Surveys play a central role in much of the research conducted in the humanities and social sciences. A common data type encountered in surveys is the ordinal variable, which differs from nominal categorical variables. Several regression methods are available for analysing ordinal data, with the cumulative logistic model being one of the most widely used. However, ordinal survey data often present challenges, particularly in studies with small sample sizes, where some response categories and levels of explanatory variables can have low response rates. In such cases, classical statistical methods can produce unreliable or incomplete estimates. Here, we investigate the use of probabilistic programming, grounded in Bayesian analysis, as a more robust alternative for estimating category probabilities of ordinal variables and other model parameters. These models are better equipped to handle uncertainty and provide more reliable estimates, even in the presence of sparse data. We validate the approach with simulated data where the ground truth is known, and demonstrate the advantages of this approach by comparing it to its classical frequentist counterpart in the context of cultural participation and access survey.