update placeholder text

Author: penelopeysm

index.qmd

@@ -33,23 +33,30 @@ Turing is written entirely in Julia, and is interoperable with its powerful ecos
 
 ::: {.example-text style="text-align:right;padding:0.5rem;"}
 
-<div class="fs-4 fw-bold pb-1">Hello, World in Turing</div>
+<div class="fs-4 fw-bold pb-1">Intuitive syntax</div>
 
-Some text about how easy it is to [get going](https://turinglang.org/docs/tutorials/00-introduction/).
+The [modelling syntax of Turing.jl](https://turinglang.org/docs/core-functionality) closely resembles the mathematical specification of a probabilistic model.
+For example, the following model describes a coin flip experiment with `N` flips, where `p` is the probability of heads.
+
+```math
+\begin{align*}
+p &\sim \text{Beta}(1, 1) \\
+y_i &\sim \text{Bernoulli}(p) \quad \text{for } i = 1, \ldots, N
+```
 
 :::
 
 ::: {.example-code style="overflow-x: scroll;"}
 ```{.julia .code-overflow-scroll}
+# Define the model
 @model function coinflip(; N::Int)
-    # Prior belief about the probability of heads
     p ~ Beta(1, 1)
-
-    # Heads or tails of a coin are drawn from `N`
-    # Bernoulli distributions with success rate `p`
     y ~ filldist(Bernoulli(p), N)
+end
 
-end;
+# Condition on data
+data = [0, 1, 1, 0, 1]
+model = coinflip(; N = length(data)) | (; y = data)
 ```
 :::
 
@@ -59,22 +66,90 @@ end;
 
 ::: {.example-text style="padding:0.5rem;"}
 
-<div class="fs-4 fw-bold pb-1">Goodbye, World in Turing</div>
+<div class="fs-4 fw-bold pb-1">Flexible parameter inference</div>
+
+Turing.jl provides full support for sampling one or more MCMC chains from the posterior distribution, including options for parallel sampling.
+
+Variational inference and point estimation methods are also available.
+
+:::
+
+::: {.example-code style="overflow-x: scroll;"}
+```{.julia .code-overflow-scroll}
+# Sample one chain
+chain = sample(model, NUTS(), 1000)
+
+# Sample four chains, one per thread
+# Note: to obtain speedups, Julia must be started
+# with multiple threads enabled, e.g. `julia -t 4`
+chains = sample(model, NUTS(), MCMCThreads(), 1000, 4)
+```
+:::
+
+:::::
+
+
+::::: {.d-flex .flex-row .flex-wrap .panel-wrapper .gap-3 .pb-2}
+
+::: {.example-text style="text-align:right;padding:0.5rem;"}
+
+<div class="fs-4 fw-bold pb-1">MCMC sampling algorithms</div>
+
+A number of MCMC sampling algorithms are available in Turing.jl, including (but not limited to) HMC, NUTS, Metropolis–Hastings, particle samplers, and Gibbs.
 
-Some text about how easy it is to interface with external packages like AbstractGPs. Learn more about modelling [Gaussian Processes](https://turinglang.org/docs/tutorials/15-gaussian-processes/) with Turing.jl.
+Turing.jl also supports ['external samplers'](https://turinglang.org/docs/usage/external-samplers/) which conform to the AbstractMCMC.jl interface, meaning that users can implement their own algorithms.
 
 :::
 
 ::: {.example-code style="overflow-x: scroll;"}
 ```{.julia .code-overflow-scroll}
-@model function putting_model(d, n; jitter=1e-4)
-    v ~ Gamma(2, 1)
-    l ~ Gamma(4, 1)
-    f = GP(v * with_lengthscale(SEKernel(), l))
-    f_latent ~ f(d, jitter)
-    binomials = Binomial.(n, logistic.(f_latent))
-    y ~ product_distribution(binomials)
-    return (fx=f(d, jitter), f_latent=f_latent, y=y)
+sample(model, NUTS(), 1000)
+
+sample(model, MH(), 1000)
+
+using SliceSampling
+sample(model, externalsampler(SliceSteppingOut(2.0)), 1000)
+```
+:::
+
+:::::
+
+::::: {.d-flex .flex-row-reverse .flex-wrap .panel-wrapper .gap-3 .pt-2 .section-end-space}
+
+::: {.example-text style="padding:0.5rem;"}
+
+<div class="fs-4 fw-bold pb-1">Composability with Julia</div>
+
+As Turing.jl models are simply Julia functions under the hood, they can contain arbitrary Julia code.
+
+For example, [differential equations](https://turinglang.org/docs/tutorials/bayesian-differential-equations/) can be added to a model using `DifferentialEquations.jl`, which is a completely independent package.
+
+:::
+
+::: {.example-code style="overflow-x: scroll;"}
+```{.julia .code-overflow-scroll}
+using DifferentialEquations
+
+# Define the system of equations
+function lotka_volterra(du, u, params, t)
+    α, β, δ, γ = params
+    x, y = u
+    du[1] = (α * x) - (β * x * y)
+    du[2] = (δ * x * y) - (γ * y)
+end
+prob = ODEProblem(lotka_volterra, ...)
+
+# Use it in a model
+@model function fit_lotka_volterra()
+    # Priors
+    α ~ Normal(0, 1)
+    # ...
+
+    # Solve the ODE
+    predictions = solve(prob, Tsit5(); p=p)
+
+    # Likelihood
+    data ~ Poisson.(predictions, ...)
 end
 ```
 :::
@@ -179,4 +254,4 @@ Placeholder text introducing the Bayesian Workflow diagram from the ACM special
 
 <div class="section-start-space">
 
-{{< include _includes//citation/cite.qmd >}}
+{{< include _includes//citation/cite.qmd >}}