# Implementing the Prowise Learn Update Algorithm

The Prowise Learn algorithm (Vermeiren et al., 2025) extends the Elo-style Maths Garden updates (`vignette("maths-garden-update")`) with paired item updates that counteract rating drift — the tendency for item difficulty estimates to slide systematically over time.

# Mathematical foundation

Abilities are updated exactly as in Maths Garden:

\[\theta_j^{new} = \theta_j + K_\theta \sum_{i \in I_j} (S_{ij} - E(S_{ij})).\]

Item difficulties, however, are updated in consecutive pairs of items administered to the same respondent. For a pair (previous item, current item),

\[\kappa = 0.5\,\big(K_b (S_{now} - E_{now}) - K_b (S_{prev} - E_{prev})\big), \qquad b_{now} \mathrel{+}= \kappa, \quad b_{prev} \mathrel{-}= \kappa.\]

Because each pair adds \(+\kappa\) to one item and \(-\kappa\) to the other, the total difficulty mass is conserved, so items keep their relative positions and do not drift en masse. Expected responses use the Rasch model, \(E(S_{ij}) = 1 / (1 + e^{-(\theta_j - b_i)})\).

# Implementation in `meow`

Paired updates are inherently order dependent, so `update_prowise_learn()` uses `meow_long(R, admin)`, which returns the administered responses ordered by respondent and then by administration order. Consecutive within-respondent rows form the pairs; the per-item contributions are aggregated with `tapply()`:

    update_prowise_learn <- function(pers, item, R, admin, K_theta = 0.1, K_b = 0.1) {
      long  <- meow_long(R, admin)
      E_Sij <- stats::plogis(pers$theta[long$id] - item$b[long$item])

      # ability update (as in Maths Garden)
      dtheta <- tapply(long$resp - E_Sij, long$id, sum)
      pers$theta[as.integer(names(dtheta))] <-
        pers$theta[as.integer(names(dtheta))] + K_theta * dtheta

      # paired item updates over consecutive administrations
      n <- nrow(long)
      if (n >= 2) {
        nxt <- 2:n; prv <- 1:(n - 1)
        pair <- which(long$id[nxt] == long$id[prv])
        if (length(pair) > 0) {
          now <- nxt[pair]; pre <- prv[pair]
          kappa <- 0.5 * (K_b * (long$resp[now] - E_Sij[now]) -
                          K_b * (long$resp[pre] - E_Sij[pre]))
          add_now <- tapply(kappa,  long$item[now], sum)
          add_pre <- tapply(-kappa, long$item[pre], sum)
          item$b[as.integer(names(add_now))] <- item$b[as.integer(names(add_now))] + add_now
          item$b[as.integer(names(add_pre))] <- item$b[as.integer(names(add_pre))] + add_pre
        }
      }
      list(pers = pers, item = item)
    }

# Using it

    sim <- meow(
      select_fun  = select_max_info,
      update_fun  = update_prowise_learn,
      data_loader = data_simple_1pl,
      data_args   = list(N_persons = 100, N_items = 50),
      update_args = list(K_theta = 0.05, K_b = 0.05)
    )
    head(sim$results[, 1:3])
    #>   iter pers_theta_1_est pers_theta_2_est
    #> 1    1        0.1250000       -0.1250000
    #> 2    2        0.2656826       -0.2657371
    #> 3    3        0.3692123       -0.3715956
    #> 4    4        0.4806109       -0.4711769
    #> 5    5        0.5559021       -0.6101828
    #> 6    6        0.6200887       -0.7172594

# Practical notes

- Paired updates require respondents to answer at least two items, so the item difficulties only begin to move once administration is under way.
- Effectiveness depends on the administration order; this is exactly why the matrix `admin` carries the order of administration.
- As with Maths Garden, keep learning rates modest and consider bounding the estimates for stability.