  Ordinal response model is a popular and commonly used regression for ordered
categorical data in a wide range of fields such as medicine and social
sciences. However, it is empirically known that the existence of ``outliers'',
combinations of the ordered categorical response and covariates that are
heterogeneous compared to other pairs, makes the inference with the ordinal
response model unreliable. In this article, we prove that the posterior
distribution in the ordinal response model does not satisfy the posterior
robustness with any link functions, i.e., the posterior cannot ignore the
influence of large outliers. Furthermore, to achieve robust Bayesian inference
in the ordinal response model, this article defines general posteriors in the
ordinal response model with two robust divergences (the density-power and
$\gamma$-divergences) based on the framework of the general posterior
inference. We also provide an algorithm for generating posterior samples from
the proposed posteriors. The robustness of the proposed methods against
outliers is clarified from the posterior robustness and the index of robustness
based on the Fisher-Rao metric. Through numerical experiments on artificial
data and two real datasets, we show that the proposed methods perform better
than the ordinary bayesian methods with and without outliers in the data for
various link functions.
