Profile Hidden Markov Models (HMMs) are graphical models that can be used to produce finite length sequences from a distribution. In fact, although they were only introduced for bioinformatics 25 years ago (by Haussler et al., Hawaii International Conference on Systems Science 1993), they are arguably the most commonly used statistical model in bioinformatics, with multiple applications, including protein structure and function prediction, classifications of novel proteins into existing protein families and superfamilies, metagenomics, and multiple sequence alignment. The standard use of profile HMMs in bioinformatics has two steps: first a profile HMM is built for a collection of molecular sequences (which may not be in a multiple sequence alignment), and then the profile HMM is used in some subsequent analysis of new molecular sequences. The construction of the profile thus is itself a statistical estimation problem, since any given set of sequences might potentially fit more than one model well. Hence a basic question about profile HMMs is whether they are \em statistically identifiable, which means that no two profile HMMs can produce the same distribution on finite length sequences. Indeed, statistical identifiability is a fundamental aspect of any statistical model, and yet it is not known whether profile HMMs are statistically identifiable. In this paper, we report on preliminary results towards characterizing the statistical identifiability of profile HMMs in one of the standard forms used in bioinformatics.