Demographic matrix models are standard tools for analyzing the dynamics of age- or stage-structured populations. Here, we present a method for estimating the average vital rates that parameterize a demographic matrix using a series of measurements of population size and structure (an “inverse problem” of demographic analysis). We join a deterministic, density-independent demographic matrix model with a stochastic observation model to write a likelihood function for the matrix parameters given the data. Adopting a Bayesian perspective, we combine this likelihood function with prior distributions for the model parameters to produce a joint posterior distribution for the parameters. We use a numerical technique (Markov chain Monte Carlo) to estimate and analyze the posterior distribution, and from this we calculate posterior distributions for functions of the demographic matrix, such as the population multiplication rate, stable stage distribution, and matrix sensitivities. Although measurements of population size and structure rarely contain enough information to estimate all the parameters in a matrix precisely, our analysis sheds light on the information that the data do contain about the vital rates by quantifying the precision of the parameter estimates and the correlations among them. Moreover, we show that matrix functions such as the population multiplication rate and matrix sensitivities can still be estimated precisely despite sizable uncertainty in the estimates of individual parameters, permitting biologically meaningful inference. We illustrate our approach for three populations of pea aphids (Acyrthosiphon pisum).