Use \widehat for Colab

Maybe colab is using old mathjax

Ref mathjax/MathJax#1913

Author: patnr

notebooks/T3 - Bayesian inference.ipynb

@@ -588,10 +588,10 @@
     "As such, barring any approximations (such as using `Bayes_rule_LG1` outside the linear-Gaussian case),\n",
     "the (full) posterior will be **optimal** from the perspective of any [proper scoring rule](https://en.wikipedia.org/wiki/Scoring_rule#Propriety_and_consistency).\n",
     "\n",
-    "*But if you must* pick a single point value estimate $\\hat{x}$\n",
-    "then you should **decide** on it by optimising (with respect to $\\hat{x}$)\n",
+    "*But if you must* pick a single point value estimate $\\widehat{x}$\n",
+    "then you should **decide** on it by optimising (with respect to $\\widehat{x}$)\n",
     "the expectation (with respect to $x$) of some utility/loss function,\n",
-    "i.e. $\\Expect\\, \\text{Loss}(x - \\hat{x})$.\n",
+    "i.e. $\\Expect\\, \\text{Loss}(x - \\widehat{x})$.\n",
     "For instance, if the posterior pdf happens to be symmetric\n",
     "(as in the linear-Gaussian context above),\n",
     "and your loss function is convex and symmetric,\n",

notebooks/T4 - Time series filtering.ipynb

@@ -237,15 +237,15 @@
     "  The following does relies only on the (\"hidden Markov model\") assumptions.\n",
     "\n",
     "  - The analysis \"assimilates\" $y_k$ according to Bayes' rule to compute $p(x_k | y_{1:k})$,\n",
-    "    where $y_{1:k} = y_1, \\ldots, y_k$ is shorthand notation.\n",
-    "    $$\n",
-    "    p(x_k | y_{1:k}) \\propto p(y_k | x_k) \\, p(x_k | x_{1:k-1}) \\,.\n",
-    "    $$\n",
+    "  where $y_{1:k} = y_1, \\ldots, y_k$ is shorthand notation.\n",
+    "  $$\n",
+    "  p(x_k | y_{1:k}) \\propto p(y_k | x_k) \\, p(x_k | x_{1:k-1}) \\,.\n",
+    "  $$\n",
     "  - The forecast \"propagates\" the uncertainty (i.e. density) according to the Chapman-Kolmogorov equation\n",
-    "    to produce $p(x_{k+1}| y_{1:k})$.\n",
-    "    $$\n",
-    "    p(x_{k+1} | y_{1:k}) = \\int p(x_{k+1} | x_k) \\, p(x_k | y_{1:k}) \\, d x_k \\,.\n",
-    "    $$\n",
+    "  to produce $p(x_{k+1}| y_{1:k})$.\n",
+    "  $$\n",
+    "  p(x_{k+1} | y_{1:k}) = \\int p(x_{k+1} | x_k) \\, p(x_k | y_{1:k}) \\, d x_k \\,.\n",
+    "  $$\n",
     "\n",
     "  It is important to appreciate the benefits of the recursive form of these computations:\n",
     "  It reflects the recursiveness (Markov property) of nature:\n",

notebooks/T7 - Geostats & Kriging [optional].ipynb

@@ -715,8 +715,8 @@
    "source": [
     "#### Kriging\n",
     "\n",
-    "Kriging finds the best (minimum) mean square error (MSE), $\\Expect (\\hat{x} - x)^2$, among all linear \"predictors\",\n",
-    "$\\hat{x} = \\vect{w}\\tr \\vect{y}$, that are unbiased (BLUP).\n",
+    "Kriging finds the best (minimum) mean square error (MSE), $\\Expect (\\widehat{x} - x)^2$, among all linear \"predictors\",\n",
+    "$\\widehat{x} = \\vect{w}\\tr \\vect{y}$, that are unbiased (BLUP).\n",
     "\n",
     "<a name='Exc-–-\"simple\"-kriging-(SK)'></a>\n",
     "\n",
@@ -725,13 +725,13 @@
     "Suppose $X(s)$ has a mean that is constant in space, $\\mu$, and known.\n",
     "Since it is easy to subtract (and later re-include) $\\mu$\n",
     "from both $x$ and the data $\\vect{y}$, we simply assume $\\mu = 0$.\n",
-    "Thus, $\\Expect \\vect{y} = \\vect{0}$ and $\\Expect \\hat{x} = 0$ for any weights $\\vect{w}$,\n",
-    "and so $\\hat{x}$ is already unbiased.\n",
+    "Thus, $\\Expect \\vect{y} = \\vect{0}$ and $\\Expect \\widehat{x} = 0$ for any weights $\\vect{w}$,\n",
+    "and so $\\widehat{x}$ is already unbiased.\n",
     "Meanwhile,\n",
     "$$\n",
     "\\begin{align}\n",
     "  \\text{MSE}\n",
-    "  % \\Expect \\big( \\hat{x} - x \\big)^2\n",
+    "  % \\Expect \\big( \\widehat{x} - x \\big)^2\n",
     "  &= \\Expect \\big( \\vect{w}\\tr \\vect{y} - x \\big)^2 \\\\\n",
     "  % &= \\Expect \\big( \\vect{w}\\tr \\vect{y} \\vect{y}\\tr \\vect{w} - 2 x \\vect{y}\\tr \\vect{w} + x^2 \\big) \\\\\n",
     "  &= \\vect{w}\\tr \\Expect \\big( \\vect{y} \\vect{y}\\tr \\big) \\vect{w}\n",
@@ -767,7 +767,7 @@
     "  which then requires that the weights sum to one, i.e. $\\vect{w}\\tr\\vect{1} = 1$.\n",
     "  Then, if $\\mat{C}_{\\vect{y}\\vect{y}}$ is the covariance of $\\vect{r}$,\n",
     "  the BLUE weights become $(\\vect{1}\\tr \\mat{C}_{\\vect{y}\\vect{y}}^{-1})/(\\vect{1}\\tr \\mat{C}_{\\vect{y}\\vect{y}}^{-1} \\vect{1})$,\n",
-    "  and so $\\hat{x}(s)$ is constant in $s$, i.e. flat.\n",
+    "  and so $\\widehat{x}(s)$ is constant in $s$, i.e. flat.\n",
     "  Thus we see the need to include the randomness of $x$ along with that of $\\vect{y}$.\n",
     "  This seems contrary to the classical BLUE framing,\n",
     "  but we have already seen it done for the Kalman gain (by augmenting the observations by the prior mean).\n",
@@ -777,7 +777,7 @@
     "\n",
     "  Now for the \"dual\" perspective of kriging, which is held by **radial basis function (RBF) interpolation**.\n",
     "  Ultimately, kriging (simple, ordinary, and universal) provides an estimate of the form\n",
-    "  $\\hat{x} = \\vect{y}\\tr (\\mat{A}_\\vect{y}^{-1} \\vect{b}_{\\vect{y} x})$,\n",
+    "  $\\widehat{x} = \\vect{y}\\tr (\\mat{A}_\\vect{y}^{-1} \\vect{b}_{\\vect{y} x})$,\n",
     "  where $\\vect{y}$ is the observations,\n",
     "  and the subscripts indicate that $\\mat{A}_\\vect{y}$ depends on the locations of $\\vect{y}$,\n",
     "  while $\\vect{b}_{\\vect{y} x}$ depends on the locations of $\\vect{y}$ and $x$ both.\n",
@@ -807,7 +807,7 @@
     "  - - -\n",
     "</details>\n",
     "\n",
-    "However, we previously derived this $\\hat{x}$ as the posterior/conditional mean of Gaussian distribution.\n",
+    "However, we previously derived this $\\widehat{x}$ as the posterior/conditional mean of Gaussian distribution.\n",
     "This perspective is that of GP regression,\n",
     "but was also highlighted by [Krige (1951)](#References),\n",
     "and makes it evident that kriging also provides an uncertainty estimate\n",
@@ -986,7 +986,7 @@
     "Let us do away with the assumption that the mean of the field is known,\n",
     "all the while retaining the assumption that it is constant in space.\n",
     "The resulting method is called ordinary kriging.\n",
-    "In this case, unbiasedness of $\\hat{x} = \\vect{w}\\tr \\vect{y}$ requires that the weights sum to one.\n",
+    "In this case, unbiasedness of $\\widehat{x} = \\vect{w}\\tr \\vect{y}$ requires that the weights sum to one.\n",
     "This can be imposed on the MSE minimization using a Lagrange multiplier $\\lambda$,\n",
     "yielding the augmented system to solve:\n",
     "$$ \\begin{pmatrix} \\mat{C}_{\\vect{y} \\vect{y}} & \\vect{1} \\\\ \\vect{1}\\tr & 0 \\end{pmatrix} \\begin{pmatrix} \\vect{w} \\\\ \\lambda \\end{pmatrix}\n",

notebooks/T8 - Monte-Carlo & cov estimation.ipynb

@@ -350,7 +350,7 @@
     "- (a). What's the difference between error and residual?\n",
     "- (b). What's the difference between error and bias?\n",
     "- (c). Show that mean-square-error (MSE) = Bias${}^2$ + Var.  \n",
-    "  *Hint: start by writing down the definitions of error, bias, and variance (of $\\hat{\\theta}$).*"
+    "  *Hint: start by writing down the definitions of error, bias, and variance (of $\\widehat{\\theta}$).*"
    ]
   },
   {

notebooks/nb_mirrors/T3 - Bayesian inference.md

@@ -389,10 +389,10 @@ It merely states how to update our quantitative belief (weighted possibilities)
 As such, barring any approximations (such as using `Bayes_rule_LG1` outside the linear-Gaussian case),
 the (full) posterior will be **optimal** from the perspective of any [proper scoring rule](https://en.wikipedia.org/wiki/Scoring_rule#Propriety_and_consistency).
 
-*But if you must* pick a single point value estimate $\hat{x}$
-then you should **decide** on it by optimising (with respect to $\hat{x}$)
+*But if you must* pick a single point value estimate $\widehat{x}$
+then you should **decide** on it by optimising (with respect to $\widehat{x}$)
 the expectation (with respect to $x$) of some utility/loss function,
-i.e. $\Expect\, \text{Loss}(x - \hat{x})$.
+i.e. $\Expect\, \text{Loss}(x - \widehat{x})$.
 For instance, if the posterior pdf happens to be symmetric
 (as in the linear-Gaussian context above),
 and your loss function is convex and symmetric,

notebooks/nb_mirrors/T3 - Bayesian inference.py

@@ -379,10 +379,10 @@ def pdf_fitted(x, mu, sigma2, dist=dist):
 # As such, barring any approximations (such as using `Bayes_rule_LG1` outside the linear-Gaussian case),
 # the (full) posterior will be **optimal** from the perspective of any [proper scoring rule](https://en.wikipedia.org/wiki/Scoring_rule#Propriety_and_consistency).
 #
-# *But if you must* pick a single point value estimate $\hat{x}$
-# then you should **decide** on it by optimising (with respect to $\hat{x}$)
+# *But if you must* pick a single point value estimate $\widehat{x}$
+# then you should **decide** on it by optimising (with respect to $\widehat{x}$)
 # the expectation (with respect to $x$) of some utility/loss function,
-# i.e. $\Expect\, \text{Loss}(x - \hat{x})$.
+# i.e. $\Expect\, \text{Loss}(x - \widehat{x})$.
 # For instance, if the posterior pdf happens to be symmetric
 # (as in the linear-Gaussian context above),
 # and your loss function is convex and symmetric,

notebooks/nb_mirrors/T4 - Time series filtering.md

@@ -197,15 +197,15 @@ the KF of eqns. (5)-(8) computes the *exact* Bayesian pdfs for $x_k$.
   The following does relies only on the ("hidden Markov model") assumptions.
 
   - The analysis "assimilates" $y_k$ according to Bayes' rule to compute $p(x_k | y_{1:k})$,
-    where $y_{1:k} = y_1, \ldots, y_k$ is shorthand notation.
-    $$
-    p(x_k | y_{1:k}) \propto p(y_k | x_k) \, p(x_k | x_{1:k-1}) \,.
-    $$
+  where $y_{1:k} = y_1, \ldots, y_k$ is shorthand notation.
+  $$
+  p(x_k | y_{1:k}) \propto p(y_k | x_k) \, p(x_k | x_{1:k-1}) \,.
+  $$
   - The forecast "propagates" the uncertainty (i.e. density) according to the Chapman-Kolmogorov equation
-    to produce $p(x_{k+1}| y_{1:k})$.
-    $$
-    p(x_{k+1} | y_{1:k}) = \int p(x_{k+1} | x_k) \, p(x_k | y_{1:k}) \, d x_k \,.
-    $$
+  to produce $p(x_{k+1}| y_{1:k})$.
+  $$
+  p(x_{k+1} | y_{1:k}) = \int p(x_{k+1} | x_k) \, p(x_k | y_{1:k}) \, d x_k \,.
+  $$
 
   It is important to appreciate the benefits of the recursive form of these computations:
   It reflects the recursiveness (Markov property) of nature:

notebooks/nb_mirrors/T4 - Time series filtering.py

@@ -194,15 +194,15 @@ def exprmt(seed=4, nTime=50, M=0.97, logR=1, logQ=1, analyses_only=False, logR_b
 #   The following does relies only on the ("hidden Markov model") assumptions.
 #
 #   - The analysis "assimilates" $y_k$ according to Bayes' rule to compute $p(x_k | y_{1:k})$,
-#     where $y_{1:k} = y_1, \ldots, y_k$ is shorthand notation.
-#     $$
-#     p(x_k | y_{1:k}) \propto p(y_k | x_k) \, p(x_k | x_{1:k-1}) \,.
-#     $$
+#   where $y_{1:k} = y_1, \ldots, y_k$ is shorthand notation.
+#   $$
+#   p(x_k | y_{1:k}) \propto p(y_k | x_k) \, p(x_k | x_{1:k-1}) \,.
+#   $$
 #   - The forecast "propagates" the uncertainty (i.e. density) according to the Chapman-Kolmogorov equation
-#     to produce $p(x_{k+1}| y_{1:k})$.
-#     $$
-#     p(x_{k+1} | y_{1:k}) = \int p(x_{k+1} | x_k) \, p(x_k | y_{1:k}) \, d x_k \,.
-#     $$
+#   to produce $p(x_{k+1}| y_{1:k})$.
+#   $$
+#   p(x_{k+1} | y_{1:k}) = \int p(x_{k+1} | x_k) \, p(x_k | y_{1:k}) \, d x_k \,.
+#   $$
 #
 #   It is important to appreciate the benefits of the recursive form of these computations:
 #   It reflects the recursiveness (Markov property) of nature:

notebooks/nb_mirrors/T7 - Geostats & Kriging [optional].md

@@ -424,8 +424,8 @@ estims["Inv-dist."][obs_indices] = observations # Fix singularities
 
 #### Kriging
 
-Kriging finds the best (minimum) mean square error (MSE), $\Expect (\hat{x} - x)^2$, among all linear "predictors",
-$\hat{x} = \vect{w}\tr \vect{y}$, that are unbiased (BLUP).
+Kriging finds the best (minimum) mean square error (MSE), $\Expect (\widehat{x} - x)^2$, among all linear "predictors",
+$\widehat{x} = \vect{w}\tr \vect{y}$, that are unbiased (BLUP).
 
 <a name='Exc-–-"simple"-kriging-(SK)'></a>
 
@@ -434,13 +434,13 @@ $\hat{x} = \vect{w}\tr \vect{y}$, that are unbiased (BLUP).
 Suppose $X(s)$ has a mean that is constant in space, $\mu$, and known.
 Since it is easy to subtract (and later re-include) $\mu$
 from both $x$ and the data $\vect{y}$, we simply assume $\mu = 0$.
-Thus, $\Expect \vect{y} = \vect{0}$ and $\Expect \hat{x} = 0$ for any weights $\vect{w}$,
-and so $\hat{x}$ is already unbiased.
+Thus, $\Expect \vect{y} = \vect{0}$ and $\Expect \widehat{x} = 0$ for any weights $\vect{w}$,
+and so $\widehat{x}$ is already unbiased.
 Meanwhile,
 $$
 \begin{align}
   \text{MSE}
-  % \Expect \big( \hat{x} - x \big)^2
+  % \Expect \big( \widehat{x} - x \big)^2
   &= \Expect \big( \vect{w}\tr \vect{y} - x \big)^2 \\
   % &= \Expect \big( \vect{w}\tr \vect{y} \vect{y}\tr \vect{w} - 2 x \vect{y}\tr \vect{w} + x^2 \big) \\
   &= \vect{w}\tr \Expect \big( \vect{y} \vect{y}\tr \big) \vect{w}
@@ -476,7 +476,7 @@ yielding $\vect{w}_{\text{SK}} = \mat{C}_{\vect{y} \vect{y}}^{-1} \, \vect{c}_{x
   which then requires that the weights sum to one, i.e. $\vect{w}\tr\vect{1} = 1$.
   Then, if $\mat{C}_{\vect{y}\vect{y}}$ is the covariance of $\vect{r}$,
   the BLUE weights become $(\vect{1}\tr \mat{C}_{\vect{y}\vect{y}}^{-1})/(\vect{1}\tr \mat{C}_{\vect{y}\vect{y}}^{-1} \vect{1})$,
-  and so $\hat{x}(s)$ is constant in $s$, i.e. flat.
+  and so $\widehat{x}(s)$ is constant in $s$, i.e. flat.
   Thus we see the need to include the randomness of $x$ along with that of $\vect{y}$.
   This seems contrary to the classical BLUE framing,
   but we have already seen it done for the Kalman gain (by augmenting the observations by the prior mean).
@@ -486,7 +486,7 @@ yielding $\vect{w}_{\text{SK}} = \mat{C}_{\vect{y} \vect{y}}^{-1} \, \vect{c}_{x
 
   Now for the "dual" perspective of kriging, which is held by **radial basis function (RBF) interpolation**.
   Ultimately, kriging (simple, ordinary, and universal) provides an estimate of the form
-  $\hat{x} = \vect{y}\tr (\mat{A}_\vect{y}^{-1} \vect{b}_{\vect{y} x})$,
+  $\widehat{x} = \vect{y}\tr (\mat{A}_\vect{y}^{-1} \vect{b}_{\vect{y} x})$,
   where $\vect{y}$ is the observations,
   and the subscripts indicate that $\mat{A}_\vect{y}$ depends on the locations of $\vect{y}$,
   while $\vect{b}_{\vect{y} x}$ depends on the locations of $\vect{y}$ and $x$ both.
@@ -516,7 +516,7 @@ yielding $\vect{w}_{\text{SK}} = \mat{C}_{\vect{y} \vect{y}}^{-1} \, \vect{c}_{x
   - - -
 </details>
 
-However, we previously derived this $\hat{x}$ as the posterior/conditional mean of Gaussian distribution.
+However, we previously derived this $\widehat{x}$ as the posterior/conditional mean of Gaussian distribution.
 This perspective is that of GP regression,
 but was also highlighted by [Krige (1951)](#References),
 and makes it evident that kriging also provides an uncertainty estimate
@@ -617,7 +617,7 @@ def simple_kriging(vg, dists_xy, dists_yy, observations, mu):
 Let us do away with the assumption that the mean of the field is known,
 all the while retaining the assumption that it is constant in space.
 The resulting method is called ordinary kriging.
-In this case, unbiasedness of $\hat{x} = \vect{w}\tr \vect{y}$ requires that the weights sum to one.
+In this case, unbiasedness of $\widehat{x} = \vect{w}\tr \vect{y}$ requires that the weights sum to one.
 This can be imposed on the MSE minimization using a Lagrange multiplier $\lambda$,
 yielding the augmented system to solve:
 $$ \begin{pmatrix} \mat{C}_{\vect{y} \vect{y}} & \vect{1} \\ \vect{1}\tr & 0 \end{pmatrix} \begin{pmatrix} \vect{w} \\ \lambda \end{pmatrix}

notebooks/nb_mirrors/T7 - Geostats & Kriging [optional].py

@@ -389,8 +389,8 @@ def sample_2D(power=1.5, transf01="expo", Range=0.3, nugget=1e-2):
 
 # #### Kriging
 #
-# Kriging finds the best (minimum) mean square error (MSE), $\Expect (\hat{x} - x)^2$, among all linear "predictors",
-# $\hat{x} = \vect{w}\tr \vect{y}$, that are unbiased (BLUP).
+# Kriging finds the best (minimum) mean square error (MSE), $\Expect (\widehat{x} - x)^2$, among all linear "predictors",
+# $\widehat{x} = \vect{w}\tr \vect{y}$, that are unbiased (BLUP).
 #
 # <a name='Exc-–-"simple"-kriging-(SK)'></a>
 #
@@ -399,13 +399,13 @@ def sample_2D(power=1.5, transf01="expo", Range=0.3, nugget=1e-2):
 # Suppose $X(s)$ has a mean that is constant in space, $\mu$, and known.
 # Since it is easy to subtract (and later re-include) $\mu$
 # from both $x$ and the data $\vect{y}$, we simply assume $\mu = 0$.
-# Thus, $\Expect \vect{y} = \vect{0}$ and $\Expect \hat{x} = 0$ for any weights $\vect{w}$,
-# and so $\hat{x}$ is already unbiased.
+# Thus, $\Expect \vect{y} = \vect{0}$ and $\Expect \widehat{x} = 0$ for any weights $\vect{w}$,
+# and so $\widehat{x}$ is already unbiased.
 # Meanwhile,
 # $$
 # \begin{align}
 #   \text{MSE}
-#   % \Expect \big( \hat{x} - x \big)^2
+#   % \Expect \big( \widehat{x} - x \big)^2
 #   &= \Expect \big( \vect{w}\tr \vect{y} - x \big)^2 \\
 #   % &= \Expect \big( \vect{w}\tr \vect{y} \vect{y}\tr \vect{w} - 2 x \vect{y}\tr \vect{w} + x^2 \big) \\
 #   &= \vect{w}\tr \Expect \big( \vect{y} \vect{y}\tr \big) \vect{w}
@@ -441,7 +441,7 @@ def sample_2D(power=1.5, transf01="expo", Range=0.3, nugget=1e-2):
 #   which then requires that the weights sum to one, i.e. $\vect{w}\tr\vect{1} = 1$.
 #   Then, if $\mat{C}_{\vect{y}\vect{y}}$ is the covariance of $\vect{r}$,
 #   the BLUE weights become $(\vect{1}\tr \mat{C}_{\vect{y}\vect{y}}^{-1})/(\vect{1}\tr \mat{C}_{\vect{y}\vect{y}}^{-1} \vect{1})$,
-#   and so $\hat{x}(s)$ is constant in $s$, i.e. flat.
+#   and so $\widehat{x}(s)$ is constant in $s$, i.e. flat.
 #   Thus we see the need to include the randomness of $x$ along with that of $\vect{y}$.
 #   This seems contrary to the classical BLUE framing,
 #   but we have already seen it done for the Kalman gain (by augmenting the observations by the prior mean).
@@ -451,7 +451,7 @@ def sample_2D(power=1.5, transf01="expo", Range=0.3, nugget=1e-2):
 #
 #   Now for the "dual" perspective of kriging, which is held by **radial basis function (RBF) interpolation**.
 #   Ultimately, kriging (simple, ordinary, and universal) provides an estimate of the form
-#   $\hat{x} = \vect{y}\tr (\mat{A}_\vect{y}^{-1} \vect{b}_{\vect{y} x})$,
+#   $\widehat{x} = \vect{y}\tr (\mat{A}_\vect{y}^{-1} \vect{b}_{\vect{y} x})$,
 #   where $\vect{y}$ is the observations,
 #   and the subscripts indicate that $\mat{A}_\vect{y}$ depends on the locations of $\vect{y}$,
 #   while $\vect{b}_{\vect{y} x}$ depends on the locations of $\vect{y}$ and $x$ both.
@@ -481,7 +481,7 @@ def sample_2D(power=1.5, transf01="expo", Range=0.3, nugget=1e-2):
 #   - - -
 # </details>
 #
-# However, we previously derived this $\hat{x}$ as the posterior/conditional mean of Gaussian distribution.
+# However, we previously derived this $\widehat{x}$ as the posterior/conditional mean of Gaussian distribution.
 # This perspective is that of GP regression,
 # but was also highlighted by [Krige (1951)](#References),
 # and makes it evident that kriging also provides an uncertainty estimate
@@ -577,7 +577,7 @@ def simple_kriging(vg, dists_xy, dists_yy, observations, mu):
 # Let us do away with the assumption that the mean of the field is known,
 # all the while retaining the assumption that it is constant in space.
 # The resulting method is called ordinary kriging.
-# In this case, unbiasedness of $\hat{x} = \vect{w}\tr \vect{y}$ requires that the weights sum to one.
+# In this case, unbiasedness of $\widehat{x} = \vect{w}\tr \vect{y}$ requires that the weights sum to one.
 # This can be imposed on the MSE minimization using a Lagrange multiplier $\lambda$,
 # yielding the augmented system to solve:
 # $$ \begin{pmatrix} \mat{C}_{\vect{y} \vect{y}} & \vect{1} \\ \vect{1}\tr & 0 \end{pmatrix} \begin{pmatrix} \vect{w} \\ \lambda \end{pmatrix}