### High-Level Example Usage

Source: https://github.com/romeov/probabilisticparameterestimators.jl/blob/master/README.md

Demonstrates how to use the ProbabilisticParameterEstimators.jl library for parameter estimation. It shows setting up an observation function, true parameters, prior beliefs, noise models, and then using different estimators (MCMCEstimator, LinearApproxEstimator, LSQEstimator) to predict parameter samples or distributions.

```julia
using Distributions
using LinearAlgebra
using ProbabilisticParameterEstimators

# observation function with multivariate observations
f(x, p) = [(x + 1)^2 - sum(p);
           (x + 1)^3 + diff(p)[1]]

# true parameter (to be estimated) and a prior belief
thtrue = [1.0, 2.0]
paramprior = MvNormal(zeros(2), 4.0 * I)

# observation noise
xs = rand(5)
obsnoises = [rand()/10 * I(2) * MvNormal(zeros(2), I) for _ in eachindex(xs)]
noisemodel = UncorrGaussianNoiseModel(obsnoises)

# noisy observations x and y
ysmeas = f.(xs, [θtrue]) .+ rand.(obsnoises)

# find a probabilistic description of θ either as samples or as a distribution
# currently we provide three methods
for est in [MCMCEstimator(),
            LinearApproxEstimator(),
            LSQEstimator()]
    # either
    samples =  predictsamples(est, f, xs, ysmeas, paramprior, noisemodel, 100)
    # or
    dist    =  predictdist(est, f, xs, ysmeas, paramprior, noisemodel; nsamples=100)
end
```

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

### Uncorrelated Product Noise Model Example

Source: https://github.com/romeov/probabilisticparameterestimators.jl/blob/master/docs/src/noisemodels.md

Illustrates the use of the UncorrProductNoiseModel for univariate noise with arbitrary distributions. The example sets up a model with truncated normal distributions and generates noisy measurements.

```julia
## UncorrProductNoiseModel
xs = eachcol(rand(2,30))
# one univariate distribution per observation
productnoise = [truncated(0.1*Normal(), 0, Inf) for _ in 1:length(xs)]
noisemodel = UncorrProductNoiseModel(productnoise)
θtrue = [5., 10]
ysmeas = f.(xs, [θtrue]) .+ rand.(productnoise)
```

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

### Uncorrelated Gaussian Noise Model Example

Source: https://github.com/romeov/probabilisticparameterestimators.jl/blob/master/docs/src/noisemodels.md

Demonstrates the usage of the UncorrGaussianNoiseModel with multivariate Gaussian noise. It shows how to create the model with a list of normal distributions and generate measurements with added noise.

```julia
## UncorrGaussianNoiseModel
xs = rand(5)
# one (uni- or multivariate) normal distribution per observation
noises = [MvNormal(zeros(2), I) for _ in eachindex(xs)]
noisemodel = UncorrGaussianNoiseModel(noises)
θtrue = [1.0, 2.0]
ysmeas = f.(xs, [θtrue]) .+ rand.(noises)
```

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

### CorrGaussianNoiseModel Example

Source: https://github.com/romeov/probabilisticparameterestimators.jl/blob/master/README.md

Shows an example of CorrGaussianNoiseModel, used when there is correlation between observations. It employs a single multivariate normal distribution for all observations, flattening multivariate observations to match the noise model.

```julia
using Distributions
using LinearAlgebra

# Assuming CorrGaussianNoiseModel is defined elsewhere

xs = 5 * eachcol(rand(2,30))
n = length(xs)
# one large Gaussian relating all observations
corrnoise = MvNormal(zeros(n), I(n) + 1/5*hermitianpart(rand(n, n)))
noisemodel = CorrGaussianNoiseModel(corrnoise)
θtrue = [5., 10]
ysmeas = f.(xs, [θtrue]) .+ rand(corrnoise)
```

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

### Correlated Gaussian Noise Model Example

Source: https://github.com/romeov/probabilisticparameterestimators.jl/blob/master/docs/src/noisemodels.md

Shows how to implement the CorrGaussianNoiseModel for correlated observations using a single multivariate normal distribution. The example constructs a correlated noise distribution and generates measurements.

```julia
## CorrGaussianNoiseModel
xs = 5 * eachcol(rand(2,30))
n = length(xs)
# one large Gaussian relating all observations
corrnoise = MvNormal(zeros(n), I(n) + 1/5*hermitianpart(rand(n, n)))
noisemodel = CorrGaussianNoiseModel(corrnoise)
θtrue = [5., 10]
ysmeas = f.(xs, [θtrue]) .+ rand(corrnoise)
```

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

### UncorrProductNoiseModel Example

Source: https://github.com/romeov/probabilisticparameterestimators.jl/blob/master/README.md

Demonstrates the UncorrProductNoiseModel, which handles uncorrelated univariate noise distributions for each observation. This model cannot model correlations within a single observation.

```julia
using Distributions

# Assuming UncorrProductNoiseModel is defined elsewhere

xs = eachcol(rand(2,30))
# one univariate distribution per observation
productnoise = [truncated(0.1*Normal(), 0, Inf) for _ in 1:length(xs)]
noisemodel = UncorrProductNoiseModel(productnoise)
θtrue = [5., 10]
# Assuming f is a defined function
ysmeas = f.(xs, [θtrue]) .+ rand.(productnoise)
```

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

### High-Level Example of Probabilistic Parameter Estimation in Julia

Source: https://github.com/romeov/probabilisticparameterestimators.jl/blob/master/docs/src/index.md

Demonstrates how to use ProbabilisticParameterEstimators.jl for parameter estimation. It defines an observation function, true parameters, prior beliefs, noise models, and then uses different estimators (MCMCEstimator, LinearApproxEstimator, LSQEstimator) to predict parameter samples or distributions from noisy observations.

```julia
using Distributions
using LinearAlgebra
using ProbabilisticParameterEstimators

# observation function with multivariate observations
f(x, p) = [(x + 1)^2 - sum(p);
           (x + 1)^3 + diff(p)[1]]

# true parameter (to be estimated) and a prior belief
θtrue = [1.0, 2.0]
paramprior = MvNormal(zeros(2), 4.0 * I)

# observation noise
xs = rand(5)
noises = [rand()/10 * I(2) for _ in eachindex(xs)]
noisemodel = UncorrGaussianNoiseModel(noises)

# noisy observations x and y
ysmeas = f.(xs, [θtrue]) .+ rand.(MvNormal.(zeros(2), noises))

# find a probabilistic description of θ either as samples or as a distribution
# currently we provide three methods
for est in [MCMCEstimator(),
            LinearApproxEstimator(),
            LSQEstimator()]
    # either
    samples =  predictsamples(est, f, xs, ysmeas, paramprior, noisemodel, 100)
    # or
    dist    =  predictdist(est, f, xs, ysmeas, paramprior, noisemodel; nsamples=100)
end
```

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

### UncorrGaussianNoiseModel Example

Source: https://github.com/romeov/probabilisticparameterestimators.jl/blob/master/README.md

Illustrates the creation and usage of UncorrGaussianNoiseModel. This model is suitable when there is no correlation between multiple observations, with each observation having its own (possibly multivariate) Gaussian noise distribution.

```julia
using Distributions
using LinearAlgebra

# Assuming MvNormal and UncorrGaussianNoiseModel are defined elsewhere

xs = rand(5)
# one (uni- or multivariate) normal distribution per observation
noises = [MvNormal(zeros(2), I) for _ in eachindex(xs)]
noisemodel = UncorrGaussianNoiseModel(noises)
θtrue = [1.0, 2.0]
# Assuming f is a defined function
ysmeas = f.(xs, [θtrue]) .+ rand.(noises)
```

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

### Utility Function Documentation

Source: https://github.com/romeov/probabilisticparameterestimators.jl/blob/master/docs/src/noisemodels.md

Provides documentation links for utility functions related to noise modeling.

```APIDOC
mvnoisedistribution
covmatrix
```

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

### Noise Model Documentation

Source: https://github.com/romeov/probabilisticparameterestimators.jl/blob/master/docs/src/noisemodels.md

Provides documentation links for the core noise model types in the ProbabilisticParameterEstimators package.

```APIDOC
ProbabilisticParameterEstimators.NoiseModel
UncorrGaussianNoiseModel
UncorrProductNoiseModel
CorrGaussianNoiseModel
```

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

### LinearApproxEstimator Usage

Source: https://github.com/romeov/probabilisticparameterestimators.jl/blob/master/README.md

Demonstrates the usage of LinearApproxEstimator for predicting distributions and samples. It highlights the efficiency of solving one optimization problem and computing one Jacobian for MvNormal creation.

```julia
predictdist(::LinearApproxEstimator, f, xs, ys, paramprior, noisemodel, nsamples)
predictsamples(::LinearApproxEstimator, f, xs, ys, paramprior, noisemodel, nsamples)
```

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

### MCMCEstimator Usage

Source: https://github.com/romeov/probabilisticparameterestimators.jl/blob/master/README.md

Details the usage of the MCMCEstimator, which frames the parameter estimation problem as a Markov-Chain Monte-Carlo inference problem solved using the NUTS algorithm from Turing.jl. It explains how `predictdist` generates samples and optionally fits a MvNormal distribution.

```julia
# predictdist(::MCMCEstimator, f, xs, ys, paramprior, noisemodel, nsamples)
# will create nsamples samples (after skipping warmup steps).
# predictdist(::MCMCEstimator, f, xs, ys, paramprior, noisemodel, nsamples)
# will do the same, and then fit a MvNormal distribution.
```

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

### Predicting Samples

Source: https://github.com/romeov/probabilisticparameterestimators.jl/blob/master/docs/src/estimators.md

Generates probabilistic predictions as a set of samples. This is useful for Monte Carlo methods or when a distributional representation is not required.

```APIDOC
ProbabilisticParameterEstimators.predictsamples
  predictsamples(method::EstimationMethod, data::AbstractArray{<:Real}, obs_noise::Real)
    Generates probabilistic predictions as samples.
    Parameters:
      method: The estimation method used (e.g., LSQEstimator, LinearApproxEstimator, MCMCEstimator).
      data: The observed data.
      obs_noise: The noise level of the observations.
```

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

### LSQEstimator Usage

Source: https://github.com/romeov/probabilisticparameterestimators.jl/blob/master/README.md

Explains the LSQEstimator, which estimates parameters by repeatedly solving a least-squares problem with modified observations. It details how noise is sampled from the noise model and how weights are determined for uncorrelated and correlated noise. It also clarifies the role of `paramprior` and the output of `predictsamples` and `predictdist`.

```julia
# The LSQEstimator works by sampling noise ε⁽ᵏ⁾ from the noise model and repeatedly solving a least-squares parameter estimation problem for modified observations y - ε⁽ᵏ⁾, i.e.
# θ = arg min_θ Σᵢ ((yᵢ - εᵢ⁽ᵏ⁾) - f(xᵢ, θ))² ⋅ wᵢ
# for uncorrelated noise, where the weights wᵢ are chosen as the inverse variance.
# For correlated noise, the weight results from the whole covariance matrix.
# The paramprior is used to sample initial guesses for θ.

# Therefore predictsamples(::LSQEstimator, f, xs, ys, paramprior, noisemodel, nsamples) will solve nsamples optimization problems and return a sample each.
# predictdist(::LSQEstimator, f, xs, ys, paramprior, noisemodel, nsamples) will do the same, and then fit a MvNormal distribution.
```

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

### Estimation Method Base Type

Source: https://github.com/romeov/probabilisticparameterestimators.jl/blob/master/docs/src/estimators.md

Defines the abstract base type for all estimation methods. This serves as a common interface for different estimation strategies.

```APIDOC
ProbabilisticParameterEstimators.EstimationMethod
  EstimationMethod
    Abstract base type for all estimation methods. Concrete estimators like LSQEstimator, LinearApproxEstimator, and MCMCEstimator must subtype this.
```

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

### MCMC Estimator

Source: https://github.com/romeov/probabilisticparameterestimators.jl/blob/master/docs/src/estimators.md

Implements estimation using Markov Chain Monte Carlo (MCMC) methods. MCMC is powerful for complex probability distributions and Bayesian inference.

```APIDOC
MCMCEstimator
  MCMCEstimator()
    MCMC Estimator.
    Initializes the MCMCEstimator.
```

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

### Linear Approximation Estimator

Source: https://github.com/romeov/probabilisticparameterestimators.jl/blob/master/docs/src/estimators.md

Implements estimation using linear approximation. This method is suitable when the relationship between parameters and observations can be approximated linearly.

```APIDOC
LinearApproxEstimator
  LinearApproxEstimator()
    Linear Approximation Estimator.
    Initializes the LinearApproxEstimator.
```

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

### Predicting Distributions

Source: https://github.com/romeov/probabilisticparameterestimators.jl/blob/master/docs/src/estimators.md

Generates probabilistic predictions as a distribution. This function is part of the core prediction capabilities of the parameter estimators.

```APIDOC
ProbabilisticParameterEstimators.predictdist
  predictdist(method::EstimationMethod, data::AbstractArray{<:Real}, obs_noise::Real)
    Generates probabilistic predictions as a distribution.
    Parameters:
      method: The estimation method used (e.g., LSQEstimator, LinearApproxEstimator, MCMCEstimator).
      data: The observed data.
      obs_noise: The noise level of the observations.
```

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

### Least Squares Estimator

Source: https://github.com/romeov/probabilisticparameterestimators.jl/blob/master/docs/src/estimators.md

Implements the Least Squares estimation method. This is a common technique for finding the best fit parameters by minimizing the sum of the squares of the residuals.

```APIDOC
LSQEstimator
  LSQEstimator()
    Least Squares Estimator.
    Initializes the LSQEstimator.
```

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