### Installing DiffEqBayes.jl Package in Julia

Source: https://github.com/sciml/diffeqbayes.jl/blob/master/docs/src/index.md

This snippet demonstrates how to install the DiffEqBayes.jl package using Julia's built-in package manager, Pkg. It first loads the Pkg module and then calls the `add` function to download and install the specified package. This is a standard procedure for adding new dependencies to a Julia environment.

```Julia
using Pkg
Pkg.add("DiffEqBayes")
```

--------------------------------

### Installing DiffEqBayes.jl

Source: https://github.com/sciml/diffeqbayes.jl/blob/master/docs/src/methods.md

This snippet demonstrates how to add and use the DiffEqBayes.jl package in Julia. It first uses the Pkg manager to add the package, then brings its functionalities into scope with `using DiffEqBayes`.

```Julia
using Pkg
Pkg.add("DiffEqBayes")
using DiffEqBayes
```

--------------------------------

### Performing Bayesian Inference for ODEs with DiffEqBayes.jl in Julia

Source: https://github.com/sciml/diffeqbayes.jl/blob/master/README.md

This comprehensive example demonstrates setting up a Lotka-Volterra ODE problem, simulating data with added noise, and then performing Bayesian parameter inference using various backends provided by DiffEqBayes.jl. It showcases integration with CmdStan.jl, Turing.jl, DynamicHMC.jl, and ApproxBayes.jl, highlighting the flexibility in choosing inference methods. The `prob1` is the `ODEProblem`, `t` are observation times, `data` are the observed values, and `priors` define the prior distributions for parameters.

```Julia
using ParameterizedFunctions, OrdinaryDiffEq, RecursiveArrayTools, Distributions
f1 = @ode_def LotkaVolterra begin
    dx = a * x - x * y
    dy = -3 * y + x * y
end a

p = [1.5]
u0 = [1.0, 1.0]
tspan = (0.0, 10.0)
prob1 = ODEProblem(f1, u0, tspan, p)

σ = 0.01                         # noise, fixed for now
t = collect(1.0:10.0)   # observation times
sol = solve(prob1, Tsit5())
priors = [Normal(1.5, 1)]
randomized = VectorOfArray([(sol(t[i]) + σ * randn(2)) for i in 1:length(t)])
data = convert(Array, randomized)

using CmdStan #required for using the Stan backend
bayesian_result_stan = stan_inference(prob1, t, data, priors)

bayesian_result_turing = turing_inference(prob1, Tsit5(), t, data, priors)

using DynamicHMC #required for DynamicHMC backend
bayesian_result_hmc = dynamichmc_inference(prob1, Tsit5(), t, data, priors)

bayesian_result_abc = abc_inference(prob1, Tsit5(), t, data, priors)
```

--------------------------------

### Checking All Package Dependencies (Manifest) in Julia

Source: https://github.com/sciml/diffeqbayes.jl/blob/master/docs/src/index.md

This Julia code uses the Pkg module to display the status of all package dependencies, including transitive ones, as recorded in the project's manifest file. By setting `mode = PKGMODE_MANIFEST`, it provides a complete overview of the exact versions of all packages used, ensuring full reproducibility of the environment. The `# hide` comments are for documentation generation.

```Julia
using Pkg # hide
Pkg.status(; mode = PKGMODE_MANIFEST) # hide
```

--------------------------------

### Displaying Julia Version and System Information

Source: https://github.com/sciml/diffeqbayes.jl/blob/master/docs/src/index.md

This snippet utilizes the `InteractiveUtils` module to print detailed information about the Julia version and the system environment. The `versioninfo()` function provides crucial context for reproducibility, including the Julia version, operating system, and CPU architecture. The `# hide` comments suggest this is part of a documentation generation process.

```Julia
using InteractiveUtils # hide
versioninfo() # hide
```

--------------------------------

### Checking Direct Package Dependencies in Julia

Source: https://github.com/sciml/diffeqbayes.jl/blob/master/docs/src/index.md

This Julia snippet uses the Pkg module to display the status of direct package dependencies. The `Pkg.status()` function lists all currently loaded packages and their versions, providing insight into the immediate dependencies used for building the documentation. The `# hide` comments indicate these lines are for internal documentation generation and might not be explicitly shown in the final output.

```Julia
using Pkg # hide
Pkg.status() # hide
```

--------------------------------

### Generating GitHub Project File Link in Julia

Source: https://github.com/sciml/diffeqbayes.jl/blob/master/docs/src/index.md

This Julia snippet dynamically constructs a URL for the project's `Project.toml` file hosted on GitHub. Similar to the manifest link generation, it reads the `Project.toml` file to extract the package `version` and `name` using the `TOML` module, then concatenates these values into a specific GitHub URL path. This provides a direct link to the project's primary configuration file for reproducibility.

```Julia
using TOML
version = TOML.parse(read("../../Project.toml", String))["version"]
name = TOML.parse(read("../../Project.toml", String))["name"]
link = "https://github.com/SciML/" * name * ".jl/tree/gh-pages/v" * version * "/assets/Project.toml"
```

--------------------------------

### Setting up Lotka-Volterra ODE and Data in Julia

Source: https://github.com/sciml/diffeqbayes.jl/blob/master/docs/src/examples.md

This snippet initializes the Lotka-Volterra ODE model, defines initial conditions, time span, and parameters. It then solves the ODE to generate synthetic data with added noise, which will be used for Bayesian inference. Dependencies include DiffEqBayes, ParameterizedFunctions, OrdinaryDiffEq, RecursiveArrayTools, and Distributions.

```Julia
using DiffEqBayes, ParameterizedFunctions, OrdinaryDiffEq, RecursiveArrayTools,
      Distributions
f1 = @ode_def LotkaVolterra begin
    dx = a * x - x * y
    dy = -3 * y + x * y
end a

p = [1.5]
u0 = [1.0, 1.0]
tspan = (0.0, 10.0)
prob1 = ODEProblem(f1, u0, tspan, p)

σ = 0.01                         # noise, fixed for now
t = collect(1.0:10.0)   # observation times
sol = solve(prob1, Tsit5())
priors = [Normal(1.5, 1)]
randomized = VectorOfArray([(sol(t[i]) + σ * randn(2)) for i in 1:length(t)])
data = convert(Array, randomized)
```

--------------------------------

### Generating GitHub Manifest File Link in Julia

Source: https://github.com/sciml/diffeqbayes.jl/blob/master/docs/src/index.md

This Julia snippet dynamically constructs a URL for the project's `Manifest.toml` file hosted on GitHub. It reads the `Project.toml` file to extract the package `version` and `name` using the `TOML` module, then concatenates these values into a specific GitHub URL path. This is used to provide a direct link to the exact dependency manifest for reproducibility.

```Julia
using TOML
version = TOML.parse(read("../../Project.toml", String))["version"]
name = TOML.parse(read("../../Project.toml", String))["name"]
link = "https://github.com/SciML/" * name * ".jl/tree/gh-pages/v" * version * "/assets/Manifest.toml"
```

--------------------------------

### Adding and Using DiffEqBayes.jl in Julia

Source: https://github.com/sciml/diffeqbayes.jl/blob/master/README.md

This snippet demonstrates how to add the DiffEqBayes.jl package to your Julia environment using `Pkg.add()` and then make its functionalities available in your current session using `using DiffEqBayes`. This is the first step required to utilize the package's features for Bayesian estimation of differential equations.

```Julia
Pkg.add("DiffEqBayes")
using DiffEqBayes
```

--------------------------------

### Importing Libraries for Pendulum Inference (Julia)

Source: https://github.com/sciml/diffeqbayes.jl/blob/master/docs/src/examples/pendulum.md

This snippet imports necessary Julia packages for defining and solving differential equations (OrdinaryDiffEq), performing Bayesian inference (DiffEqBayes, CmdStan, DynamicHMC, TransformVariables), handling array tools (RecursiveArrayTools), working with distributions (Distributions), plotting (Plots, StatsPlots), and benchmarking code execution (BenchmarkTools). These libraries collectively provide the tools required for the entire Bayesian inference workflow.

```Julia
using DiffEqBayes, OrdinaryDiffEq, RecursiveArrayTools, Distributions, Plots, StatsPlots,
      BenchmarkTools, TransformVariables, CmdStan, DynamicHMC
```

--------------------------------

### Performing Bayesian Inference with Stan.jl in Julia

Source: https://github.com/sciml/diffeqbayes.jl/blob/master/docs/src/examples.md

This snippet demonstrates how to use the Stan.jl backend for Bayesian inference. It requires the `CmdStan` package and utilizes the `stan_inference` function with the ODE problem, solver, observation times, data, and priors as inputs. The result is stored in `bayesian_result_stan`.

```Julia
using CmdStan #required for using the Stan backend
bayesian_result_stan = stan_inference(prob1, :rk45, t, data, priors)
```

--------------------------------

### Visualizing Generated Data (Julia)

Source: https://github.com/sciml/diffeqbayes.jl/blob/master/docs/src/examples/pendulum.md

This snippet overlays the newly generated noisy data onto the existing plot of the true solution. The `scatter!(data')` command adds scatter points representing the observed data, allowing for a visual comparison between the simulated true behavior and the noisy measurements that will be used for parameter estimation.

```Julia
scatter!(data')
```

--------------------------------

### Generating Dummy Data for Estimation (Julia)

Source: https://github.com/sciml/diffeqbayes.jl/blob/master/docs/src/examples/pendulum.md

This code generates synthetic observational data by sampling the solved pendulum ODE at specific time points and adding random noise. It creates a vector of time points `t` from 1 to 10. For each time point, it evaluates the true solution `sol(t[i])` and adds Gaussian noise (`0.01randn(2)`) to simulate measurement errors, storing the results in a `VectorOfArray` and then converting it to a standard `Array`.

```Julia
t = collect(range(1, stop = 10, length = 10))
randomized = VectorOfArray([(sol(t[i]) + 0.01randn(2)) for i in 1:length(t)])
data = convert(Array, randomized)
```

--------------------------------

### Benchmarking DynamicHMC.jl Inference (Julia)

Source: https://github.com/sciml/diffeqbayes.jl/blob/master/docs/src/examples/pendulum.md

This code benchmarks the Bayesian inference process using the DynamicHMC.jl backend via the `dynamichmc_inference` function. It measures the execution time for 10,000 samples, providing a performance comparison against the Turing.jl and Stan.jl backends. This allows users to evaluate the efficiency of different MCMC algorithms and implementations for their specific problem.

```Julia
@btime bayesian_result = dynamichmc_inference(prob1, Tsit5(), t, data, priors;
    num_samples = 10_000)
```

--------------------------------

### Benchmarking Stan.jl Inference (Julia)

Source: https://github.com/sciml/diffeqbayes.jl/blob/master/docs/src/examples/pendulum.md

This snippet benchmarks the Bayesian inference process using the Stan.jl backend via the `stan_inference` function. Similar to the Turing.jl benchmark, it measures the execution time for 10,000 samples, but uses Stan's `rk45` solver and passes sampling arguments through a dictionary. The `print_summary = false` argument suppresses Stan's default summary output, focusing solely on the timing.

```Julia
@btime bayesian_result = stan_inference(prob1, :rk45, t, data, priors;
sample_kwargs = Dict(:num_samples => 10_000), print_summary = false)
```

--------------------------------

### Solving and Plotting the Pendulum ODE (Julia)

Source: https://github.com/sciml/diffeqbayes.jl/blob/master/docs/src/examples/pendulum.md

This snippet solves the previously defined `ODEProblem` (`prob1`) using the `Tsit5` algorithm, which is a 5th-order Tsitouras Runge-Kutta method. The resulting solution object `sol` contains the time series of the pendulum's state. The `plot(sol)` command then visualizes this solution, allowing for an understanding of the pendulum's behavior with the given parameters.

```Julia
sol = solve(prob1, Tsit5())
plot(sol)
```

--------------------------------

### Performing Bayesian Inference with DynamicHMC.jl in Julia

Source: https://github.com/sciml/diffeqbayes.jl/blob/master/docs/src/examples.md

This snippet illustrates the use of DynamicHMC.jl as the backend for sampling in Bayesian inference. It invokes the `dynamichmc_inference` function, providing the ODE problem, a solver (Tsit5()), observation times, data, and priors. The output of the inference is stored in `bayesian_result_hmc`.

```Julia
bayesian_result_hmc = dynamichmc_inference(prob1, Tsit5(), t, data, priors)
```

--------------------------------

### Default Sampler Arguments for Turing.jl Inference in Julia

Source: https://github.com/sciml/diffeqbayes.jl/blob/master/docs/src/methods.md

This snippet shows the default values for the sampler arguments used in `turing_inference`. By default, it uses the NUTS sampler with a target acceptance rate of 0.65, performs sampling serially, collects 1000 samples, and runs a single MCMC chain.

```Julia
sampler = Turing.NUTS(0.65)
parallel_type = MCMCSerial()
num_samples = 1000
n_chains = 1
```

--------------------------------

### Defining the Simple Pendulum ODE (Julia)

Source: https://github.com/sciml/diffeqbayes.jl/blob/master/docs/src/examples/pendulum.md

This code defines the ordinary differential equation (ODE) for a simple pendulum with a drag term (ω) and length (L). The `pendulum` function takes the state derivatives `du`, current state `u` (position and velocity), parameters `p`, and time `t` as input. It then initializes the initial conditions `u0`, the time span `tspan`, and creates an `ODEProblem` instance with the defined function, initial conditions, time span, and initial parameter values.

```Julia
function pendulum(du, u, p, t)
    ω, L = p
    x, y = u
    du[1] = y
    du[2] = -ω * y - (9.8 / L) * sin(x)
end

u0 = [1.0, 0.1]
tspan = (0.0, 10.0)
prob1 = ODEProblem(pendulum, u0, tspan, [1.0, 2.5])
```

--------------------------------

### Benchmarking Turing.jl Inference (Julia)

Source: https://github.com/sciml/diffeqbayes.jl/blob/master/docs/src/examples/pendulum.md

This code snippet uses the `@btime` macro from `BenchmarkTools.jl` to measure the execution time of the Bayesian inference process when using the Turing.jl backend. It performs the same `turing_inference` call as before, with the specified ODE problem, solver, data, priors, and sampling arguments, providing a precise timing measurement for performance comparison.

```Julia
@btime bayesian_result = turing_inference(prob1, Tsit5(), t, data, priors;
    syms = [:omega, :L], sample_args = (num_samples = 10_000,))
```

--------------------------------

### Performing Bayesian Inference with Turing.jl (Julia)

Source: https://github.com/sciml/diffeqbayes.jl/blob/master/docs/src/examples/pendulum.md

This snippet executes the Bayesian parameter inference using the `turing_inference` function from DiffEqBayes.jl, leveraging the Turing.jl backend. It takes the `ODEProblem` (`prob1`), the solver (`Tsit5`), time points `t`, observed `data`, and defined `priors` as inputs. Additionally, it assigns symbolic names to the parameters (`:omega`, `:L`) and specifies the number of samples for the MCMC chain (`10_000`), returning the `bayesian_result` object containing the posterior samples.

```Julia
bayesian_result = turing_inference(prob1, Tsit5(), t, data, priors;
    syms = [:omega, :L], sample_args = (num_samples = 10_000,))
```

--------------------------------

### Performing Bayesian Inference with Stan.jl in Julia

Source: https://github.com/sciml/diffeqbayes.jl/blob/master/docs/src/methods.md

`stan_inference` facilitates Bayesian inference for differential equations using Stan.jl. It requires a `DEProblem`, a Stan-compatible algorithm (`:rk45` or `:bdf`), time points, observed data, and optional prior distributions. It supports custom likelihoods, hyperparameter priors, and extensive keyword arguments for solver and sampler configuration.

```Julia
stan_inference(prob::DiffEqBase.DEProblem, alg, t, data, priors = nothing;
    stanmodel = nothing, likelihood = Normal, vars = (StanODEData(), InverseGamma(3, 3)), sample_u0 = false, solve_kwargs = Dict(), diffeq_string = nothing, sample_kwargs = Dict(), output_format = :mcmcchains, print_summary = true, tmpdir = mktempdir())
```

--------------------------------

### Performing Bayesian Inference with Turing.jl in Julia

Source: https://github.com/sciml/diffeqbayes.jl/blob/master/docs/src/examples.md

This snippet shows how to perform Bayesian inference using the Turing.jl backend. It calls the `turing_inference` function, passing the ODE problem, a solver (Tsit5()), observation times, data, and priors. The inference result is assigned to `bayesian_result_turing`.

```Julia
bayesian_result_turing = turing_inference(prob1, Tsit5(), t, data, priors)
```

--------------------------------

### Performing Bayesian Inference with DynamicHMC.jl in Julia

Source: https://github.com/sciml/diffeqbayes.jl/blob/master/docs/src/methods.md

`dynamichmc_inference` uses DynamicHMC.jl for Bayesian parameter estimation of differential equations. It requires a `DEProblem`, an algorithm, time points, observed data, prior distributions, and transformations for parameter domains. Optional arguments include initial step size (`ϵ`) for the NUTS algorithm and initial parameter values.

```Julia
dynamichmc_inference(prob::DEProblem, alg, t, data, priors, transformations;
    σ = 0.01, ϵ = 0.001, initial = Float64[])
```

--------------------------------

### Performing Bayesian Inference with Turing.jl in Julia

Source: https://github.com/sciml/diffeqbayes.jl/blob/master/docs/src/methods.md

`turing_inference` performs parameter inference for differential equations using Turing.jl. It takes a `DEProblem`, an algorithm, time points, observed data, and prior distributions. It allows for detailed configuration of the differential equation solver via `solve_kwargs` and the MCMC sampler via `sample_args` and `sample_kwargs`.

```Julia
turing_inference(prob::DiffEqBase.DEProblem, alg, t, data, priors;
    likelihood_dist_priors, likelihood, syms, sample_u0 = false, progress = false, solve_kwargs = Dict(), sample_args = NamedTuple(), sample_kwargs= Dict())
```

--------------------------------

### Plotting MCMC Chains for Diagnostics (Julia)

Source: https://github.com/sciml/diffeqbayes.jl/blob/master/docs/src/examples/pendulum.md

This snippet generates diagnostic plots of the MCMC (Markov Chain Monte Carlo) chains for the inferred parameters. The `plot(bayesian_result, colordim = :parameter)` command creates trace plots where each parameter's sampled values over iterations are shown, colored by parameter. These plots are crucial for assessing chain convergence, mixing, and identifying potential issues like non-stationarity or poor exploration of the parameter space.

```Julia
plot(bayesian_result, colordim = :parameter)
```

--------------------------------

### Plotting Posterior Distributions (Julia)

Source: https://github.com/sciml/diffeqbayes.jl/blob/master/docs/src/examples/pendulum.md

This code visualizes the posterior distributions of the inferred parameters obtained from the Bayesian inference. By simply calling `plot(bayesian_result)`, the function generates plots that typically show histograms or kernel density estimates of the marginal posterior distributions for each parameter, allowing for an assessment of the learned parameter values and their uncertainties.

```Julia
plot(bayesian_result)
```

--------------------------------

### Performing Bayesian Inference on Specific Observables with DiffEqBayes.jl in Julia

Source: https://github.com/sciml/diffeqbayes.jl/blob/master/README.md

This snippet illustrates how to perform Bayesian inference when data is only available for a subset of the model's variables. By utilizing the `save_idxs` keyword argument in both the `solve` function and the `*_inference` calls, users can specify which variables are observed. This is crucial for models with latent variables, allowing the inference to focus only on the measured outputs.

```Julia
sol = solve(prob1, Tsit5(), save_idxs = [1])
randomized = VectorOfArray([(sol(t[i]) + σ * randn(1)) for i in 1:length(t)])
data = convert(Array, randomized)

using CmdStan #required for using the Stan backend
bayesian_result_stan = stan_inference(prob1, t, data, priors, save_idxs = [1])

bayesian_result_turing = turing_inference(prob1, Tsit5(), t, data, priors, save_idxs = [1])

using DynamicHMC #required for DynamicHMC backend
bayesian_result_hmc = dynamichmc_inference(prob1, Tsit5(), t, data, priors, save_idxs = [1])

bayesian_result_abc = abc_inference(prob1, Tsit5(), t, data, priors, save_idxs = [1])
```

--------------------------------

### Performing ABC Inference for Differential Equations in Julia

Source: https://github.com/sciml/diffeqbayes.jl/blob/master/docs/src/methods.md

The `abc_inference` function performs Approximate Bayesian Computation (ABC) for parameter inference on `DEProblem`s. It leverages `ApproxBayes.jl` for the ABC algorithms, `Distributions.jl` for defining prior distributions, and `Distances.jl` for specifying distance metrics. Key parameters include the differential equation problem (`prob`), algorithm (`alg`), time points (`t`), observed data (`data`), prior distributions (`priors`), target distance (`ϵ`), number of samples (`num_samples`), and maximum iterations (`maxiterations`). The function supports `ABCSMC` and `ABCRejection` algorithms and passes extra keyword arguments to the internal differential equation solver.

```Julia
abc_inference(prob::DEProblem, alg, t, data, priors; ϵ = 0.001,
    distancefunction = euclidean, ABCalgorithm = ABCSMC, progress = false,
    num_samples = 500, maxiterations = 10^5, kwargs...)
```

--------------------------------

### Defining Priors for Bayesian Inference (Julia)

Source: https://github.com/sciml/diffeqbayes.jl/blob/master/docs/src/examples/pendulum.md

This code defines the prior probability distributions for the pendulum's parameters, `ω` (drag) and `L` (length). It uses `truncated(Normal(mean, std), lower=min_val)` to specify normal distributions that are truncated at a lower bound of 0.0, reflecting the physical constraint that these parameters must be non-negative. The priors represent initial beliefs about the parameter values before observing any data.

```Julia
priors = [
    truncated(Normal(0.1, 1.0), lower = 0.0),
    truncated(Normal(3.0, 1.0), lower = 0.0)
]
```

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