AbstractThere has been a recent surge of interest in estimation theory based on very simple noise descriptions; for example, the absolute value of a noise sample is simply bounded. to date, this line of work has not been critically compared with pre‐existing work on stochastic estimation theory which uses more complicated noise descriptions. the present paper attempts to redress this by examining the rapprochement between the two schools of work. For example, we show that for many problems a bounded‐error estimation approach is precisely equivalent in terms of the final result to the stochastic approach of Bayesian estimation. We also show that in spite of having the advantages of being simple and intuitive, bounded‐error estimation theory is demanding on the quantitative accuracy of prior information. In contrast, we discuss how the assumptions underlying stochastic estimation theory are more complex but have the key feature that qualitative assumptions on the nature of a typical disturbance sequence can be made to reduce the importance of quantitative assumptions being correct. We also discuss how stochastic theory can be extended to deal with a problem at present tackled only by bounded‐error estimation methods: the quantification of estimation errors arising from the presence of undermo