# Locally Scaled and Stochastic Volatility Metropolis-Hastings Algorithms
## 4 Results and Discussion  
We implemented all the models and algorithms in PyTorch. All the experiments were run on a machine with a 64 bit CPU. The sampling performance of the algorithms for the simulation study and real world benchmark datasets is shown in Figure 1. The \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Dataset** & **Mean** & **Standard Deviation** & **Skew** & **Kurtosis** \\ \hline MTN & \(-0.00088\) & \(0.03013\) & \(-1.620\) & \(20.147\) \\ S\&P 500 Index & \(0.00043\) & \(0.01317\) & \(-1.159\) & \(21.839\) \\ Bitcoin & \(0.00187\) & \(0.04447\) & \(-0.925\) & \(13.586\) \\ USDZAR & \(0.00019\) & \(0.00854\) & \(0.117\) & \(1.673\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Descriptive statistics for the jump diffusion process datasets. detailed results for the financial market data and real world classification datasets across different metrics is shown in Tables 3 and 4, respectively. Note that the execution time \(t\) in Tables 3 and 4 is in seconds. Figure 2 shows the diagnostic negative log-likelihood trace plots for each sampler across various targets. Figure 3 shows the autocorrelations produced by each MCMC method for the first dimension across various targets. Figure 4 shows the predictive performance of each of the sampling methods on the real world classification datasets modeled using BLR. The first two rows in Figure 1 show the \(m\)ESS and \(m\)ESS normalised by execution time for the multivariate Gaussian distributions and Neal's funnel used in the simulation study  
Figure 1: Inference results across each dataset over thirty iterations of each algorithm. The first row is the effective sample size while the second row is the time-normalised effective sample size for each dataset. with increasing dimensionality \(D\). The third and fourth row in Figure 1 show the \(m\)ESS and \(m\)ESS normalised by execution time for the financial market datasets modeled using jump diffusion processes, while rows five and six show the results for real world classification datasets modeled using Bayesian logistic regression. The results in Tables 3 and 4 are the mean results over the thirty iterations for each algorithm. Figure 2: Negative log-likelihood diagnostic trace plots for various targets. These are traces plots from a single run of the MCMC chain. (**a**) Diagnostic negative log-likelihood trace plots for the Fraud dataset and (**b**) Diagnostic negative log-likelihood trace plots for the multivariate Gaussian with \(D=10\). The other targets produce similar convergence behavior. Figure 3: Autocorrelation plots for the first dimension across the various targets. These are autocorrelations from a single run of the MCMC chain. (**a**) Autocorrelation plot for the first dimension on the Neal funnel for \(D=20\) and (**b**) Autocorrelation plot for the first dimension on the multivariate Gaussian with \(D=20\). The first couple rows of Figure 1 show that in all the values of \(D\) on both targets, SVMH and LSMH outperform MH in terms of effective sample sizes with SVMH and LSMH having similar effective sample sizes. When execution time is taken into consideration, SVMH outperforms LSMH and MH. The results also show that the NUTS algorithm  
Figure 4: Predictive performance for all the sampling methods on the real world classification datasets. The results were averaged over thirty runs of each algorithm. (**a**) ROC curves for the Heart dataset and (**b**) ROC curves for Australian credit dataset. outperforms all the MCMC methods on an effective sample size basis, while the MALA outperforms all the other MCMC methods on a normalised effective sample size basis. Rows three to six in Figure 1 show that across all the real world datasets, SVMH and LSMH have similar effective sample sizes, with MH producing the smallest effective sample size. NUTS outperforms all the algorithms on an effective sample size basis. On a normalised effective sample size basis, SVMH outperforms all the algorithms on the jump diffusion datasets except for the USDZAR dataset.