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

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: "Turing's Core Functionality"
+title: "Core Functionality"
 engine: julia
 ---