and symmetric. But daily shares are: Count data (0, 1, 2, 3...), not continuous Zero-inflated and right-skewed Non-negative by definition Predicting negative shares is nonsensical, but linear regression doesn't know that. What you'd actually use A Negative Binomial regression (since we already established shares are overdispersed count data) where: Outcome: avg daily shares over the next 14 days Key predictor: today's share count (or better, their historical average) Other predictors you'd add: days active on platform, avg shares over past 30 days, user tenure The histor

(3) Cohort with 2 shares on day 1 2 shares is slightly above the median (which is 0) but not extreme. These users are mildly active. Their day 1 value is a mix of: Some genuine mild-sharers (true signal) Some typical 0-share users who just happened to share twice (noise) Over days 2–30, the noise users revert to 0, dragging the cohort average down. The genuine mild-sharers stay around 1–2. So the trend looks like: Starts at 2 → drops in the first few days → stabilizes at some value below 2, maybe around 0.8–1.2 The drop is moderate and the stabilization happens relatively quickly. Cohort with 5 shares on day 1 5 shares is quite extreme — well into the right tail. This cohort is a mix of: A few genuine power users (true signal) Many moderate users who had an unusually active day (noise) The noise component is much larger here because 5 shares is so far from the typical user's behavior. So over days 2–30: Starts at 5 → drops sharply in the first few days → stabilizes much lower, maybe around 1.5–2.5 The drop is steeper and more dramatic than the 2-share cohort. This is regression to the mean hitting harder because the starting point was more extreme. Visualizing the trend--- Which cohort has larger variance? The 5-share cohort, and by a lot. Here's why: The 5-share cohort is a much more mixed bag of people. It contains: Genuine power users who'll stay at 5–8 shares/day Moderate users who'll drop to 1–2 Typical users who had a freak day and will drop to 0 That wide spread of und