Reorganise introductory docs (#520)

* Remove quick start -- it's repeated content

* Restructure documentation

* Streamline introductory tutorial

The excised material is more appropriate for the very first page that
people click on.

* Streamline 'Getting Started' page

1. Remove the section on posterior checks; this is the landing page and
   it's not necessary for people reading about the library for the first
   time to go through that.
2. Signpost the way to the rest of the documentation at the bottom.
3. Minor wording changes

* Update _quarto.yml

Co-authored-by: Hong Ge <3279477+yebai@users.noreply.github.com>

---------

Co-authored-by: Hong Ge <3279477+yebai@users.noreply.github.com>

Author: penelopeysm

_quarto.yml

@@ -50,16 +50,15 @@ website:
     - text: documentation
       collapse-level: 1
       contents: 
-        - section: "Documentation"
+        - section: "Users"
           # href: tutorials/index.qmd, This page will be added later so keep this line commented
           contents:
-          - section: "Using Turing - Modelling Syntax and Interface"
+          - tutorials/docs-00-getting-started/index.qmd
+          - tutorials/docs-12-using-turing-guide/index.qmd
+
+          - section: "Usage Tips"
             collapse-level: 1
             contents: 
-              - tutorials/docs-00-getting-started/index.qmd
-              - text: "Quick Start"
-                href: tutorials/docs-14-using-turing-quick-start/index.qmd
-              - tutorials/docs-12-using-turing-guide/index.qmd
               - text: "Mode Estimation"
                 href: tutorials/docs-17-mode-estimation/index.qmd
               - tutorials/docs-09-using-turing-advanced/index.qmd
@@ -70,7 +69,7 @@ website:
               - text: "External Samplers"
                 href: tutorials/docs-16-using-turing-external-samplers/index.qmd
 
-          - section: "Using Turing - Tutorials"
+          - section: "Tutorials"
             contents: 
               - tutorials/00-introduction/index.qmd
               - text: Gaussian Mixture Models
@@ -97,25 +96,27 @@ website:
               - text: "Gaussian Process Latent Variable Models"
                 href: tutorials/12-gplvm/index.qmd
 
-          - section: "Developers: Contributing"
+        - section: "Developers"
+          contents:
+          - section: "Contributing"
             collapse-level: 1
             contents:
               - text: "How to Contribute"
                 href: tutorials/docs-01-contributing-guide/index.qmd
 
-          - section: "Developers: PPL"
+          - section: "DynamicPPL in Depth"
             collapse-level: 1
             contents:
               - tutorials/docs-05-for-developers-compiler/index.qmd
               - text: "A Mini Turing Implementation I: Compiler"
                 href: tutorials/14-minituring/index.qmd
               - text: "A Mini Turing Implementation II: Contexts"
                 href: tutorials/16-contexts/index.qmd
-              - tutorials/docs-06-for-developers-interface/index.qmd
 
-          - section: "Developers: Inference"
+          - section: "Inference (note: outdated)"
             collapse-level: 1
             contents:
+              - tutorials/docs-06-for-developers-interface/index.qmd
               - tutorials/docs-04-for-developers-abstractmcmc-turing/index.qmd
               - tutorials/docs-07-for-developers-variational-inference/index.qmd
               - text: "Implementing Samplers"

tutorials/00-introduction/index.qmd

@@ -1,5 +1,5 @@
 ---
-title: Introduction to Turing
+title: "Introduction: Coin Flipping"
 engine: julia
 aliases: 
   - ../
@@ -12,23 +12,12 @@ using Pkg;
 Pkg.instantiate();
 ```
 
-### Introduction
+This is the first of a series of guided tutorials on the Turing language.
+In this tutorial, we will use Bayesian inference to estimate the probability that a coin flip will result in heads, given a series of observations.
 
-This is the first of a series of tutorials on the universal probabilistic programming language **Turing**.
+### Setup
 
-Turing is a probabilistic programming system written entirely in Julia. It has an intuitive modelling syntax and supports a wide range of sampling-based inference algorithms.
-
-Familiarity with Julia is assumed throughout this tutorial. If you are new to Julia, [Learning Julia](https://julialang.org/learning/) is a good starting point.
-
-For users new to Bayesian machine learning, please consider more thorough introductions to the field such as [Pattern Recognition and Machine Learning](https://www.springer.com/us/book/9780387310732). This tutorial tries to provide an intuition for Bayesian inference and gives a simple example on how to use Turing. Note that this is not a comprehensive introduction to Bayesian machine learning.
-
-### Coin Flipping Without Turing
-
-The following example illustrates the effect of updating our beliefs with every piece of new evidence we observe.
-
-Assume that we are unsure about the probability of heads in a coin flip. To get an intuitive understanding of what "updating our beliefs" is, we will visualize the probability of heads in a coin flip after each observed evidence.
-
-First, let us load some packages that we need to simulate a coin flip
+First, let us load some packages that we need to simulate a coin flip:
 
 ```{julia}
 using Distributions
@@ -43,8 +32,7 @@ and to visualize our results.
 using StatsPlots
 ```
 
-Note that Turing is not loaded here — we do not use it in this example. If you are already familiar with posterior updates, you can proceed to the next step.
-
+Note that Turing is not loaded here — we do not use it in this example.
 Next, we configure the data generating model. Let us set the true probability that a coin flip turns up heads
 
 ```{julia}
@@ -63,13 +51,20 @@ We simulate `N` coin flips by drawing N random samples from the Bernoulli distri
 data = rand(Bernoulli(p_true), N);
 ```
 
-Here is what the first five coin flips look like:
+Here are the first five coin flips:
 
 ```{julia}
 data[1:5]
 ```
 
-Next, we specify a prior belief about the distribution of heads and tails in a coin toss. Here we choose a [Beta](https://en.wikipedia.org/wiki/Beta_distribution) distribution as prior distribution for the probability of heads. Before any coin flip is observed, we assume a uniform distribution $\operatorname{U}(0, 1) = \operatorname{Beta}(1, 1)$ of the probability of heads. I.e., every probability is equally likely initially.
+
+### Coin Flipping Without Turing
+
+The following example illustrates the effect of updating our beliefs with every piece of new evidence we observe.
+
+Assume that we are unsure about the probability of heads in a coin flip. To get an intuitive understanding of what "updating our beliefs" is, we will visualize the probability of heads in a coin flip after each observed evidence.
+
+We begin by specifying a prior belief about the distribution of heads and tails in a coin toss. Here we choose a [Beta](https://en.wikipedia.org/wiki/Beta_distribution) distribution as prior distribution for the probability of heads. Before any coin flip is observed, we assume a uniform distribution $\operatorname{U}(0, 1) = \operatorname{Beta}(1, 1)$ of the probability of heads. I.e., every probability is equally likely initially.
 
 ```{julia}
 prior_belief = Beta(1, 1);

tutorials/docs-00-getting-started/index.qmd

@@ -16,96 +16,71 @@ Pkg.instantiate();
 
 To use Turing, you need to install Julia first and then install Turing.
 
-### Install Julia
+You will need to install Julia 1.7 or greater, which you can get from [the official Julia website](http://julialang.org/downloads/).
 
-You will need to install Julia 1.3 or greater, which you can get from [the official Julia website](http://julialang.org/downloads/).
-
-### Install Turing.jl
-
-Turing is an officially registered Julia package, so you can install a stable version of Turing by running the following in the Julia REPL:
+Turing is officially registered in the [Julia General package registry](https://github.com/JuliaRegistries/General), which means that you can install a stable version of Turing by running the following in the Julia REPL:
 
 ```{julia}
+#| eval: false
 #| output: false
 using Pkg
 Pkg.add("Turing")
 ```
 
-You can check if all tests pass by running `Pkg.test("Turing")` (it might take a long time)
-
-### Example
-
-Here's a simple example showing Turing in action.
+### Example usage
 
-First, we can load the Turing and StatsPlots modules
+First, we load the Turing and StatsPlots modules.
+The latter is required for visualising the results.
 
 ```{julia}
 using Turing
 using StatsPlots
 ```
 
-Then, we define a simple Normal model with unknown mean and variance
+We then specify our model, which is a simple Gaussian model with unknown mean and variance.
+Models are defined as ordinary Julia functions, prefixed with the `@model` macro.
+Each statement inside closely resembles how the model would be defined with mathematical notation.
+Here, both `x` and `y` are observed values, and are therefore passed as function parameters.
+`m` and `s²` are the parameters to be inferred.
 
 ```{julia}
 @model function gdemo(x, y)
     s² ~ InverseGamma(2, 3)
     m ~ Normal(0, sqrt(s²))
     x ~ Normal(m, sqrt(s²))
-    return y ~ Normal(m, sqrt(s²))
+    y ~ Normal(m, sqrt(s²))
 end
 ```
 
-Then we can run a sampler to collect results. In this case, it is a Hamiltonian Monte Carlo sampler
-
-```{julia}
-chn = sample(gdemo(1.5, 2), NUTS(), 1000, progress=false)
-```
-
-We can plot the results
+Suppose we observe `x = 1.5` and `y = 2`, and want to infer the mean and variance.
+We can pass these data as arguments to the `gdemo` function, and run a sampler to collect the results.
+Here, we collect 1000 samples using the No U-Turn Sampler (NUTS) algorithm.
 
 ```{julia}
-plot(chn)
+chain = sample(gdemo(1.5, 2), NUTS(), 1000, progress=false)
 ```
 
-In this case, because we use the normal-inverse gamma distribution as a conjugate prior, we can compute its updated mean as follows:
+We can plot the results:
 
 ```{julia}
-s² = InverseGamma(2, 3)
-m = Normal(0, 1)
-data = [1.5, 2]
-x_bar = mean(data)
-N = length(data)
-
-mean_exp = (m.σ * m.μ + N * x_bar) / (m.σ + N)
+plot(chain)
 ```
 
-We can also compute the updated variance
+and obtain summary statistics by indexing the chain:
 
 ```{julia}
-updated_alpha = shape(s²) + (N / 2)
-updated_beta =
-    scale(s²) +
-    (1 / 2) * sum((data[n] - x_bar)^2 for n in 1:N) +
-    (N * m.σ) / (N + m.σ) * ((x_bar)^2) / 2
-variance_exp = updated_beta / (updated_alpha - 1)
+mean(chain[:m]), mean(chain[:s²])
 ```
 
-Finally, we can check if these expectations align with our HMC approximations from earlier. We can compute samples from a normal-inverse gamma following the equations given [here](https://en.wikipedia.org/wiki/Normal-inverse-gamma_distribution#Generating_normal-inverse-gamma_random_variates).
+### Where to go next
 
-```{julia}
-function sample_posterior(alpha, beta, mean, lambda, iterations)
-    samples = []
-    for i in 1:iterations
-        sample_variance = rand(InverseGamma(alpha, beta), 1)
-        sample_x = rand(Normal(mean, sqrt(sample_variance[1]) / lambda), 1)
-        samples = append!(samples, sample_x)
-    end
-    return samples
-end
+::: {.callout-note title="Note on prerequisites"}
+Familiarity with Julia is assumed throughout the Turing documentation.
+If you are new to Julia, [Learning Julia](https://julialang.org/learning/) is a good starting point.
 
-analytical_samples = sample_posterior(updated_alpha, updated_beta, mean_exp, 2, 1000);
-```
+The underlying theory of Bayesian machine learning is not explained in detail in this documentation.
+A thorough introduction to the field is [*Pattern Recognition and Machine Learning*](https://www.springer.com/us/book/9780387310732) (Bishop, 2006); an online version is available [here (PDF, 18.1 MB)](https://www.microsoft.com/en-us/research/uploads/prod/2006/01/Bishop-Pattern-Recognition-and-Machine-Learning-2006.pdf).
+:::
 
-```{julia}
-density(analytical_samples; label="Posterior (Analytical)")
-density!(chn[:m]; label="Posterior (HMC)")
-```
+The next page on [Turing's core functionality](../../tutorials/docs-12-using-turing-guide/) explains the basic features of the Turing language.
+From there, you can either look at [worked examples of how different models are implemented in Turing](../../tutorials/00-introduction/), or [specific tips and tricks that can help you get the most out of Turing](../../tutorials/docs-17-mode-estimation/).

tutorials/docs-12-using-turing-guide/index.qmd

@@ -1,5 +1,5 @@
 ---
-title: Guide
+title: "Core Functionality"
 engine: julia
 ---
 
@@ -10,6 +10,8 @@ using Pkg;
 Pkg.instantiate();
 ```
 
+This article provides an overview of the core functionality in Turing.jl, which are likely to be used across a wide range of models.
+
 ## Basics
 
 ### Introduction