### abstract ###
Walley's Imprecise Dirichlet Model (IDM) for categorical  iid  \ data extends the classical Dirichlet model to a set of priors
It overcomes several fundamental problems which other approaches to uncertainty suffer from
Yet, to be useful in practice, one needs efficient ways for computing the imprecise=robust sets or intervals
The main objective of this work is to derive exact, conservative, and approximate, robust and credible interval estimates under the IDM for a large class of statistical estimators, including the entropy and mutual information
### introduction ###
This work derives interval estimates under the Imprecise Dirichlet Model (IDM)  CITATION  for a large class of statistical estimators
In the IDM one considers an  iid  \ process with unknown chances  SYMBOL  for outcome  SYMBOL
The prior uncertainty about  SYMBOL  is modeled by a set of Dirichlet priors  SYMBOL , where%  %  SYMBOL , and  SYMBOL  is a hyper-parameter, typically chosen between 1 and 2
Sets of probability distributions are often called Imprecise probabilities, hence the name IDM for this model
We avoid the term  imprecise  and use  robust  instead, or capitalize  Imprecise
The IDM overcomes several fundamental problems which other approaches to uncertainty suffer from  CITATION
For instance, the IDM satisfies the representation invariance principle and the symmetry principle, which are mutually exclusive in a pure Bayesian treatment with proper prior  CITATION
SYMBOL  The counts  SYMBOL  for  SYMBOL  form a minimal sufficient statistic of the data of size  SYMBOL
Statistical estimators  SYMBOL  usually also depend on the chosen prior: so a set of priors leads to a set of estimators  SYMBOL
For instance, the expected chances  SYMBOL  lead to a robust interval estimate  SYMBOL
Robust intervals for the variance  SYMBOL   CITATION  and for the mean and variance of linear-combinations  SYMBOL  have also been derived  CITATION
Bayesian estimators (like expectations) depend on  SYMBOL  and  SYMBOL  only through  SYMBOL  (and  SYMBOL  which we suppress), i e \  SYMBOL
The main objective of this work is to derive approximate, conservative, and exact intervals  SYMBOL  for general  SYMBOL , and for the expected (also called predictive) entropy and the expected mutual information in particular
These results are key building blocks for applying the IDM
Walley suggests, for instance, to use  SYMBOL  for inference problems and  SYMBOL  for decision problems  CITATION , where  SYMBOL  is some function of  SYMBOL
One application is the inference of robust tree-dependency structures  CITATION , in which edges are partially ordered based on Imprecise mutual information
Section  gives a brief introduction to the IDM and describes our problem setup
In Section  we derive exact robust intervals for concave functions  SYMBOL , such as the entropy
Section  derives approximate robust intervals for arbitrary  SYMBOL
In Section  we show how bounds of elementary functions can be used to get bounds for composite function, especially for sums and products of functions
The results are used in Section  for deriving robust intervals for the mutual information
The issue of how to set up IDM models on product spaces is discussed in Section
Section  addresses the problem of how to combine Bayesian credible intervals with the robust intervals of the IDM
Conclusions are given in Section
Appendix  lists properties of the  SYMBOL  function, which occurs in the expressions for the expected entropy and mutual information
Appendix  contains a table of used notation
