update eqn

Author: swo

docs/lpl.md

@@ -4,50 +4,44 @@ This model proposes a latent true coverage curve, which is subject to observatio
 
 ## Terminology and notation
 
-- _feature_: A categorical feature of the data (e.g., season, geography) that partially determines coverage
-- _level_: A value taken on by a feature (e.g., New Jersey, or 2018/2019)
-- _group_: A unique combination of features (e.g., New Jersey in 2018/2019)
-- $t$: time since the start of the season, measured in $\text{year}^{-1}$
-- $n_{gk}$: number of people in group $g$, surveyed at timepoint $k$. Drawn from the `sample_size` column of the NIS data; called `N_tot` in the codebase.
-- $x_{gk}$: number of people in group $g$, surveyed at timepoint $k$, who are vaccinated. Approximated as $\mathrm{round}(\hat{v}_{gk}, n_{gk})$, where $\hat{v}_{gk}$ is the `estimate` column.
-- $v_g(t_k)$: latent true coverage among group $g$ at time $t$
-- $z_{gj}$: integer index indicating the level of the $j$-th feature for group $g$.
-
-For example, let the features be season and geography, in that order. Let group 5 be associated with the fourth season and the third geography. Then $z_{51} = 4$ and $z_{52} = 3$.
-
-## Model overview
-
-For each group $g$, the latent coverage $v_g(t)$ is assumed to be a sum of a logistic curve and a line with intercept at $t=0$. The shape parameter $K$ and midpoint $\tau$ of the logistic curve are assumed to be common to all groups. The height $A_g$ of the logistic curve is a grand mean $\mu_A$ plus effects $\delta_{A,j,z_{gj}}$ for each feature $j$. For example, the $A_g$ for Alaska in 2018/2019 will be the grand mean $\mu_A$, plus the Alaska effect, plus the 2018/2019 effect. In fact, the model uses a third interaction term season-geography, so there is also an Alaska-in-2018/2019 term.
-
-The slopes $M_g$ follow a similar pattern.
-
-The observations $x_{gt}$ are beta binomial-distributed around the mean $v_g(t) \cdot n_{gt}$, with variance modified by an extra parameter $D$.
+- $C_{sgm}$: reported, estimated coverage (proportion of people vaccinated) in season $s$, geography (i.e., state) $g$, and month $m$ (`estimate` in the data)
+- $N_{sgm}$: number of people surveyed (`sample_size` in the data)
+- $v_{sg}(t)$: latent true coverage at time $t$ (measured in years)
 
 ## Model equations
 
 ```math
 \begin{align*}
-x_{gk} &\sim \mathrm{BetaBinom}\big( n_{gk}, v_g(t_k) \cdot D, [1-v_g(t_k)] \cdot D \big) \\
-v_g(t) &= \frac{A_g}{1 + \exp\{- K \cdot (t - \tau)\}} + M_g t \\
-A_g &= \mu_A + \sum_j \delta_{Ai z_{gi}} \\
-M_g &= \mu_M + \sum_j \delta_{Mi z_{gi}} \\
-\mu_A &\sim \text{Beta}(100.0, 180.0) \\
-\mu_M &\sim \text{Gamma}(\text{shape} = 1.0, \text{rate} = 10.0) \\
-\delta_{Aij} &\sim \mathcal{N}(0, \sigma_{Ai}) \\
-\delta_{Mij} &\sim \mathcal{N}(0, \sigma_{Mi}) \\
-\sigma_{Ai} &\sim \text{Exp}(40.0) \\
-\sigma_{Mi} &\sim \text{Exp}(40.0) \\
+X_{sgm} &= \mathrm{round}(C_{sgm} \cdot N_{sgm}) \\
+v_{sg}(t) &= \frac{A_{sg}}{1 + \exp\{- K \cdot (t - \tau)\}} + B_{sg} t \\
+A_{sg} &= \beta^{(A)} + \beta_s^{(AS)} + \beta_g^{(AG)} + \beta_{sg}^{(ASG)} \\
+B_{sg} &= \beta^{(B)} + \beta_s^{(BS)} + \beta_g^{(BG)} + \beta_{sg}^{(BSG)} \\
+\\
+X_{sgm} &\sim \mathrm{BetaBinom}\big( N_{sgm}, v_{sg}(t_m) \cdot D, [1-v_{sg}(t_m)] \cdot D \big) \\
 K &\sim \text{Gamma}(\text{shape} = 25.0, \text{rate} = 1.0) \\
 \tau &\sim \text{Beta}(100.0, 225.0) \\
 D &\sim \text{Gamma}(\text{shape} = 350.0, \text{rate} = 1.0) \\
+\beta^{(A)} &\sim \text{Beta}(100.0, 180.0) \\
+\beta_s^{(AS)} &\sim \mathcal{N}\left(0, \sigma^{(AS)}\right) \\
+\beta_g^{(AG)} &\sim \mathcal{N}\left(0, \sigma^{(AG)}\right) \\
+\beta_{sg}^{(ASG)} &\sim \mathcal{N}\left(0, \sigma^{(ASG)}\right) \\
+\beta^{(B)} &\sim \text{Gamma}(\text{shape} = 1.0, \text{rate} = 10.0) \\
+\beta_s^{(BS)} &\sim \mathcal{N}\left(0, \sigma^{(BS)}\right) \\
+\beta_g^{(BG)} &\sim \mathcal{N}\left(0, \sigma^{(BG)}\right) \\
+\beta_{sg}^{(BSG)} &\sim \mathcal{N}\left(0, \sigma^{(BSG)}\right) \\
+\sigma^{(\bullet)} &\sim \text{Exp}(40.0) \\
 \end{align*}
 ```
 
-Note that:
+Notes:
+
+- The latent coverage $v_{sg}(t)$ is assumed to be a sum of a logistic curve and a line with intercept at $t=0$
+- The shape parameter $K$ and midpoint $\tau$ of the logistic curve are assumed to be common to all groups
+- The height $A_{sg}$ of the logistic curve is a grand mean $\beta^{(A)}$ plus effects for the season, state, and season-state interaction. The slopes $M_g$ follow a similar pattern.
 
 ```math
 \begin{align*}
-\mathbb{E}[x_{gk}] &= v_g(t_k) \cdot n_{gk} \\
-\mathrm{Var}[x_{gk}] &= v_g(t_k) \cdot [1-v_g(t_k)] \cdot \frac{n_{gk} (n_{gk} + D)}{D+1}
+\mathbb{E}[X_{sgk}] &= v_{sg}(t_k) \cdot N_{sgk} \\
+\mathrm{Var}[X_{sgk}] &= v_{sg}(t_k) \cdot [1-v_{sg}(t_k)] \cdot \frac{N_{sgk} (N_{sgk} + D)}{D+1}
 \end{align*}
 ```