Standardize observation process equation notation between READMEs and tutorials. (#715)

* update observation process tutorials

* Update docs/tutorials/observation_processes_measurements.qmd

---------

Co-authored-by: Dylan H. Morris <dylanhmorris@users.noreply.github.com>

Author: cdc-mitzimorris

docs/tutorials/observation_processes_counts.qmd

@@ -36,14 +36,13 @@ Count observation processes transform infections into predicted counts by applyi
 
 The predicted observations on day $t$ are:
 
-$$\lambda_t = \alpha \cdot \sum_{d=0}^{D} I_{t-d} \cdot p_d$$
+$$\mu(t) = \alpha \sum_{s} I(t-s) \, \pi(s)$$
 
 where:
 
-- $I_{t-d}$ is the number of incident (new) infections on day $t-d$ (i.e., $d$ days before day $t$)
+- $I(t-s)$ is the number of incident (new) infections on day $t-s$ (i.e., $s$ days before day $t$)
 - $\alpha$ is the rate of ascertained counts per infection (e.g., infection-to-hospital admission rate). This can model a mix of biological effects (e.g. some percentage of infections lead to hospital admissions, but not all) and reporting effects (e.g. some percentage of admissions that occur are reported, but not all).
-- $p_d$ is the delay distribution from infection to observation, conditional on an infection leading to an observation
-- $D$ is the maximum delay
+- $\pi(s)$ is the delay distribution from infection to observation, conditional on an infection leading to an observation
 
 Discrete observations are generated by sampling from a noise distribution—e.g. Poisson or negative binomial—to model reporting variability.
 Poisson assumes variance equals the mean; negative binomial accommodates the overdispersion common in surveillance data.
@@ -53,11 +52,11 @@ Poisson assumes variance equals the mean; negative binomial accommodates the ove
 ## Hospital admissions example
 
 For hospital admissions data, we construct a `Counts` observation process.
-The delay is the key mechanism: infections from $d$ days ago ($I_{t-d}$) contribute to today's hospital admissions ($\lambda_t$) weighted by the probability ($p_d$) that an infection leads to hospitalization after exactly $d$ days. The convolution sums these contributions across all past days.
+The delay is the key mechanism: infections from $s$ days ago ($I(t-s)$) contribute to today's expected hospital admissions ($\mu(t)$) weighted by the probability ($\pi(s)$) that an infection leads to hospitalization after exactly $s$ days. The convolution sums these contributions across all past days.
 
 The process generates hospital admissions by sampling from a negative binomial distribution:
 
-$$Y_t \sim \text{NegativeBinomial}(\mu = \lambda_t, \text{concentration} = \phi)$$
+$$Y_t \sim \text{NegativeBinomial}(\text{mean} = \mu(t), \text{concentration} = \phi)$$
 
 The concentration parameter $\phi$ (sometimes called $k$ or the dispersion parameter) controls overdispersion: as $\phi \to \infty$, the distribution approaches Poisson; smaller values allow greater overdispersion.
 

docs/tutorials/observation_processes_measurements.qmd

@@ -45,12 +45,11 @@ The `Measurements` class models continuous signals derived from infections. Unli
 
 All measurement observation processes follow the same pattern:
 
-$$\text{observed} \sim \text{Noise}\bigl(\text{predicted}(\text{infections})\bigr)$$
+$$y_t \sim \text{Noise}(\mu(t))$$
 
-where:
+where $\mu(t)$ is the predicted measurement value on day $t$, computed from infections via a domain-specific transformation, and the noise model adds stochastic variation around predictions.
 
-1. **`_predicted_obs(infections)`**: Transforms infections into predicted measurement values (you implement this)
-2. **Noise model**: Adds stochastic variation around predictions (provided by PyRenew)
+Subclasses implement `_predicted_obs()` to compute $\mu(t)$ from infections. PyRenew provides the noise model.
 
 The `Measurements` base class provides:
 
@@ -159,7 +158,7 @@ class MyMeasurement(Measurements):
 ## Measurement Example: Wastewater
 
 To illustrate the framework, we specify a wastewater viral concentration observation process,
-based on the [PyRenew-HEW](https://github.com/cdcgov/pyrenew-hew) family of models.
+based on the [PyRenew-HEW](https://github.com/cdcgov/pyrenew-multisignal) family of models.
 
 **The wastewater signal**
 
@@ -172,18 +171,18 @@ The predicted concentration depends on:
 
 The predicted log-concentration on day $t$ is:
 
-$$\log(\lambda_t) = \log\left(\frac{G}{V} \cdot \sum_{d=0}^{D} I_{t-d} \cdot p_d\right)$$
+$$\log(\mu(t)) = \log\left(\frac{G}{V} \cdot \sum_{s} I(t-s) \, \pi(s)\right)$$
 
 where:
 
-- $I_{t-d}$ is infections on day $t-d$
-- $p_d$ is the shedding kinetics PMF (fraction shed on day $d$ post-infection)
+- $I(t-s)$ is infections on day $t-s$
+- $\pi(s)$ is the shedding kinetics PMF (fraction shed on day $s$ post-infection)
 - $G$ is genome copies shed per infection
 - $V$ is wastewater volume per person per day
 
 Observations are log-concentrations with normal noise:
 
-$$y_t \sim \text{Normal}(\log(\lambda_t) + \text{sensor\_mode}, \text{sensor\_sd})$$
+$$y_t \sim \text{Normal}(\log(\mu(t)) + \text{sensor\_mode}, \text{sensor\_sd})$$
 
 ### Implementing the Wastewater class