Dont use mathcal which mathjax default renders as squares on Colab+macOS

Explanation
https://math.meta.stackexchange.com/questions/35298/why-using-mathcal-command-produce-a-square

Issue tracker
googlecolab/colabtools#3192

Permanent fix (unlikely)
googlecolab/colabtools#2822

notebooks/T2 - Gaussian distribution.ipynb

@@ -64,14 +64,14 @@
    "metadata": {},
    "source": [
     "## The univariate (a.k.a. 1-dimensional, scalar) case\n",
-    "Consider the Gaussian random variable $x \\sim \\mathcal{N}(\\mu, \\sigma^2)$.  \n",
+    "Consider the Gaussian random variable $x \\sim \\mathscr{N}(\\mu, \\sigma^2)$.  \n",
     "Its probability density function (**pdf**),\n",
     "$\n",
-    "p(x) = \\mathcal{N}(x \\mid \\mu, \\sigma^2)\n",
+    "p(x) = \\mathscr{N}(x \\mid \\mu, \\sigma^2)\n",
     "$ for $x \\in (-\\infty, +\\infty)$,\n",
     "is given by\n",
     "$$\\begin{align}\n",
-    "\\mathcal{N}(x \\mid \\mu, \\sigma^2) = (2 \\pi \\sigma^2)^{-1/2} e^{-(x-\\mu)^2/2 \\sigma^2} \\,. \\tag{G1}\n",
+    "\\mathscr{N}(x \\mid \\mu, \\sigma^2) = (2 \\pi \\sigma^2)^{-1/2} e^{-(x-\\mu)^2/2 \\sigma^2} \\,. \\tag{G1}\n",
     "\\end{align}$$\n",
     "\n",
     "Run the cell below to define a function to compute the pdf (G1) using the `scipy` library."
@@ -157,7 +157,7 @@
    "id": "94b6d541",
    "metadata": {},
    "source": [
-    "**Exc -- Derivatives:** Recall $p(x) = \\mathcal{N}(x \\mid \\mu, \\sigma^2)$ from eqn (G1).  \n",
+    "**Exc -- Derivatives:** Recall $p(x) = \\mathscr{N}(x \\mid \\mu, \\sigma^2)$ from eqn (G1).  \n",
     "Use pen, paper, and calculus to answer the following questions,  \n",
     "which derive some helpful mnemonics about the distribution.\n",
     "\n",
@@ -201,7 +201,7 @@
    "metadata": {},
    "source": [
     "#### Exc (optional) -- Integrals\n",
-    "Recall $p(x) = \\mathcal{N}(x \\mid \\mu, \\sigma^2)$ from eqn (G1). Abbreviate it using $c = (2 \\pi \\sigma^2)^{-1/2}$.  \n",
+    "Recall $p(x) = \\mathscr{N}(x \\mid \\mu, \\sigma^2)$ from eqn (G1). Abbreviate it using $c = (2 \\pi \\sigma^2)^{-1/2}$.  \n",
     "Use pen, paper, and calculus to show that\n",
     " - (i) the first parameter, $\\mu$, indicates its **mean**, i.e. that $$\\mu = \\Expect[x] \\,.$$\n",
     "   *Hint: you can rely on the result of (iii)*\n",

notebooks/T3 - Bayesian inference.ipynb

@@ -48,7 +48,7 @@
     "studied the Gaussian probability density function (pdf), defined by:\n",
     "\n",
     "$$\\begin{align}\n",
-    "\\mathcal{N}(x \\mid \\mu, \\sigma^2) &= (2 \\pi \\sigma^2)^{-1/2} e^{-(x-\\mu)^2/2 \\sigma^2} \\,,\\tag{G1} \\\\\n",
+    "\\mathscr{N}(x \\mid \\mu, \\sigma^2) &= (2 \\pi \\sigma^2)^{-1/2} e^{-(x-\\mu)^2/2 \\sigma^2} \\,,\\tag{G1} \\\\\n",
     "\\NormDist(\\x \\mid  \\mathbf{\\mu}, \\mathbf{\\Sigma})\n",
     "&=\n",
     "|2 \\pi \\mathbf{\\Sigma}|^{-1/2} \\, \\exp\\Big(-\\frac{1}{2}\\|\\x-\\mathbf{\\mu}\\|^2_\\mathbf{\\Sigma} \\Big) \\,, \\tag{GM}\n",
@@ -476,12 +476,12 @@
     "In response to this computational difficulty, we try to be smart and do something more analytical (\"pen-and-paper\"): we only compute the parameters (mean and (co)variance) of the posterior pdf.\n",
     "\n",
     "This is doable and quite simple in the Gaussian-Gaussian case, when $\\ObsMod$ is linear (i.e. just a number):  \n",
-    "- Given the prior of $p(x) = \\mathcal{N}(x \\mid x\\supf, P\\supf)$\n",
-    "- and a likelihood $p(y|x) = \\mathcal{N}(y \\mid \\ObsMod x,R)$,  \n",
+    "- Given the prior of $p(x) = \\mathscr{N}(x \\mid x\\supf, P\\supf)$\n",
+    "- and a likelihood $p(y|x) = \\mathscr{N}(y \\mid \\ObsMod x,R)$,  \n",
     "- $\\implies$ posterior\n",
     "$\n",
     "p(x|y)\n",
-    "= \\mathcal{N}(x \\mid x\\supa, P\\supa) \\,,\n",
+    "= \\mathscr{N}(x \\mid x\\supa, P\\supa) \\,,\n",
     "$\n",
     "where, in the 1-dimensional/univariate/scalar (multivariate is discussed in [T5](T5%20-%20Multivariate%20Kalman%20filter.ipynb)) case:\n",
     "\n",
@@ -501,7 +501,7 @@
     "- (a) Actually derive the first term of the RHS, i.e. eqns. (5) and (6).  \n",
     "  *Hint: you can simplify the task by first \"hiding\" $\\ObsMod$ by astutely multiplying by $1$ somewhere.*\n",
     "- (b) *Optional*: Derive the full RHS (i.e. also the second term).\n",
-    "- (c) Derive $p(x|y) = \\mathcal{N}(x \\mid x\\supa, P\\supa)$ from eqns. (5) and (6)\n",
+    "- (c) Derive $p(x|y) = \\mathscr{N}(x \\mid x\\supa, P\\supa)$ from eqns. (5) and (6)\n",
     "  using part (a), Bayes' rule (BR2), and the Gaussian pdf (G1)."
    ]
   },
@@ -522,11 +522,11 @@
    "source": [
     "**Exc -- Temperature example:**\n",
     "The statement $x = \\mu \\pm \\sigma$ is *sometimes* used\n",
-    "as a shorthand for $p(x) = \\mathcal{N}(x \\mid \\mu, \\sigma^2)$. Suppose\n",
+    "as a shorthand for $p(x) = \\mathscr{N}(x \\mid \\mu, \\sigma^2)$. Suppose\n",
     "- you think the temperature $x = 20°C \\pm 2°C$,\n",
     "- a thermometer yields the observation $y = 18°C \\pm 2°C$.\n",
     "\n",
-    "Show that your posterior is $p(x|y) = \\mathcal{N}(x \\mid 19, 2)$"
+    "Show that your posterior is $p(x|y) = \\mathscr{N}(x \\mid 19, 2)$"
    ]
   },
   {

notebooks/T4 - Time series filtering.ipynb

@@ -208,7 +208,7 @@
     "Formulae (5) and (6) are called the **forecast step** of the KF.\n",
     "But when $y_1$ becomes available, according to eqn. (Obs),\n",
     "then we can update/condition our estimate of $x_1$, i.e. compute the posterior,\n",
-    "$p(x_1 | y_1) = \\mathcal{N}(x_1 \\mid x\\supa_1, P\\supa_1) \\,,$\n",
+    "$p(x_1 | y_1) = \\mathscr{N}(x_1 \\mid x\\supa_1, P\\supa_1) \\,,$\n",
     "using the formulae we developed for Bayes' rule with\n",
     "[Gaussian distributions](T3%20-%20Bayesian%20inference.ipynb#Gaussian-Gaussian-Bayes'-rule-(1D)).\n",
     "\n",

notebooks/T8 - Monte-Carlo & ensembles.ipynb

@@ -109,7 +109,7 @@
     "\n",
     "**Exc -- Multivariate Gaussian sampling:**\n",
     "Suppose $\\z$ is a standard Gaussian,\n",
-    "i.e. $p(\\z) = \\mathcal{N}(\\z \\mid \\bvec{0},\\I_{\\xDim})$,\n",
+    "i.e. $p(\\z) = \\mathscr{N}(\\z \\mid \\bvec{0},\\I_{\\xDim})$,\n",
     "where $\\I_{\\xDim}$ is the $\\xDim$-dimensional identity matrix.  \n",
     "Let $\\x = \\mat{L}\\z + \\mu$.\n",
     "\n",

notebooks/resources/answers.py

@@ -592,7 +592,7 @@ def setup_typeset():
 $$ \begin{align}
 p\, (y_1, \ldots, y_K \;|\; a)
 &= \prod_{k=1}^K \, p\, (y_k \;|\; a) \tag{each obs. is indep. of others, knowing $a$.} \\\
-&= \prod_k \, \mathcal{N}(y_k \mid a k,R) \tag{inserted eqn. (3) and then (1).} \\\
+&= \prod_k \, \NormDist{N}(y_k \mid a k,R) \tag{inserted eqn. (3) and then (1).} \\\
 &= \prod_k \,  (2 \pi R)^{-1/2} e^{-(y_k - a k)^2/2 R} \tag{inserted eqn. (G1) from T2.} \\\
 &= c \exp\Big(\frac{-1}{2 R}\sum_k (y_k - a k)^2\Big) \\\
 \end{align} $$

notebooks/resources/macros.py

@@ -16,7 +16,7 @@
 macros=r'''
 \newcommand{\Reals}{\mathbb{R}}
 \newcommand{\Expect}[0]{\mathbb{E}}
-\newcommand{\NormDist}{\mathcal{N}}
+\newcommand{\NormDist}{\mathscr{N}}
 
 \newcommand{\DynMod}[0]{\mathscr{M}}
 \newcommand{\ObsMod}[0]{\mathscr{H}}

notebooks/scripts/T2 - Gaussian distribution.py

@@ -43,14 +43,14 @@
 
 
 # ## The univariate (a.k.a. 1-dimensional, scalar) case
-# Consider the Gaussian random variable $x \sim \mathcal{N}(\mu, \sigma^2)$.  
+# Consider the Gaussian random variable $x \sim \mathscr{N}(\mu, \sigma^2)$.  
 # Its probability density function (**pdf**),
 # $
-# p(x) = \mathcal{N}(x \mid \mu, \sigma^2)
+# p(x) = \mathscr{N}(x \mid \mu, \sigma^2)
 # $ for $x \in (-\infty, +\infty)$,
 # is given by
 # $$\begin{align}
-# \mathcal{N}(x \mid \mu, \sigma^2) = (2 \pi \sigma^2)^{-1/2} e^{-(x-\mu)^2/2 \sigma^2} \,. \tag{G1}
+# \mathscr{N}(x \mid \mu, \sigma^2) = (2 \pi \sigma^2)^{-1/2} e^{-(x-\mu)^2/2 \sigma^2} \,. \tag{G1}
 # \end{align}$$
 #
 # Run the cell below to define a function to compute the pdf (G1) using the `scipy` library.
@@ -89,7 +89,7 @@ def plot_pdf(mu=0, sigma=5):
 # show_answer('pdf_G1')
 # -
 
-# **Exc -- Derivatives:** Recall $p(x) = \mathcal{N}(x \mid \mu, \sigma^2)$ from eqn (G1).  
+# **Exc -- Derivatives:** Recall $p(x) = \mathscr{N}(x \mid \mu, \sigma^2)$ from eqn (G1).  
 # Use pen, paper, and calculus to answer the following questions,  
 # which derive some helpful mnemonics about the distribution.
 #
@@ -115,7 +115,7 @@ def plot_pdf(mu=0, sigma=5):
 # -
 
 # #### Exc (optional) -- Integrals
-# Recall $p(x) = \mathcal{N}(x \mid \mu, \sigma^2)$ from eqn (G1). Abbreviate it using $c = (2 \pi \sigma^2)^{-1/2}$.  
+# Recall $p(x) = \mathscr{N}(x \mid \mu, \sigma^2)$ from eqn (G1). Abbreviate it using $c = (2 \pi \sigma^2)^{-1/2}$.  
 # Use pen, paper, and calculus to show that
 #  - (i) the first parameter, $\mu$, indicates its **mean**, i.e. that $$\mu = \Expect[x] \,.$$
 #    *Hint: you can rely on the result of (iii)*

notebooks/scripts/T3 - Bayesian inference.py

@@ -34,7 +34,7 @@
 # studied the Gaussian probability density function (pdf), defined by:
 #
 # $$\begin{align}
-# \mathcal{N}(x \mid \mu, \sigma^2) &= (2 \pi \sigma^2)^{-1/2} e^{-(x-\mu)^2/2 \sigma^2} \,,\tag{G1} \\
+# \mathscr{N}(x \mid \mu, \sigma^2) &= (2 \pi \sigma^2)^{-1/2} e^{-(x-\mu)^2/2 \sigma^2} \,,\tag{G1} \\
 # \NormDist(\x \mid  \mathbf{\mu}, \mathbf{\Sigma})
 # &=
 # |2 \pi \mathbf{\Sigma}|^{-1/2} \, \exp\Big(-\frac{1}{2}\|\x-\mathbf{\mu}\|^2_\mathbf{\Sigma} \Big) \,, \tag{GM}
@@ -299,12 +299,12 @@ def Bayes2(  corr_R =.6,                 y1=1,          R1=4**2,
 # In response to this computational difficulty, we try to be smart and do something more analytical ("pen-and-paper"): we only compute the parameters (mean and (co)variance) of the posterior pdf.
 #
 # This is doable and quite simple in the Gaussian-Gaussian case, when $\ObsMod$ is linear (i.e. just a number):  
-# - Given the prior of $p(x) = \mathcal{N}(x \mid x\supf, P\supf)$
-# - and a likelihood $p(y|x) = \mathcal{N}(y \mid \ObsMod x,R)$,  
+# - Given the prior of $p(x) = \mathscr{N}(x \mid x\supf, P\supf)$
+# - and a likelihood $p(y|x) = \mathscr{N}(y \mid \ObsMod x,R)$,  
 # - $\implies$ posterior
 # $
 # p(x|y)
-# = \mathcal{N}(x \mid x\supa, P\supa) \,,
+# = \mathscr{N}(x \mid x\supa, P\supa) \,,
 # $
 # where, in the 1-dimensional/univariate/scalar (multivariate is discussed in [T5](T5%20-%20Multivariate%20Kalman%20filter.ipynb)) case:
 #
@@ -324,7 +324,7 @@ def Bayes2(  corr_R =.6,                 y1=1,          R1=4**2,
 # - (a) Actually derive the first term of the RHS, i.e. eqns. (5) and (6).  
 #   *Hint: you can simplify the task by first "hiding" $\ObsMod$ by astutely multiplying by $1$ somewhere.*
 # - (b) *Optional*: Derive the full RHS (i.e. also the second term).
-# - (c) Derive $p(x|y) = \mathcal{N}(x \mid x\supa, P\supa)$ from eqns. (5) and (6)
+# - (c) Derive $p(x|y) = \mathscr{N}(x \mid x\supa, P\supa)$ from eqns. (5) and (6)
 #   using part (a), Bayes' rule (BR2), and the Gaussian pdf (G1).
 
 # +
@@ -333,11 +333,11 @@ def Bayes2(  corr_R =.6,                 y1=1,          R1=4**2,
 
 # **Exc -- Temperature example:**
 # The statement $x = \mu \pm \sigma$ is *sometimes* used
-# as a shorthand for $p(x) = \mathcal{N}(x \mid \mu, \sigma^2)$. Suppose
+# as a shorthand for $p(x) = \mathscr{N}(x \mid \mu, \sigma^2)$. Suppose
 # - you think the temperature $x = 20°C \pm 2°C$,
 # - a thermometer yields the observation $y = 18°C \pm 2°C$.
 #
-# Show that your posterior is $p(x|y) = \mathcal{N}(x \mid 19, 2)$
+# Show that your posterior is $p(x|y) = \mathscr{N}(x \mid 19, 2)$
 
 # +
 # show_answer('GG BR example')

notebooks/scripts/T4 - Time series filtering.py

@@ -153,7 +153,7 @@ def exprmt(seed=4, nTime=50, M=0.97, logR=1, logQ=1, analyses_only=False, logR_b
 # Formulae (5) and (6) are called the **forecast step** of the KF.
 # But when $y_1$ becomes available, according to eqn. (Obs),
 # then we can update/condition our estimate of $x_1$, i.e. compute the posterior,
-# $p(x_1 | y_1) = \mathcal{N}(x_1 \mid x\supa_1, P\supa_1) \,,$
+# $p(x_1 | y_1) = \mathscr{N}(x_1 \mid x\supa_1, P\supa_1) \,,$
 # using the formulae we developed for Bayes' rule with
 # [Gaussian distributions](T3%20-%20Bayesian%20inference.ipynb#Gaussian-Gaussian-Bayes'-rule-(1D)).
 #

notebooks/scripts/T8 - Monte-Carlo & ensembles.py

@@ -77,7 +77,7 @@ def pdf_reconstructions(seed=5,       nbins=10,      bw=.3):
 #
 # **Exc -- Multivariate Gaussian sampling:**
 # Suppose $\z$ is a standard Gaussian,
-# i.e. $p(\z) = \mathcal{N}(\z \mid \bvec{0},\I_{\xDim})$,
+# i.e. $p(\z) = \mathscr{N}(\z \mid \bvec{0},\I_{\xDim})$,
 # where $\I_{\xDim}$ is the $\xDim$-dimensional identity matrix.  
 # Let $\x = \mat{L}\z + \mu$.
 #