# Probabilistic Mixture Model-Based Spectral Unmixing
## 4 Results  
After generating 100 simulated datasets of \(N=500\) observations each in the aforementioned manner, we infer endmember coordinates in two ways: first, by using the model distribution function to sample from the posterior over model parameters with the no-U turn (NUTS) HMC sampler; second, with mean-field stochastic variational inference (SVI) using the ADAM optimizer and a learning rate of 0.005. To mitigate failures to converge (a common issue with mixture models), we run multiple rounds of inference with different random initializations. We select between 3 starts per dataset for HMC and 10 for VI; in each case, we retain only the inference round that produces the highest log likelihood for the dataset under the posterior samples (HMC) or approximation (VI). Per dataset, the sampling and inference are run for a total of 10 min on 12 CPU cores. Figure 2 juxtaposes the endmember coordinates determined through N-FINDR and the hierarchical Bayesian model using HMC with the NUTS sampler. Specifically, it shows representative outcomes of these two methods under the four possible extreme conditions of hyperparameters \(\alpha_{1}\) and \(\sigma\): low \(\alpha_{1}\) with low \(\sigma\), low \(\alpha_{1}\) with high \(\sigma\), high \(\alpha_{1}\) with low \(\sigma\), and high \(\alpha_{1}\) with high \(\sigma\). This comparison provides a qualitative contrast of the data distribution and the respective behaviors of the two endmember extraction approaches in different corners of the space of parameters \(\alpha_{1}\) and \(\sigma\), which, we reiterate, capture the degree of mixing and severity of noise in the formulation. In this comparison, N-FINDR serves not as an example of a state-of-the-art model but rather as a representative instance of deterministic endmember extraction. Further, Figure 3 compares the reconstruction accuracy for 100 instances of this simulation-inference procedure, with 100 distinct combinations of the hyperparameters \(\alpha_{1}\) and \(\sigma\). For VI and HMC, we use Monte Carlo samples from the posterior to estimate the central tendency (i.e., estimated mean from MC samples) for the prediction. This is used as the approximation for the point prediction for the Bayesian approaches. With N finder, we use Monte Carlo samples over different seeds, and average these samples for the mean N-FINDR prediction. In each of the cases, due diligence is carried out to ensure that the number of MC samples is sufficient for a robust final prediction. Subsequently, we use the Frobenius norm of the difference between the point prediction and ground truth as our figure of merit for the reconstruction accuracy. Figure 3 compares the reconstruction accuracy of HMC-NUTS, mean field VI, and the N-FINDR baseline under varying conditions of mixture diversity (\(\alpha_{1}\)) and noise (\(\sigma\)). High \(\alpha_{1}\) and \(\sigma\) values pose the most difficulty for all approaches, while low values yield better performance. However, when either \(\alpha_{1}\) or \(\sigma\) is large, HMC-NUTS significantly outperforms N-FINDR. The superior performance of HMC-NUTS in these scenarios aligns with the expected benefits of the Bayesian mixture model approach. N-FINDR assumes the presence of pure endmembers and relies on finding the maximum-volume simplex enclosing the data, struggling when pure endmembers are absent (high \(\alpha_{1}\)). In contrast, the Bayesian approach explicitly models mixing and uncertainty, handling low mixture diversity more effectively. Similarly, the deterministic nature of N-FINDR makes it sensitive to high noise levels (large \(\sigma\)), while the Bayesian model incorporates noise through the likelihood function, providing robustness. The results in Figure 3 confirm these expectations, highlighting the advantages of the probabilistic mixture model approach in challenging conditions where deterministic methods like N-FINDR may falter. Finally, we investigate posterior approximation with VI as a computationally cheaper alternative to HMC.