# Probabilistic Mixture Model-Based Spectral Unmixing
## 5 Summary and Future Directions  
In summary, we applied a probabilistic mixture model for the extraction of endmembers from spectral mixtures. Our approach formulates the retrieval of endmember spectra and per-observation endmember weights as a Bayesian inference problem using relatively uninformative priors for the phase-mixing and observation processes. Using simulated datasets, we sampled the model posterior with HMC and found that the resulting recovered endmember parameters are more robust to observation noise and absence, in the data, of pure endmember spectra compared to the popular N-FINDR algorithm. This Bayesian approach provides a principled framework for incorporating relevant prior information on the posterior.  
Figure 4: Scatter plots of reconstruction error (horizontal axis) and ground truth (vertical axis), concatenated over the individual scalar components of the endmember coordinates under HMC sampling (**right**), mean field variational inference (**center**), and N-FINDR (**left**). The blue curve in each panel is a smoothed density estimate along the horizonal axis—it represents the empirical probability distribution, over all datasets, for the scalar difference between the endmember coordinate estimates and ground truth.  
Figure 3: L2 reconstruction error for endmember coordinates via HMC posterior samples (**left**), mean field variational inference (**center**), compared to N-FINDR (**right**), with varying values of the hyperparameters \(\alpha_{1}\) and \(\sigma\). The reconstruction error is a sum in quadrature of the L2 distance between the point prediction and the ground truth. The log likelihood of the predictions is also reported in the figure.  
mation without introducing undue assumptions and opens the door to other intrinsically probabilistic analyses, such as uncertainty quantification.  
Both the baseline model (N-FINDR) and our Bayesian approach assume that the number of endmembers is known. Unlike deterministic approaches such as N-FINDR, the probabilistic formulation would allow one to relax this potentially onerous assumption by replacing the fixed number of mixture components and corresponding endmember weights by a set of samples from a stick-breaking distribution [24] as in a Dirichlet process mixture model. Assumptions about \(K\) would thus be condensed into a single hyperparameter, the concentration parameter of the stick-breaking process. Additionally, our approach can be generalized to incorporate more flexible, non-linear mappings using Bayesian Neural Networks (BNNs) to infer embeddings.  
Conceptualization, O.H., A.A.M. and A.M.; methodology, O.H. and A.A.M.; software, O.H. and A.A.M.; validation, O.H., A.A.M. and A.M.; formal analysis, O.H., A.A.M. and A.M.; investigation, O.H., A.A.M. and A.M.; resources, O.H., A.A.M. and A.M.; data curation, O.H.; writing--original draft preparation, O.H., A.A.M. and A.M.; writing--review and editing, O.H., A.A.M. and A.M.; visualization, O.H.; supervision, A.M.; project administration, A.M.; funding acquisition, A.M. All authors have read and agreed to the published version of the manuscript.  
This work was performed and partially supported by the US Department of Energy (DOE), Office of Science, Office of Basic Energy Sciences Data, Artificial Intelligence and Machine Learning at the DOE Scientific User Facilities program under the MLExchange Project (award No. 107514). [PERSON] was partially supported by the SLAC ML Initiative.  
The raw data supporting the conclusions of this article will be made available by the authors on request.  
The authors declare no conflicts of interest.