Fix broken links and tags in docs (#367)

README.md

@@ -35,23 +35,11 @@ If you are interested in using AdvancedHMC.jl through a probabilistic programmin
 
 This section demonstrates a minimal example of sampling from a multivariate Gaussian (10-dimensional) using the no U-turn sampler (NUTS). Below we describe the major components of the Hamiltonian system which are essential to sample using this approach:
 
-- **Metric**: In many sampling problems the sample space is usually associated with a metric that allows us to measure the distance between any two points, and other similar quantities. In the example in this section, we use a special metric called the **Euclidean Metric**, represented with a `D × D` matrix from which we can compute distances. 
-<details>
- <summary>Further details about the Metric component</summary>
- The Euclidean metric is also known as the mass matrix in the physical perspective. For available metrics refer <a href="#hamiltonian-mass-matrix-metric">Hamiltonian mass matrix</a>.
-</details>
-
-- **Leapfrog integration**: Leapfrog integration is a second-order numerical method for integrating differential equations (In this case they are equations of motion for the relative position of one particle with respect to the other). The order of this integration signifies its rate of convergence. Any algorithm with a finite time step size will have numerical errors, and the order is related to this error. For a second-order algorithm, this error scales as the second power of the time step, hence, the name second-order. High-order integrators are usually complex to code and have a limited region of convergence; hence they do not allow arbitrarily large time steps. A second-order integrator is suitable for our purpose. Hence we opt for the leapfrog integrator. It is called `leapfrog` due to the ways this algorithm is written, where the positions and velocities of particles `leap over` each other. 
-<details>
- <summary>About the leapfrog integration scheme</summary>
- Suppose ${\bf x}$ and ${\bf v}$ are the position and velocity of an individual particle respectively; $i$ and $i+1$ are the indices for time values $t_i$ and $t_{i+1}$ respectively; $dt = t_{i+1} - t_i$ is the time step size (constant and regularly spaced intervals), and ${\bf a}$ is the acceleration induced on a particle by the forces of all other particles. Furthermore, suppose positions are defined at times $t_i, t_{i+1}, t_{i+2}, \dots $, spaced at constant intervals $dt$, the velocities are defined at halfway times in between, denoted by $t_{i-1/2}, t_{i+1/2}, t_{i+3/2}, \dots $, where $t_{i+1} - t_{i + 1/2} = t_{i + 1/2} - t_i = dt / 2$, and the accelerations ${\bf a}$ are defined only on integer times, just like the positions. Then the leapfrog integration scheme is given as: $x_{i} = x_{i-1} + v_{i-1/2} dt; \quad v_{i+1/2} = v_{i-1/2} + a_i dt$. For available integrators refer <a href="#integrator-integrator">Integrator</a>.
-</details>
-
-- **Kernel for trajectories (static or dynamic)**: Different kernels, which may be static or dynamic, can be used. At each iteration of any variant of the HMC algorithm, there are two main steps - the first step changes the momentum and the second step may change both the position and the momentum of a particle. 
-<details>
- <summary>More about the kernels</summary>
- In the classical HMC approach, during the first step, new values for the momentum variables are randomly drawn from their Gaussian distribution, independently of the current values of the position variables. A Metropolis update is performed during the second step, using Hamiltonian dynamics to provide a new state. For available kernels refer <a href="#kernel-kernel">kernel</a>.
-</details>
+- **Metric**: In many sampling problems the sample space is associated with a metric that allows us to measure the distance between any two points, and other similar quantities. In the example in this section, we use a special metric called the **Euclidean Metric**, represented with a `D × D` matrix from which we can compute distances.[^1]
+
+- **Leapfrog integration**: Leapfrog integration is a second-order numerical method for integrating differential equations (In this case they are equations of motion for the relative position of one particle with respect to the other). The order of this integration signifies its rate of convergence. Any algorithm with a finite time step size will have numerical errors, and the order is related to this error. For a second-order algorithm, this error scales as the second power of the time step, hence, the name second-order. High-order integrators are usually complex to code and have a limited region of convergence; hence they do not allow arbitrarily large time steps. A second-order integrator is suitable for our purpose. Hence we opt for the leapfrog integrator. It is called `leapfrog` due to the ways this algorithm is written, where the positions and velocities of particles "leap over" each other.[^2]
+
+- **Kernel for trajectories (static or dynamic)**: Different kernels, which may be static or dynamic, can be used. At each iteration of any variant of the HMC algorithm, there are two main steps - the first step changes the momentum and the second step may change both the position and the momentum of a particle.[^3]
 
 ```julia
 using AdvancedHMC, ForwardDiff
@@ -97,7 +85,7 @@ samples, stats = sample(hamiltonian, kernel, initial_θ, n_samples, adaptor, n_a
 ### Parallel sampling 
 
 AdvancedHMC enables parallel sampling (either distributed or multi-thread) via Julia's [parallel computing functions](https://docs.julialang.org/en/v1/manual/parallel-computing/).
-It also supports vectorized sampling for static HMC and has been discussed in more detail in the documentation [here](https://turinglang.github.io/AdvancedHMC.jl/stable/#Parallel-sampling).
+It also supports vectorized sampling for static HMC.
 
 The below example utilizes the `@threads` macro to sample 4 chains across 4 threads.
 
@@ -359,3 +347,9 @@ with the following BibTeX entry:
 5. Betancourt, M. (2016). Identifying the optimal integration time in Hamiltonian Monte Carlo. [arXiv preprint arXiv:1601.00225](https://arxiv.org/abs/1601.00225).
 
 6. Hoffman, M. D., & Gelman, A. (2014). The No-U-Turn Sampler: adaptively setting path lengths in Hamiltonian Monte Carlo. Journal of Machine Learning Research, 15(1), 1593-1623. ([arXiv](http://arxiv.org/abs/1111.4246))
+
+## Footnotes
+
+[^1]: The Euclidean metric is also known as the mass matrix in the physical perspective. See [Hamiltonian mass matrix](#Hamiltonian-mass-matrix-(metric)) for available metrics.
+[^2]: About the leapfrog integration scheme: Suppose ${\bf x}$ and ${\bf v}$ are the position and velocity of an individual particle respectively; $i$ and $i+1$ are the indices for time values $t_i$ and $t_{i+1}$ respectively; $dt = t_{i+1} - t_i$ is the time step size (constant and regularly spaced intervals), and ${\bf a}$ is the acceleration induced on a particle by the forces of all other particles. Furthermore, suppose positions are defined at times $t_i, t_{i+1}, t_{i+2}, \dots $, spaced at constant intervals $dt$, the velocities are defined at halfway times in between, denoted by $t_{i-1/2}, t_{i+1/2}, t_{i+3/2}, \dots $, where $t_{i+1} - t_{i + 1/2} = t_{i + 1/2} - t_i = dt / 2$, and the accelerations ${\bf a}$ are defined only on integer times, just like the positions. Then the leapfrog integration scheme is given as: $x_{i} = x_{i-1} + v_{i-1/2} dt; \quad v_{i+1/2} = v_{i-1/2} + a_i dt$. For available integrators refer to [Integrator](#Integrator-(integrator)).
+[^3]: On kernels: In the classical HMC approach, during the first step, new values for the momentum variables are randomly drawn from their Gaussian distribution, independently of the current values of the position variables. A Metropolis update is performed during the second step, using Hamiltonian dynamics to provide a new state. For available kernels refer to [Kernel](#Kernel-(kernel)).