# Probabilistic Mixture Model-Based Spectral Unmixing
## 3 Methods  
We begin with a physical mixing model, where each endmember \(k\in 1,\ldots,K\) is characterized by a latent absolute spectral signature \(\vec{c}_{k}\) and a global relative concentration \(w_{k}\), with \(\sum_{k}w_{k}=1\). We assume a linear mixing process, where the observed D-dimensional spectrum \(\vec{y}_{i}\) of mixture \(i\) is:  
\[\vec{y}_{i}=\sum_{k}w_{ik}\vec{c}_{k}+\vec{e}_{i}\]  
Here, \(w_{ik}\) is the relative concentration of endmember \(k\) in mixture \(i\), satisfying \(\sum_{k}w_{ik}=1\), and \(\vec{e}_{i}\sim\mathcal{N}(0,\Sigma)\) is Gaussian observation noise. The standard linear mixing model assumes fixed endmember signatures \(\vec{c}_{k}\), limiting its flexibility to capture spectral variability across mixtures.  
To address this limitation, we propose a hierarchical Bayesian model that treats the \(\vec{c}_{k}\) as random variables with prior distributions representing the variability of each endmember. The model priors are:  
\[\begin{split}\vec{c}_{k}&\sim\mathcal{N}(\vec{\mu} _{k},\Sigma_{k}),\quad\vec{\mu}_{k}=\vec{0},\quad\Sigma_{k}=I\\ \vec{w}_{i}&\sim\text{Dir}(a_{1}\vec{w}),\quad\vec{ w}\sim\text{Dir}(a_{0}\mathbf{1})\\ \vec{e}_{i}&\sim\mathcal{N}(0,\Sigma),\quad\Sigma =\text{diag}(\vec{s}),\quad s_{d}\sim\mathcal{U}(\sigma/2,3\sigma/2)\end{split} \tag{1}\]The key components of this model, as shown in the plate diagram of Figure 1, are:  
Endmember spectra \(\vec{c}_{k}\) are drawn from multivariate Gaussian priors with zero mean and identity covariance. This allows for variation in the spectral signatures across mixtures.  
Mixture-specific abundances \(\vec{w}_{i}\) are drawn from a Dirichlet distribution with parameters \(\alpha_{1}\vec{w}\), where \(\alpha_{1}\) controls the diversity of mixtures, and \(\vec{w}\) are global relative concentrations.  
Global concentrations \(\vec{w}\) are drawn from a symmetric Dirichlet prior with concentration parameter \(\alpha_{0}\). Observation noise \(\vec{e}_{i}\) is Gaussian with zero mean and diagonal covariance \(\Sigma\) parameterized by scale variables \(s_{d}\) with uniform priors.  
The use of two Dirichlet distributions in our hierarchical model reflects the non-negativity and sum-to-one constraints on the endmember global weights and mixture fractions. The first Dirichlet distribution serves as a prior for the global endmember frequencies, while the second governs the variability of mixing fractions within each observed mixture.  
In contrast to non-probabilistic methods--including the simplex-type approach N-FINDR, which relies on a strong assumption of noise-free data--this probabilistic framework explicitly includes observation noise, allows for incomplete sampling of the endmember simplex, and models the endmember parameters as distributions. By incorporating epistemic uncertainty and supporting the inference of full posterior distributions over endmember parameters, our approach offers a more flexible and principled avenue for unmixing tasks compared to traditional methods that rely on fixed endmember signatures.  
The proposed approach can be seen as a Bayesian generalization of simplex-based unmixing methods like N-FINDR, which normally assume the presence of at least one pure pixel per endmember. Our model relaxes this assumption and allows endmembers to vary according to the specified priors, while still enabling the posterior estimates of \(\vec{c}_{k}\) to adapt to the data at hand.