  Topological data analysis has emerged as a powerful tool for extracting the
metric, geometric and topological features underlying the data as a
multi-resolution summary statistic, and has found applications in several areas
where data arises from complex sources. In this paper, we examine the use of
topological summary statistics through the lens of statistical inference. We
investigate necessary and sufficient conditions under which \textit{valid
statistical inference} is possible using {topological summary statistics}.
Additionally, we provide examples of models that demonstrate invariance with
respect to topological summaries.
