Is an HMM an appropriate model for a case like this? If not, what is? I hope to build a fully empirical model of Total Daily Energy Expenditure (TDEE), i.e. the number of calories an individual must eat in order to neither lose nor gain weight.
As data, I have individuals' weight (measured daily) and calorie intake (also daily).
As the TDEE is not directly observed, it occurred to me to use a Hidden Markov Model (HMM) for this. However, there are continuously many states that the system can be in -- one's TDEE could be any nonnegative real number. Does that mean an HMM is the wrong tool? If so, what might serve better here?
 A: HMMs can be used with a continuous state space, in the sense that you can extend the concepts (Bayesian reasoning, hidden/observed random variables, ...) and the use that same framework with continuous hidden variables. Quoting Wikipedia:

Hidden Markov models can also be generalized to allow continuous state spaces. Examples of such models are those where the Markov process over hidden variables is a linear dynamical system, with a linear relationship among related variables and where all hidden and observed variables follow a Gaussian distribution. In simple cases, such as the linear dynamical system just mentioned, exact inference is tractable (in this case, using the Kalman filter); however, in general, exact inference in HMMs with continuous latent variables is infeasible, and approximate methods must be used, such as the extended Kalman filter or the particle filter.

This tutorial is a good introduction on the topic.
As for your precise application, I am not familiar with it and there is too few details for me to judge. But you might give HMMs a try!
