Addressing Dynamic and Sparse Qualitative Data: A Hilbert Space
  Embedding of Categorical Variables
Anirban Mukherjee Hannah H. Chang
We propose a novel framework for incorporating qualitative data into
quantitative models for causal estimation. Previous methods use categorical
variables derived from qualitative data to build quantitative models. However,
this approach can lead to data-sparse categories and yield inconsistent
(asymptotically biased) and imprecise (finite sample biased) estimates if the
qualitative information is dynamic and intricate. We use functional analysis to
create a more nuanced and flexible framework. We embed the observed categories
into a latent Baire space and introduce a continuous linear map -- a Hilbert
space embedding -- from the Baire space of categories to a Reproducing Kernel
Hilbert Space (RKHS) of representation functions. Through the Riesz
representation theorem, we establish that the canonical treatment of
categorical variables in causal models can be transformed into an identified
structure in the RKHS. Transfer learning acts as a catalyst to streamline
estimation -- embeddings from traditional models are paired with the kernel
trick to form the Hilbert space embedding. We validate our model through
comprehensive simulation evidence and demonstrate its relevance in a real-world
study that contrasts theoretical predictions from economics and psychology in
an e-commerce marketplace. The results confirm the superior performance of our
model, particularly in scenarios where qualitative information is nuanced and
complex.