Using a Continuous Time Markov Chain for Discrete Times As I understand, there some major differences between Discrete Time Markov Chains compared to Continuous Time Markov Chains.
For example:

*

*Discrete Time Markov Chain: Characterized by a constant transition probability matrix "P"

*Continuous Time Markov Chain: Characterized by a time dependent transition probability matrix "P(t)" and a constant infinitesimal generator matrix "Q". The Continuous Time Markov Chain is based on the Exponential Distribution and thereby must obey the Memoryless Property.

Suppose I collect data at discrete time points (e.g. the health of a patient at the start of every month, e.g. "healthy", "sick", "very sick") - technically speaking, there is nothing stopping me from "tricking" my computer and saying that these time measurements are actually continuous, and then estimating the P(t) and Q matrix - even though these concepts (i.e. P(t) and Q) are not defined in a Discrete Time Markov Chain.
Based on this set-up (i.e. creating a Continuous Time Markov Chain using fundamentally Discrete data), I anticipate that the P(t) matrix will be likely be "constant" between time intervals (i.e. staircase/stepwise function) - e.g. the probability of transitioning between two states remains identical from the start to the end of any given month.
My Question:  Is what I have described (e.g. piecewise approximation) a somewhat valid approach insofar as treating discrete times as continuous when using Markov Chains? Or is this fundamentally incorrect?
The reason I am interested in specifically using a Continuous Time Markov Chain (even if there is discrete data) is that this allows me in theory to obtain time-dependent transition probabilities (even if they are constant between time intervals) as opposed to constant transition probabilities - and these time-dependent transition probabilities can be useful in different real-world applications.
Thanks!
Note: I understand the risks in using such an approach - in real life, the health of a medical patient might deteriorate or improve within a given month, but the Continuous Time Markov Chain created on discrete data would not be able to pick up on this. However, all things being equal - this kind of information would fundamentally not be captured within the discrete data and the Discrete Time Markov Chain would fundamentally not be able to pick up on this. Therefore, I was interested in learning about whether or not analyzing discrete times using a Continuous Time Markov Chain might provide some benefits over a Discrete Time Markov Chain while likely not incurring any additional disadvantages (i.e. best case scenario - slightly better; worse case scenario - roughly equivalent).
 A: Yes, you can use a Continuous Time Markov Chain to model discrete time data by making a piecewise approximation of the transition probabilities between states. In this approach, you would assume that the transition probabilities remain constant within each time interval and calculate them based on the frequencies of state transitions in the discrete time data.
However, this approach has certain drawbacks that I believe ought to be mentioned. It assumes that the transition probabilities are constant within each time interval, which might not be accurate if the state transitions occur frequently or irregularly within a given time interval. It also doesn't account for the fact that the state transition times are not necessarily evenly spaced.
Despite these limitations, this approach can still provide useful insights and information about the state transitions in the discrete time data, especially when the data is limited or the state transition times are approximately evenly spaced. In such cases, the piecewise approximation of the transition probabilities can be a good approximation of the underlying process.
A: This approach does make sense, and is reminiscent of some of the analysis that I've seen used in modeling cancer mutations (discrete time models can be simulated on a computer, but continuous time models are analytically tractable, usually give basically the same results). You probably don't want to model the state of the patient, however, as being "healthy" for the whole month if they come up as healthy at the start of the month, but treat the data as an observation model at the start of the month.
That is, just because you measured that they went from "sick" to "healthy" at the start of the month, that doesn't mean that that's when they transitioned between the states. Rather, treat the process as continuous time, and make sure that you're using the data as an observation of the state rather than as an observation of a transition.
This sort of model also allows for the possibility that a patient's health deteriorates and then improves within the same month (which could be somewhat realistic).
Like mentioned in the answer above, however, even though I think a continuous time with course grained observation model is better it's much more complicated and could be much more difficult to fit to data. If all you want is time dependent transition probabilities, you can still do that in discrete time (and you can include features in predicting that transition matrix, making it something like P(t,hospital_features,patient_covariates,etc)).
