“…7 Third, bounded distributions such as the beta or the mixture of uniforms take literally the z ij that are zero, in that they place no probability mass on bins where the respondents place no mass. More broadly, the approach outlined in expression (1) does not directly address the issue of rounding: it solves the minimization problem taking all the Z ij 's literally even though the respondent may be reporting approximate probabilities (Dominitz and Manski, 1996;D'Amico and Orphanides, 2008;Boero et al, 2008Boero et al, , 2014Engelberg et al, 2009;Manski and Molinari, 2010;Manski, 2011;Giustinelli et al, 2020;Glas and Hartmann, 2022, among others, discuss the issue of rounding; Binder, 2017 uses rounding to measure uncertainty). 8 There have been attempts…”