Widespread use of antibiotics has resulted in an increase in antimicrobial-resistant microorganisms. Although not all bacterial contact results in infection, patients can become asymptomatically colonized, increasing the risk of infection and pathogen transmission. Consequently, many institutions have begun active surveillance, but in non-research settings, the resulting data are often incomplete and may include non-random testing, making conventional epidemiological analysis problematic. We describe a mathematical model and inference method for in-hospital bacterial colonization and transmission of carbapenem-resistant Enterobacteriaceae that is tailored for analysis of active surveillance data with incomplete observations. The model and inference method make use of the full detailed state of the hospital unit, which takes into account the colonization status of each individual in the unit and not only the number of colonized patients at any given time. The inference method computes the exact likelihood of all possible histories consistent with partial observations (despite the exponential increase in possible states that can make likelihood calculation intractable for large hospital units), includes techniques to improve computational efficiency, is tested by computer simulation, and is applied to active surveillance data from a 13-bed rehabilitation unit in New York City. The inference method for exact likelihood calculation is applicable to other Markov models incorporating incomplete observations. The parameters that we identify are the patient-patient transmission rate, pre-existing colonization probability, and prior-to-new-patient transmission probability. Besides identifying the parameters, we predict the effects on the total prevalence (0.07 of the total colonized patient-days) of changing the parameters and estimate the increase in total prevalence attributable to patient-patient transmission (0.02) above the baseline pre-existing colonization (0.05). Simulations with a colonized versus uncolonized long-stay patient had 44% higher total prevalence, suggesting that the long-stay patient may have been a reservoir of transmission. High-priority interventions may include isolation of incoming colonized patients and repeated screening of long-stay patients.