Let's say "pre-modern" means "before 1500", while allowing evidence from post-1500 cases to inform pre-modern cases.
I'm especially interested in cases where we have no or very limited reports from censuses. Otherwise, I'm deliberately leaving the question broad since I don't know what constraints would be useful, but my primary interest is the Middle East and Central Asia, for presumably obvious reasons. But I'm interested in anything that helps inform us of the reliability of the methods used.
Hi,
It's an open question. Estimates for Italy at the time of Augustus range from 5,500,000 people to 16,000,000 people. (Saskia Hin, The Demography of Roman Italy, page 20.) And that's with a census!
Some approaches include:
Using census data, where available, but it isn't always obvious who each census tries to count.
Using tax data, where available. For example, estimates for post-plague England rely on poll tax data from the 1370s.
Using army sizes for smaller groups. Given the small sizes and high mobilization rates of ancient Greek poleis, Mogens Herman Hansen uses army sizes to estimate some populations-- most often 12x the number of hoplite soldiers specifically, but potentially 6x the number of all soldiers. (Mogens Herman Hansen, "How to Convert an Army Figure into a Population Figure.")
Using archaeological field survey. Besides trying to count the number of small settlements, it may be useful to try to compare the extent of settlement and use of marginal land in different periods. If more marginal land is being farmed, then population is probably higher.
Using urban areas. Josiah Russell proposed extrapolation from the largest city in each region, assuming that the largest city usually accounted for 1.5% of the population of the region. Problems involve defining regions, and accounting for "flat" and "primate" city size distributions. (Late Ancient and Medieval Population and Medieval Regions and their Cities).
Using comparison with other areas with similar conditions and better data.