Analyzing the risks of anthropogenic climate change requires sound probabilistic projections of CO2 emissions. Previous projections have broken important new ground, but many rely on out-of-range projections, are limited to the 21st century, or provide only implicit probabilistic information. Here we take a step towards resolving these problems by assimilating globally aggregated observations of population size, economic output, and CO2 emissions over the last three centuries into a simple economic model. We use this model to derive probabilistic projections of business-as-usual CO2 emissions to the year 2150. We demonstrate how the common practice to limit the calibration timescale to decades can result in biased and overconfident projections. The range of several CO2 emission scenarios (e.g., from the Special Report on Emission Scenarios) misses potentially important tails of our projected probability density function. Studies that have interpreted the range of CO2 emission scenarios as an approximation for the full forcing uncertainty may well be biased towards overconfident climate change projections.