## 3 Proposed Methodology
 \tag{11}\]\[\text{EI}(\varkappa) =\int_{\eta_{0}}^{\infty}\max(f(\varkappa)-f(\varkappa^{*}),0)\varphi( \eta)\,\text{d}\eta\] \[=\int_{\eta_{0}}^{\infty}(\mu+\sigma\eta-f(\varkappa^{*}))\varphi( \eta)\,\text{d}\eta\] \[\tau_{1} =\int_{\eta_{0}}^{\infty}(\mu-f(\varkappa^{*}))\varphi(\eta)\, \text{d}\eta\] \[=\tau_{1}+\int_{\eta_{0}}^{\infty}\sigma\eta\frac{1}{\sqrt{2\pi} }e^{-\eta^{2}/2}\,\text{d}\eta\] \[\tau_{1} =(\mu-f(\varkappa^{*}))\underbrace{\int_{\eta_{0}}^{\infty} \varphi(\eta)\,\text{d}\eta}_{1-\[PERSON](\eta_{0})\equiv 1-\text{CDF}(\eta_{0})}\] \[=\tau_{1}+\frac{\sigma}{\sqrt{2\pi}}\int_{\eta_{0}}^{\infty}\eta e ^{-\eta^{2}/2}\,\text{d}\eta\] \[\tau_{1} =(\mu-f(\varkappa^{*}))(1-\[PERSON](\eta_{0}))\] \[=\tau_{1}-\frac{\sigma}{\sqrt{2\pi}}\int_{\eta_{0}}^{\infty}\left( e^{-\eta^{2}/2}\right)^{\prime}\text{d}\eta \tag{12}\] \[\tau_{1} =(\mu-f(\varkappa^{*}))(1-\[PERSON](\eta_{0}))\] \[\tau_{1} =(\mu-f(\varkappa^{*}))(1-\[PERSON](\eta_{0}))\] \[=\int_{\eta_{0}}^{\infty}(\mu+\sigma\eta-f(\varkappa^{*}))\varphi( \eta)\,\text{d}\eta\]0 \[=\int_{\eta_{0}}^{\infty}(\mu+\sigma\eta-f(\varkappa^{*}))\varphi( \eta)\,\text{d}\eta\]1 \[=\int_{\eta_{0}}^{\infty}(\mu+\sigma\eta-f(\varkappa^{*}))\varphi( \eta)\,\text{d}\eta\]2 \[=\int_{\eta_{0}}^{\infty}(\mu+\sigma\eta-f(\varkappa^{*}))\varphi( \eta)\,\text{d}\eta\]3 \[=\int_{\eta_{0}}^{\infty}(\mu+\sigma\eta-f(\varkappa^{*}))\varphi( \eta)\,\text{d}\eta\]4 \[=\int_{\eta_{0}}^{\infty}(\mu+\sigma\eta-f(\varkappa^{*}))\varphi( \eta)\,\text{d}\eta\]5  
where \(\mathfrak{f}_{opt}\) shows the optimized hyperparameter values, and \(\mu_{P}(f_{opt})\) is the posterior mean. \(\mu(\mathbf{f})\) and \(\sigma(\mathbf{f})\) are the mean and the standard deviation of the GP posterior predictive at \(\mathbf{f}\), respectively. \(\Phi\) and \(\phi\) are the CDF and PDF of the standard normal distribution, respectively. The depicted Figure 1 has list of hyperparameters _\"nu\" as numbers of Neurons, \"af\" as Activation Functions (relu','sigmoid','softplus','softsign', 'tanh','selu', 'elu', 'exponential', LeakyReLU,'relu'), \"opt\" as Optimizer ('SGD', 'Adam', 'RMSprop', 'Adadelta', 'Adagrad', 'Adamax', 'Nadam', 'Ftrl', 'SGD') \"l\" as learning rate (0.01-1), bs as Batch Size (4-256),_ ep as epochs (20-100), lyr as layers (1-3), norm as Normalization (0-1), drop as the drop-out rate (0-0.3) and k as Kernel (3-9). For getting optimized hyperparameters, we used a combined dataset and evolved with a single hyperparameter (learning rate) and acquired two optimal learning rates (0.001 and 0.0001). In this study, we analyze that Gaussian Process (GP) works inefficiently in categorical hyperparameters such as activation functions and Optimizers. When using several kernels in Gaussian Process (GP), it can be challenging to choose optimal hyperparameters. In Tree Parzen Estimators (TPE), the selection of hyperparameters is independent and there is zero correlation between hyperparameters. It starts model overfitting and increases validation error if we train our model without regularization with more epochs. We have proposed a novel and hybrid approach for acquiring optimized both numerical and categorical hyperparameters as well. In this approach, we take a sample of the dataset (BraTS21) and forward to a neural network black box model with a list of hyperparameters and pass these hyperparameters to both Gaussian Process (GP) and Tree Parzen Estimator (TPE) and obtain the optimal values with our proposed approach to handle the categorical and create a correlation between hyperparemters and then pass the optimal values to acquisition function and reiterate the procedure. The posterior distribution improves with each iteration, and the algorithm gets more secure in determining which portions of the parameter domain are worth examining and which are not. AHT additionally employs an acquisition function (exploration strategy or infill sampling criteria) to aid in the selection of the next location to be assessed.