diff --git a/src/fitter/recursive.rs b/src/fitter/recursive.rs
index 0bfddd7..348d688 100644
--- a/src/fitter/recursive.rs
+++ b/src/fitter/recursive.rs
@@ -33,7 +33,7 @@ impl RecursiveFitter {
         let m = kernel.order();
         let h = kernel.measurement_vector();
 
-        RecursiveFitter {
+        Self {
             ts_new: Vec::new(),
             kernel,
             ts: Vec::new(),
@@ -215,6 +215,7 @@ impl Fitter for RecursiveFitter {
         self.is_fitted = true;
     }
 
+    #[allow(clippy::many_single_char_names)]
     fn predict(&self, ts: &[f64]) -> (Vec<f64>, Vec<f64>) {
         if !self.is_fitted {
             panic!("new data since last call to `fit()`");
diff --git a/src/observation.rs b/src/observation.rs
index e5382f6..a0e0fb5 100644
--- a/src/observation.rs
+++ b/src/observation.rs
@@ -110,7 +110,45 @@ impl Core {
         diff
     }
 
-    fn kl_update(&mut self, _lr: f64, _storage: &mut Storage) -> f64 {
-        todo!();
+    pub fn kl_update<F>(&mut self, storage: &mut Storage, lr: f64, cvi_expectations: F) -> f64
+    where
+        F: Fn(f64, f64) -> (f64, f64, f64),
+    {
+        // Mean and variance in function space.
+        let mut f_mean: f64 = 0.0;
+        let mut f_var: f64 = 0.0;
+
+        for i in 0..self.m {
+            let item = storage.item_mut(self.items[i]);
+            let idx = self.indices[i];
+            let coeff = self.coeffs[i];
+
+            // Adjust the function-space mean & variance.
+            f_mean += coeff * item.fitter.ms(idx);
+            f_var += coeff * coeff * item.fitter.vs(idx);
+        }
+
+        // Compute the derivatives of the exp. log-lik. w.r.t. mean parameters.
+        let (exp_ll, alpha, beta) = cvi_expectations(f_mean, f_var);
+
+        for i in 0..self.m {
+            let item = storage.item_mut(self.items[i]);
+            let idx = self.indices[i];
+            let coeff = self.coeffs[i];
+
+            // Update the elements' parameters.
+            let x = -2.0 * coeff * coeff * beta;
+            let n = coeff * (alpha - 2.0 * item.fitter.ms(idx) * coeff * beta);
+
+            *item.fitter.xs_mut(idx) = (1.0 - lr) * item.fitter.xs(idx) + lr * x;
+            *item.fitter.ns_mut(idx) = (1.0 - lr) * item.fitter.ns(idx) + lr * n;
+        }
+
+        let diff = (self.exp_ll - exp_ll).abs();
+
+        // Save the expected log-likelihood.
+        self.exp_ll = exp_ll;
+
+        diff
     }
 }
diff --git a/src/observation/ordinal.rs b/src/observation/ordinal.rs
index ef0e6de..289f049 100644
--- a/src/observation/ordinal.rs
+++ b/src/observation/ordinal.rs
@@ -1,11 +1,14 @@
+use std::f64::consts::TAU;
+
 use crate::storage::Storage;
-use crate::utils::{logphi, normcdf, normpdf};
+use crate::utils::{logphi, normcdf, normpdf, ROOTS_HERMITENORM_WS, ROOTS_HERMITENORM_XS};
 
 use super::{f_params, Core, Observation};
 
 fn mm_probit_win(mean_cav: f64, cov_cav: f64) -> (f64, f64, f64) {
     // Adapted from the GPML function `likErf.m`.
     let z = mean_cav / (1.0 + cov_cav).sqrt();
+
     let (logpart, val) = logphi(z);
     let dlogpart = val / (1.0 + cov_cav).sqrt(); // 1st derivative w.r.t. mean.
     let d2logpart = -val * (z + val) / (1.0 + cov_cav);
@@ -33,6 +36,45 @@ fn mm_probit_tie(mean_cav: f64, cov_cav: f64, margin: f64) -> (f64, f64, f64) {
     (logpart, dlogpart, d2logpart)
 }
 
+fn ll_probit_win(x: f64, margin: f64) -> f64 {
+    logphi(x - margin).0
+}
+
+fn ll_probit_tie(x: f64, margin: f64) -> f64 {
+    let x = -x.abs();
+    let z = logphi(x + margin).0;
+    let a = logphi(x - margin).0 - z;
+
+    if a > -0.693 {
+        z + (-a.exp_m1()).ln()
+    } else {
+        z + (-a.exp()).ln_1p()
+    }
+}
+
+fn cvi_expectations<F>(mean: f64, var: f64, ll_fct: F) -> (f64, f64, f64)
+where
+    F: Fn(f64) -> f64,
+{
+    const N: usize = 30;
+
+    let std = var.sqrt();
+    let mut exp_ll = 0.0;
+    let mut alpha = 0.0;
+    let mut beta = 0.0;
+
+    for i in 0..N {
+        let val =
+            (ROOTS_HERMITENORM_WS[i] / TAU.sqrt()) * ll_fct(std * ROOTS_HERMITENORM_XS[i] + mean);
+
+        exp_ll += val;
+        alpha += (ROOTS_HERMITENORM_XS[i] / std) * val;
+        beta += ((ROOTS_HERMITENORM_XS[i].powi(2) - 1.0) / (2.0 * var)) * val;
+    }
+
+    (exp_ll, alpha, beta)
+}
+
 pub struct ProbitWinObservation {
     core: Core,
     margin: f64,
@@ -40,7 +82,7 @@ pub struct ProbitWinObservation {
 
 impl ProbitWinObservation {
     pub fn new(storage: &mut Storage, elems: &[(usize, f64)], t: f64, margin: f64) -> Self {
-        ProbitWinObservation {
+        Self {
             core: Core::new(storage, elems, t),
             margin,
         }
@@ -58,14 +100,17 @@ impl Observation for ProbitWinObservation {
     fn ep_update(&mut self, storage: &mut Storage, lr: f64) -> f64 {
         let margin = self.margin;
 
-        self.core
-            .ep_update(storage, lr, |mean_cav: f64, cov_cav: f64| {
-                mm_probit_win(mean_cav - margin, cov_cav)
-            })
+        self.core.ep_update(storage, lr, |mean_cav, cov_cav| {
+            mm_probit_win(mean_cav - margin, cov_cav)
+        })
     }
 
     fn kl_update(&mut self, storage: &mut Storage, lr: f64) -> f64 {
-        self.core.kl_update(lr, storage)
+        let margin = self.margin;
+
+        self.core.kl_update(storage, lr, |mean, var| {
+            cvi_expectations(mean, var, |x| ll_probit_win(x, margin))
+        })
     }
 }
 
@@ -94,26 +139,29 @@ impl Observation for ProbitTieObservation {
     fn ep_update(&mut self, storage: &mut Storage, lr: f64) -> f64 {
         let margin = self.margin;
 
-        self.core
-            .ep_update(storage, lr, |mean_cav: f64, cov_cav: f64| {
-                mm_probit_tie(mean_cav, cov_cav, margin)
-            })
+        self.core.ep_update(storage, lr, |mean_cav, cov_cav| {
+            mm_probit_tie(mean_cav, cov_cav, margin)
+        })
     }
 
-    fn kl_update(&mut self, _storage: &mut Storage, _lr: f64) -> f64 {
-        todo!();
+    fn kl_update(&mut self, storage: &mut Storage, lr: f64) -> f64 {
+        let margin = self.margin;
+
+        self.core.kl_update(storage, lr, |mean, var| {
+            cvi_expectations(mean, var, |x| ll_probit_tie(x, margin))
+        })
     }
 }
 
 pub struct LogitWinObservation {
-    _core: Core,
+    core: Core,
     _margin: f64,
 }
 
 impl LogitWinObservation {
     pub fn new(storage: &mut Storage, elems: &[(usize, f64)], t: f64, margin: f64) -> Self {
-        LogitWinObservation {
-            _core: Core::new(storage, elems, t),
+        Self {
+            core: Core::new(storage, elems, t),
             _margin: margin,
         }
     }
@@ -124,18 +172,22 @@ impl Observation for LogitWinObservation {
         todo!();
     }
 
-    fn kl_update(&mut self, _storage: &mut Storage, _lr: f64) -> f64 {
-        todo!();
+    fn kl_update(&mut self, storage: &mut Storage, lr: f64) -> f64 {
+        self.core.kl_update(storage, lr, |mean, var| todo!())
     }
 }
 
 pub struct LogitTieObservation {
-    //
+    core: Core,
+    _margin: f64,
 }
 
 impl LogitTieObservation {
-    pub fn new(_storage: &mut Storage, _elems: &[(usize, f64)], _t: f64, _margin: f64) -> Self {
-        todo!();
+    pub fn new(storage: &mut Storage, elems: &[(usize, f64)], t: f64, margin: f64) -> Self {
+        Self {
+            core: Core::new(storage, elems, t),
+            _margin: margin,
+        }
     }
 }
 
@@ -144,8 +196,8 @@ impl Observation for LogitTieObservation {
         todo!();
     }
 
-    fn kl_update(&mut self, _storage: &mut Storage, _lr: f64) -> f64 {
-        todo!();
+    fn kl_update(&mut self, storage: &mut Storage, lr: f64) -> f64 {
+        self.core.kl_update(storage, lr, |mean, var| todo!())
     }
 }
 
@@ -176,4 +228,13 @@ mod tests {
         assert_relative_eq!(b, -0.20881357058382308);
         assert_relative_eq!(c, -0.1698273481633205);
     }
+
+    #[test]
+    fn test_cvi_expectations() {
+        let (a, b, c) = cvi_expectations(0.3, 2.7, |x| logphi(x).0);
+
+        assert_relative_eq!(a, -1.198109740470403);
+        assert_relative_eq!(b, 0.8970390100143342);
+        assert_relative_eq!(c, -0.2565392538236815);
+    }
 }
diff --git a/src/utils.rs b/src/utils.rs
index b50a153..9b4ee72 100644
--- a/src/utils.rs
+++ b/src/utils.rs
@@ -87,3 +87,69 @@ pub fn logphi(z: f64) -> (f64, f64) {
         (res, dres)
     }
 }
+
+pub const ROOTS_HERMITENORM_XS: [f64; 30] = [
+    -9.706236,
+    -8.68083772,
+    -7.82505174,
+    -7.05539687,
+    -6.33999769,
+    -5.66238185,
+    -5.0126006,
+    -4.38402037,
+    -3.77189442,
+    -3.17263464,
+    -2.5834021,
+    -2.00185861,
+    -1.42600566,
+    -0.85407335,
+    -0.28443876,
+    0.28443876,
+    0.85407335,
+    1.42600566,
+    2.00185861,
+    2.5834021,
+    3.17263464,
+    3.77189442,
+    4.38402037,
+    5.0126006,
+    5.66238185,
+    6.33999769,
+    7.05539687,
+    7.82505174,
+    8.68083772,
+    9.706236,
+];
+
+pub const ROOTS_HERMITENORM_WS: [f64; 30] = [
+    4.11289324e-21,
+    3.97441190e-17,
+    4.07096517e-14,
+    1.14638786e-11,
+    1.29804729e-09,
+    7.22454173e-08,
+    2.23317741e-06,
+    4.15598507e-05,
+    4.92584902e-04,
+    3.87200709e-03,
+    2.07943554e-02,
+    7.79856428e-02,
+    2.07515826e-01,
+    3.96164962e-01,
+    5.46444893e-01,
+    5.46444893e-01,
+    3.96164962e-01,
+    2.07515826e-01,
+    7.79856428e-02,
+    2.07943554e-02,
+    3.87200709e-03,
+    4.92584902e-04,
+    4.15598507e-05,
+    2.23317741e-06,
+    7.22454173e-08,
+    1.29804729e-09,
+    1.14638786e-11,
+    4.07096517e-14,
+    3.97441190e-17,
+    4.11289324e-21,
+];
diff --git a/tests/binary-1.rs b/tests/binary-1.rs
index c541984..0e5dd1e 100644
--- a/tests/binary-1.rs
+++ b/tests/binary-1.rs
@@ -2,7 +2,7 @@ extern crate intel_mkl_src;
 
 use kickscore as ks;
 
-// #[test]
+#[test]
 fn binary_1() {
     let mut model = ks::BinaryModel::new(ks::BinaryModelObservation::Probit);