How the response is interpreted
AddiVortes() looks only at y:
- A factor, character or
logical vector is classification. Two levels give a
binary model; three or more give a multinomial model.
- A numeric vector with exactly two unique values in
{0, 1} is binary classification (1 is the event
class).
- Any other numeric vector is Gaussian regression, as in previous
versions of the package.
The first factor level is the reference class, matching
glm(). For numeric 0/1 data, 0 is the reference. Character
vectors are converted with factor(), so levels are
alphabetical.
Missing values in y are not allowed, and a
classification response must have at least two classes.
Binary classification
We simulate a two-dimensional problem whose labels follow a simple
linear boundary.
set.seed(2026)
n <- 120
x <- data.frame(
x1 = runif(n, -1, 1),
x2 = runif(n, -1, 1)
)
y <- factor(ifelse(x$x1 + x$x2 > 0, "active", "inactive"),
levels = c("inactive", "active"))
fit_bin <- AddiVortes(
y, x,
m = 200,
totalMCMCIter = 2000,
mcmcBurnIn = 500,
showProgress = FALSE
)
fit_bin$task
#> [1] "binary"
fit_bin$classLevels
#> [1] "inactive" "active"
fit_bin$inSampleAccuracy
#> [1] 0.9916667
predict(..., type = "response") returns the posterior
mean of \(\Phi(G(x))\), the probability
of the event class (active here).
type = "class" returns labels, and
type = "quantile" gives credible intervals for those
probabilities.
p_hat <- predict(fit_bin, x, type = "response", showProgress = FALSE)
y_hat <- predict(fit_bin, x, type = "class", showProgress = FALSE)
p_int <- predict(fit_bin, x,
type = "quantile",
quantiles = c(0.05, 0.95),
showProgress = FALSE
)
mean(y_hat == y)
#> [1] 0.9916667
range(p_hat)
#> [1] 0.01799616 0.96853534
head(cbind(prob = round(p_hat, 3), lower = round(p_int[, 1], 3),
upper = round(p_int[, 2], 3), class = as.character(y_hat)))
#> prob lower upper class
#> [1,] "0.724" "0.467" "0.921" "active"
#> [2,] "0.048" "0.004" "0.141" "inactive"
#> [3,] "0.091" "0.011" "0.243" "inactive"
#> [4,] "0.713" "0.438" "0.92" "active"
#> [5,] "0.3" "0.088" "0.565" "inactive"
#> [6,] "0.033" "0" "0.13" "inactive"
Observations with probability near 0.5 are the most uncertain. The
90% intervals typically cover 0.5 for those points.
plot(p_hat, as.integer(y) - 1,
xlab = "Predicted P(active)",
ylab = "Observed class",
pch = 19, col = "steelblue")
abline(v = 0.5, lty = 2, col = "grey40")
Multi-category classification
With three or more classes, AddiVortes fits \(K-1\) latent ensembles. The argument
m is the number of tessellations per
latent, so a three-class model with m = 15 uses 30
tessellations in total.
set.seed(2026)
n3 <- 90
x3 <- data.frame(
x1 = rnorm(n3),
x2 = rnorm(n3)
)
y3 <- cut(x3$x1, breaks = c(-Inf, -0.5, 0.5, Inf), labels = c("low", "mid", "high"))
fit_multi <- AddiVortes(
y3, x3,
m = 200,
totalMCMCIter = 2000,
mcmcBurnIn = 500,
showProgress = FALSE
)
fit_multi$task
#> [1] "multinomial"
fit_multi$nLatents
#> [1] 2
fit_multi$inSampleAccuracy
#> [1] 0.9888889
type = "response" is now an \(n \times K\) matrix of class probabilities.
Rows sum to 1. type = "class" returns the class with the
largest probability.
p3 <- predict(fit_multi, x3, type = "response", showProgress = FALSE)
cls3 <- predict(fit_multi, x3, type = "class", showProgress = FALSE)
head(round(p3, 3))
#> low mid high
#> [1,] 0.129 0.400 0.472
#> [2,] 0.682 0.230 0.088
#> [3,] 0.126 0.685 0.189
#> [4,] 0.195 0.671 0.135
#> [5,] 0.453 0.389 0.158
#> [6,] 0.843 0.107 0.050
mean(abs(rowSums(p3) - 1) < 1e-8)
#> [1] 1
table(predicted = cls3, observed = y3)
#> observed
#> predicted low mid high
#> low 30 0 0
#> mid 0 34 1
#> high 0 0 25
Notes on the classification prior
On the latent probit scale the residual variance is fixed at \(\sigma = 1\), so nu,
q and InitialSigma are ignored. Cell means
use
\[
\sigma_\mu = \frac{3}{k\sqrt{m}},
\]
which keeps \(G(x)\) typically
inside \((-3, 3)\) and shrinks class
probabilities towards 0.5 (or towards equal class probabilities in the
multinomial case). The default k = 3 is a reasonable
starting point.
The response itself is not scaled. Covariates are still centred and
scaled exactly as in regression.
For further details of the binary model see Stone, Ogundimu and
Gosling (2026). The multi-category construction follows the
latent-utility setup of Kindo, Wang and Peña (2016), with independent
latents (\(\Sigma = I\)).