Changes as per @willtebbutt's suggestions

Co-authored-by: Will Tebbutt <[email protected]>

Author: penelopeysm

developers/transforms/bijectors/index.qmd

@@ -46,7 +46,7 @@ However, we specified that the bijection $y = \exp(x)$ maps values of $x \in (-\
 :::
 
 Specifically, one of the primary purposes of Bijectors.jl is used to construct _bijections which map constrained distributions to unconstrained ones_.
-For example, the log-normal distribution which we saw above is constrained: its _support_, i.e. the range over which $p(x) \geq 0$, is $(0, \infty)$.
+For example, the log-normal distribution which we saw above is constrained: its _support_, i.e. the range over which $p(x) > 0$, is $(0, \infty)$.
 However, we can transform that to an unconstrained distribution (the normal distribution) using the transformation $y = \log(x)$.
 
 ::: {.callout-note}

developers/transforms/distributions/index.qmd

@@ -36,7 +36,7 @@ histogram(samples, bins=50)
 ```
 
 (Calling `Normal()` without any arguments, as we do here, gives us a normal distribution with mean 0 and standard deviation 1.)
-If you want to know the probability of observing any of the samples, you can use `logpdf`:
+If you want to know the log probability density of observing any of the samples, you can use `logpdf`:
 
 ```{julia}
 println("sample: $(samples[1])")
@@ -53,6 +53,8 @@ so we could also have calculated this manually using:
 log(1 / sqrt(2π) * exp(-samples[1]^2 / 2))
 ```
 
+(or more efficiently, `-(samples[1]^2 + log2π) / 2`, where `log2π` is from the [IrrationalConstants.jl package](https://github.com/JuliaMath/IrrationalConstants.jl)).
+
 ## Sampling from a transformed distribution
 
 Say that $x$ is distributed according to `Normal()`, and we want to draw samples of $y = \exp(x)$.
@@ -115,7 +117,7 @@ If we think about the normal distribution as a continuous curve, what the probab
 
 $$\int_a^b p(x) \, \mathrm{d}x.$$
 
-For example, if $(a, b) = (-\infty, \infty)$, then the probability of drawing a sample from the entire distribution is 1.
+For example, if $(a, b) = (-\infty, \infty)$, then the probability of drawing a sample between $a$ and $b$ is 1.
 
 Let's say that the probability density function of the log-normal distribution is $q(y)$.
 Then, the area under the curve between the two points $\exp(a)$ and $\exp(b)$ is:
@@ -163,7 +165,7 @@ println("Expected : $(logpdf(LogNormal(), samples_lognormal[1]))")
 println("Actual   : $(logpdf(MyLogNormal(), samples_lognormal[1]))")
 ```
 
-The same process can be applied to _any_ kind of transformation.
+The same process can be applied to any kind of (invertible) transformation.
 If we have some transformation from $x$ to $y$, and the probability density functions of $x$ and $y$ are $p(x)$ and $q(y)$ respectively, then we have a general formula that:
 
 $$q(y) = p(x) \left| \frac{\mathrm{d}x}{\mathrm{d}y} \right|.$$
@@ -277,7 +279,7 @@ $$\frac{\partial x_2}{\partial y_1} = \frac{1}{2\pi} \left(\frac{1}{1 + (y_2/y_1
 
 Putting together the Jacobian matrix, we have:
 
-$$\mathcal{J} = \begin{pmatrix}
+$$\mathbf{J} = \begin{pmatrix}
 -y_1 x_1 & -y_2 x_1 \\
 -cy_2/y_1^2 & c/y_1 \\
 \end{pmatrix},$$
@@ -286,7 +288,7 @@ where $c = [2\pi(1 + (y_2/y_1)^2)]^{-1}$.
 The determinant of this matrix is
 
 $$\begin{align}
-\det(\mathcal{J}) &= -cx_1 - cx_1(y_2/y_1)^2 \\
+\det(\mathbf{J}) &= -cx_1 - cx_1(y_2/y_1)^2 \\
 &= -cx_1\left[1 + \left(\frac{y_2}{y_1}\right)^2\right] \\
 &= -\frac{1}{2\pi} x_1 \\
 &= -\frac{1}{2\pi}\exp{\left(-\frac{y_1^2}{2}\right)}\exp{\left(-\frac{y_2^2}{2}\right)},
@@ -295,7 +297,7 @@ $$\begin{align}
 Coming right back to our probability density, we have that
 
 $$\begin{align}
-q(y_1, y_2) &= p(x_1, x_2) \cdot |\det(\mathcal{J})| \\
+q(y_1, y_2) &= p(x_1, x_2) \cdot |\det(\mathbf{J})| \\
 &= \frac{1}{2\pi}\exp{\left(-\frac{y_1^2}{2}\right)}\exp{\left(-\frac{y_2^2}{2}\right)},
 \end{align}$$