\(p(X_{n+1} \in \cdot | X_1..X_n) = \alpha/(\alpha+n) G_0 +
(\alpha+n)^{-1}\sum_k n_k \delta_{X_k^*}\) (Sec. 4.1.4). Closed
form: conjugacy (Sec. 4.1.3) makes the posterior
\(DP(\alpha G_0 + \sum \delta_{X_i})\).
Usage
morie_dp_predictive(x, alpha = 1, grid = NULL)
Arguments
- x
numeric observations.
- alpha
positive concentration.
- grid
evaluation points for the continuous part.
Value
list: grid, base_density, atoms, atom_probs, base_weight,
atom_weight, n_distinct, is_density, limit_note, n, method.
Details
The predictive is NOT a density – it has an atom at every
distinct observed value carrying total mass \(n/(\alpha+n)\),
because a DP draw is almost surely discrete however smooth
\(G_0\) is. Mirrors morie.fn.gh_dp_post_ex.
References
Ghosal and van der Vaart, Sec. 4.1.3-4.1.4.
Examples
set.seed(1)
morie_dp_predictive(rnorm(30), alpha = 2)$atom_weight
#> [1] 0.9375
