Autoregressive Hidden Markov Model
Gaussian AR-HMMs and their structure in equations.
Firstly the HMM piece. States $s$ at time $t$ are $s_{t}$. Transition matrix is $p_{tr}$.
$$s_{t} | s_{t-1} \sim \text{Categorical}(p_{tr}[s_{t-1}])$$
(Just expressing that we slice out the row belonging to state $s_{t-1}$ from the transition matrix.)
Now, we have a conditional distribution: emission $y$ given state $s$.
$$y_t | s_t \sim \text{Normal}(\mu[s_t] + ky_{t-1}, \sigma[s_t])$$
This is the most common. The mean depends on the previous time state. We can also make the variance depend on the previous time state too:
$$y_t | s_t \sim \text{Normal}(\mu[s_t] + ky_{t-1}, (\sigma[s_t]) k \sigma_{t-1})$$
There's many other ways to establish the autoregressive dependency, as long as the autoregressive dependency is expressed in the general form of the current emission distribution parameters depending on the previous emission's output values.
The goal of scientific model building is high explanatory power
Why does mechanistic thinking matter? In The end goals of research data science, we are in pursuit of the invariants, i.e. knowledge that stands the test of time. (How our business contexts exploit that knowledge for win-win benefit of society and the business is a matter to discuss another day).
When we build models, particularly of natural systems, predictive power matters only in the context of explanatory power, where we can map phenomena of interest to key parameters in a model. For example, in an , the autoregressive coefficient may correspond to a meaningful properly in our research context.
Being able to look at a natural system and find the most appropriate model for the system is a key skill for winning the trust of the non-quantitative researchers that we serve. (ref: Finding the appropriate model to apply is key)