McGough 2020 - NobBS Model and Priors

Summary

NobBS models each reporting-triangle cell as Poisson or negative-binomial with mean ${\lambda }_{t,d}=e^{{\alpha }_t}{\beta }_d$, a geometric random walk on ${\alpha }_t$ and a Dirichlet prior on the delay vector $\beta$. Priors are weak by default but can encode a known delay shape; the total nowcast is $N_t={\sum }_d{n}_{t,d}$ including unobserved cells drawn from the posterior predictive.

Overview

The model is a multiplicative (log-additive) decomposition: a time signal times a delay profile. It is “hierarchical” in the sense that all weeks share $\beta$ and the random-walk variance (see Hierarchical Models), and a state-space model in the sense that ${\alpha }_t$ is a latent random walk (Linear-Gaussian State-Space Models).

Main Content

Observation model

Poisson (dengue): $n_{t,d}\sim \text{Pois}({\lambda }_{t,d})$. Negative binomial (ILI, extra-Poisson variation): $n_{t,d}\sim \text{NB}(r,p_{t,d})$ with $p_{t,d}=r/(r+{\lambda }_{t,d})$, so the mean is ${\lambda }_{t,d}$ and the variance exceeds it.

Mean structure

$\log {\lambda }_{t,d}={\alpha }_t+\log {\beta }_d$, i.e. ${\lambda }_{t,d}=e^{{\alpha }_t}{\beta }_d$. ${\alpha }_t$ is the true epidemiologic signal for week $t$ (so $e^{{\alpha }_t}$ is the expected total $N_t$); ${\beta }_d$ is the probability of reporting at delay $d$, ${\sum }_{d=0}^D{\beta }_d=1$.

Priors

• Delay vector: $\beta \sim \text{Dirichlet}({\theta }_0,\dots ,{\theta }_D)$. Default ${\theta }_d=0.1$ (weak, equal weight per delay). In the paper’s analyses $\theta$ encodes a pseudo-count of $n=10$ hypothetical cases spread by the empirical delay pattern in the training period; an equal-weight prior made no material difference. ${\theta }_D$ can represent “delay $\ge D$“.
• Signal: first-order random walk, ${\alpha }_1\sim N(0,\text{precision }0.001)$, ${\alpha }_t\sim N({\alpha }_{t-1},{\tau }_{\alpha }^{-1})$ for $t>1$. Because ${\alpha }_t$ is on the log scale this is a geometric random walk.
• Random-walk precision: ${\tau }_{\alpha }\sim \text{Ga}(0.01,0.01)$ (default). A stronger prior such as Ga(10, 0.1) forces small week-to-week changes and a smoother curve.
• NB dispersion: $r\sim \text{Ga}(60,20)$, mildly informative (moderate departure from Poisson).

Nowcast quantity

At nowcast time $T$ the posterior of $N_t={\sum }_{d=0}^D{n}_{t,d}$ is obtained by drawing the unobserved cells ($t+d>T$) from their predictive distribution given the posterior of $(\alpha ,\beta ,{\tau }_{\alpha },r)$; observed cells enter as known. For the latest week only $n_{T,0}$ is known, and the nowcast is driven by $n_{T,0}/{\beta }_0$ shrunk toward ${\alpha }_{T-1}$ by the random walk.

Implementation: JAGS in R with 10,000 posterior samples; trace plots checked visually. The CRAN NobBS package accepts one record per case (onset date, report date), default Poisson with no moving window, $D=$ series length $-1$, and customisable priors. A prior such as $\theta =(3,30,50,10,4,3)$ expresses “3% same week, 30% after one week, …” (paper’s dengue example).

Examples

Why the random walk helps

Suppose delay-0 reports jump from 13 to 29 in one week with ${\beta }_0=0.29$. A pure scale-up gives $29/0.29=100$. The random walk on ${\alpha }_t$ says last week’s total was near $13/0.29\approx 45$ and pulls the new estimate toward it in proportion to the learned step variance, so the nowcast lands between 45 and 100. The paper documents exactly this behaviour on 3 Aug 1998 and 16 Aug 2010: an outlier at $d=0$ drove an overestimate corrected the following week.

import numpy as np
rng=np.random.default_rng(0)
T,D=60,6; beta=np.array([.05,.3,.3,.2,.1,.04,.01])
alpha=np.cumsum(rng.normal(0,.1,T))+4
lam=np.exp(alpha)[:,None]*beta[None,:]
n=rng.poisson(lam)
obs=np.add.outer(np.arange(T),np.arange(D+1))<=T-1   # observed cells
nowcast_naive=np.where(obs,n,0).sum(1)
print(nowcast_naive[-3:], n.sum(1)[-3:])               # truncated vs truth

Connections

McGough 2020 - Reporting Triangle and Chain-Ladder Setup defines the cells; results in McGough 2020 - Evaluation Design and Results; limitations (delay assumed constant inside the window) in McGough 2020 - Delay Dynamics, Moving Windows and Limitations. The delay-hazard alternative for E38 is in Nowcasting Concepts - Delay Hazard Models and the Renewal Equation.

See Also

McGough 2020 - Nowcasting by Bayesian Smoothing - Overview · Bayesian Structural Time-Series Model