diff --git a/BLcourse2.3/03_one_dim_SVI.ipynb b/BLcourse2.3/03_one_dim_SVI.ipynb
index 0520fa0366033d88482c76a09f7bf646e7ba4eed..f60508eba28f67c8992c710c4d837f9ba1187282 100644
--- a/BLcourse2.3/03_one_dim_SVI.ipynb
+++ b/BLcourse2.3/03_one_dim_SVI.ipynb
@@ -17,6 +17,8 @@
     "$\\newcommand{\\predve}[1]{\\mathbf{#1}}$\n",
     "$\\newcommand{\\test}[1]{#1_*}$\n",
     "$\\newcommand{\\testtest}[1]{#1_{**}}$\n",
+    "$\\newcommand{\\dd}{{\\rm{d}}}$\n",
+    "$\\newcommand{\\lt}[1]{_{\\text{#1}}}$\n",
     "$\\DeclareMathOperator{\\diag}{diag}$\n",
     "$\\DeclareMathOperator{\\cov}{cov}$"
    ]
@@ -141,6 +143,9 @@
     "2015](https://proceedings.mlr.press/v38/hensman15.html). The model is\n",
     "\"sparse\" since it works with a set of *inducing* points $(\\ma Z, \\ve u),\n",
     "\\ve u=f(\\ma Z)$ which is much smaller than the train data $(\\ma X, \\ve y)$.\n",
+    "See also [the GPJax\n",
+    "docs](https://docs.jaxgaussianprocesses.com/_examples/uncollapsed_vi) for a\n",
+    "nice introduction.\n",
     "\n",
     "We have the same hyper parameters as before\n",
     "\n",
@@ -260,16 +265,31 @@
    "source": [
     "# Fit GP to data: optimize hyper params\n",
     "\n",
-    "Now we optimize the GP hyper parameters by doing a GP-specific variational inference (VI),\n",
-    "where we optimize not the log marginal likelihood (ExactGP case),\n",
-    "but an ELBO (evidence lower bound) objective\n",
+    "Now we optimize the GP hyper parameters by doing a GP-specific variational\n",
+    "inference (VI), where we don't maximize the log marginal likelihood (ExactGP\n",
+    "case), but an ELBO (\"evidence lower bound\") objective -- a lower bound on the\n",
+    "marginal likelihood (the \"evidence\"). In variational inference, an ELBO objective\n",
+    "shows up when minimizing the KL divergence between\n",
+    "an approximate and the true posterior\n",
     "\n",
     "$$\n",
-    "\\max_\\ve\\zeta\\left(\\mathbb E_{q_{\\ve\\psi}(\\ve u)}\\left[\\ln p(\\ve y|\\ve u) \\right] -\n",
-    "D_{\\text{KL}}(q_{\\ve\\psi}(\\ve u)\\Vert p(\\ve u))\\right)\n",
+    "    p(w|y) = \\frac{p(y|w)\\,p(w)}{\\int p(y|w)\\,p(w)\\,\\dd w}\n",
+    "           = \\frac{p(y|w)\\,p(w)}{p(y)}\n",
     "$$\n",
     "\n",
-    "with respect to\n",
+    "$$\n",
+    "  \\ve\\zeta^* = \\text{arg}\\min_{\\ve\\zeta} D\\lt{KL}(q_{\\ve\\zeta}(w)\\,\\Vert\\, p(w|y))\n",
+    "$$\n",
+    "\n",
+    "to obtain the optimal variational parameters $\\ve\\zeta^*$ to approximate\n",
+    "$p(w|y)$ with $q_{\\ve\\zeta^*}(w)$.\n",
+    "\n",
+    "In our case the two distributions are the approximate\n",
+    "\n",
+    "$$q_{\\ve\\zeta}(\\mathbf f)=\\int p(\\mathbf f|\\ve u)\\,q_{\\ve\\psi}(\\ve u)\\,\\dd\\ve u\\quad(\\text{\"variational strategy\"})$$\n",
+    "\n",
+    "and the true $p(\\mathbf f|\\mathcal D)$ posterior over function values. We\n",
+    "optimize with respect to\n",
     "\n",
     "$$\\ve\\zeta = [\\ell, \\sigma_n^2, s, c, \\ve\\psi] $$\n",
     "\n",
diff --git a/BLcourse2.3/03_one_dim_SVI.py b/BLcourse2.3/03_one_dim_SVI.py
index 644e61a37b818301dd4604bb25d74b64cb5be511..8f34d40034dbda1a5a20c03fe4b1e55608d26e81 100644
--- a/BLcourse2.3/03_one_dim_SVI.py
+++ b/BLcourse2.3/03_one_dim_SVI.py
@@ -25,6 +25,8 @@
 # $\newcommand{\predve}[1]{\mathbf{#1}}$
 # $\newcommand{\test}[1]{#1_*}$
 # $\newcommand{\testtest}[1]{#1_{**}}$
+# $\newcommand{\dd}{{\rm{d}}}$
+# $\newcommand{\lt}[1]{_{\text{#1}}}$
 # $\DeclareMathOperator{\diag}{diag}$
 # $\DeclareMathOperator{\cov}{cov}$
 
@@ -108,6 +110,9 @@ ax.legend()
 # 2015](https://proceedings.mlr.press/v38/hensman15.html). The model is
 # "sparse" since it works with a set of *inducing* points $(\ma Z, \ve u),
 # \ve u=f(\ma Z)$ which is much smaller than the train data $(\ma X, \ve y)$.
+# See also [the GPJax
+# docs](https://docs.jaxgaussianprocesses.com/_examples/uncollapsed_vi) for a
+# nice introduction.
 #
 # We have the same hyper parameters as before
 #
@@ -183,16 +188,31 @@ likelihood.noise_covar.noise = 0.3
 
 # # Fit GP to data: optimize hyper params
 #
-# Now we optimize the GP hyper parameters by doing a GP-specific variational inference (VI),
-# where we optimize not the log marginal likelihood (ExactGP case),
-# but an ELBO (evidence lower bound) objective
+# Now we optimize the GP hyper parameters by doing a GP-specific variational
+# inference (VI), where we don't maximize the log marginal likelihood (ExactGP
+# case), but an ELBO ("evidence lower bound") objective -- a lower bound on the
+# marginal likelihood (the "evidence"). In variational inference, an ELBO objective
+# shows up when minimizing the KL divergence between
+# an approximate and the true posterior
 #
 # $$
-# \max_\ve\zeta\left(\mathbb E_{q_{\ve\psi}(\ve u)}\left[\ln p(\ve y|\ve u) \right] -
-# D_{\text{KL}}(q_{\ve\psi}(\ve u)\Vert p(\ve u))\right)
+#     p(w|y) = \frac{p(y|w)\,p(w)}{\int p(y|w)\,p(w)\,\dd w}
+#            = \frac{p(y|w)\,p(w)}{p(y)}
 # $$
 #
-# with respect to
+# $$
+#   \ve\zeta^* = \text{arg}\min_{\ve\zeta} D\lt{KL}(q_{\ve\zeta}(w)\,\Vert\, p(w|y))
+# $$
+#
+# to obtain the optimal variational parameters $\ve\zeta^*$ to approximate
+# $p(w|y)$ with $q_{\ve\zeta^*}(w)$.
+#
+# In our case the two distributions are the approximate
+#
+# $$q_{\ve\zeta}(\mathbf f)=\int p(\mathbf f|\ve u)\,q_{\ve\psi}(\ve u)\,\dd\ve u\quad(\text{"variational strategy"})$$
+#
+# and the true $p(\mathbf f|\mathcal D)$ posterior over function values. We
+# optimize with respect to
 #
 # $$\ve\zeta = [\ell, \sigma_n^2, s, c, \ve\psi] $$
 #