3 Hidden-state models
Sequential monte carlo (SMC) methods are a class of methods used to solve hidden-state models1.
It is helpful to define some general notation:
- \(X_t\) - hidden-states (such as the reproduction number \(R_t\) or infection incidence \(I_t\))
- \(y_t\) - observed data (such as reported cases \(C_t\))
- \(\theta\) - model parameters
Depending on your goal, you may only care about some of these terms, while the other terms are nuisance-terms. Nevertheless, it is important to consider all three.
If your goal is forecasting, then future observed data \(Y_{t+k}\) are your quantity of interest, and you only care about \(X_t\) and \(\theta\) to the extent that they allow you to estimate \(Y_{t+k}\).
If your goal is reproduction number estimation (or the estimation of any hidden-state), then \(\theta\) is often a nuisance parameter which only care about to the extent that it allows you to estimate \(R_t\).
If your goal is to learn about \(\theta\), say when modelling the effect of a non-pharmaceutical intervention, then \(\theta\) is likely your quantity of interest, and you only care about \(X_t\) to the extent that it allows you to estimate this effect.
3.1 Definition
A (sequential) hidden-state model is composed of two parts:
A state-space model:
\[ P(X_t | X_{1:t-1}, \theta) \tag{3.1}\]
which dictates how the hidden-states vary over time.
An observation model: \[ P(y_t | X_{1:t-1}, y_{1:t-1}, \theta) \tag{3.2}\]
which relates the observed data to the hidden-states.
These two distributions wholly define the model.
This structure makes it clear why hidden-state models are so popular in epidemiology. The underlying epidemic is often unobserved (represented by the state-space model), while reported cases (or other data) are generated through some observation process.
We note that we make no Markov-type assumption, although Markovian models (a.k.a Hidden Markov Models) can be viewed as a special-case of these hidden-state models.
3.2 Filtering and smoothing distributions
Borrowing language from signal processing, we highlight two different posterior distributions for \(R_t\) that we may be interested in.
The conditional2 filtering distribution is defined as:
\[ P(X_t | y_{1:t}, \theta) \tag{3.3}\]
while the conditional smoothing distribution is defined as:
\[ P(X_t | y_{1:T}, \theta) \tag{3.4}\]
The filtering distribution uses only past observations to estimate the hidden-states, whereas the smoothing distribution uses both past and future observations.
[DIAGRAM HERE]
One of the strengths of EpiFilter (Parag 2021) is its ability to find the smoothing distribution, allowing more data to inform \(R_t\) estiamtes, particularly improving inference in low-incidence scenarios.
We demonstrate SMC methods suitable for finding both distributions.
3.3 Example 1: the unsmoothed model
In Section 1.2 we introduced a simple model for \(R_t\) estimation. While it’s overkill, we can write this as hidden-state model.
[Diagram here]
We first placed a Gamma\((a_0, b_0)\) prior distribution on \(R_t\) and made no further assumptions about the dynamics. This is the state-space model:
\[ R_t | a_0, b_0 \sim \text{Gamma}(a_0, b_0) \tag{3.5}\]
Then we assumed that cases today were Poisson distributed with mean \(R_t \Lambda_t\). This is the observation model:
\[ C_t | R_t, C_{1:t-1} \sim \text{Poisson}(R_t \Lambda_t^c) \tag{3.6}\]
The hidden-states are the collection of \(R_t\) values, observed data are reported cases \(C_{1:T}\), and model parameters are \(a_0\), \(b_0\), and \(\{\omega_u\}_{u=1}^{u_{max}}\). Altogether this forms our hidden-state model.
We previously solved this hidden-state model in Section 1.2 by leveraging the conjugacy of the state-space model with the observation model. This conjugacy is rare, and often we need to make unrealistic assumptions to obtain it, hence state-space models are often solved using SMC (or other simulation-based methods).
3.4 Example 2: EpiFilter
[Diagram here]
EpiFilter (Parag 2021) is an example of a typical state-space model. Autocorrelation in \(R_t\) is modelled using a Gaussian random-walk, defining the state-space model as:
\[ R_t | R_{t-1} \sim \text{Normal}\left(R_{t-1}, \eta \sqrt{R_{t-1}}\right) \tag{3.7}\]
with initial condition \(R_0 \sim P(R_0)\). The gaussian random walk acts to smooth \(R_t\) (see Section 1.2.2). Like Example 1 above, EpiFilter also uses the Poisson renewal model as the observation distribution:
\[ C_t | R_t, C_{1:t-1} \sim \text{Poisson}(R_t \Lambda_t^c) \tag{3.8}\]
As before, the hidden-states are the collection of \(R_t\) values, and the observed data are reported cases \(C_{1:T}\). Model parameters are now \(\eta\) (which controls the smoothness of \(R_t\)), \(\{\omega_u\}_{u=1}^{u_{max}}\), and \(P(R_0)\).
No analytical solution to the posterior distribution for \(R_t\) exists, however Parag (2021) avoid the need for simulation-based methods through the use of grid-based approximations.
3.5 Example 3: Modelling infection incidence
In Section 1.4 we highlighted that the renewal model can be placed on either reported cases (as in the examples so far) or infection incidence.
When the renewal model is placed on infections it forms part of the hidden-state model, rather than the observation model. We then need to specify some observation mechanism. For this example we assume that each infection has an indepdendent \(p = 0.5\) chance of
3.6 Concluding remarks
All three examples are focussed on \(R_t\) estimation. This is simply a consequence of the popularity of renewal models for this purpose. In all three cases we have constructed an epidemic model that could equally be used for forecasting or inference about model parameters.
SMC methods can also be used to solve more standard parameter-estimation problems, but this isn’t the focus of this book.↩︎
The specification of a conditional filtering/smoothing distribution is used to highlight that model parameters have been chosen instead of estimated. We later demonstrate how to find the marginal filtering/smoothing distribution which we denote with \(P(X_t|y_{1:t})\), to highlight that model parameters have been marginalised out.↩︎