Recently, there has been an increase in literature about the Double Descent phenomenon for heavily over-parameterized models. Double Descent refers to the shape of the test risk curve, which can show a second descent in the over-parameterized regime, resulting in the remarkable combination of both low training and low test risk. However, much is still unknown about this behaviour. In this thesis we consider Double Descent and more specifically 'beneficial overfitting', meaning that the lowest test risk as a function of the number of parameters is achieved in the over-parameterized regime. We are mainly interested in under what conditions beneficial overfitting occurs. We start by exploring the test risk behaviour for simple linear regression models, with isotropic Gaussian, general Gaussian and sub-Gaussian covariates. For random feature selection and isotropic covariance, beneficial overfitting occurs for a large signal-to-noise ratio. For deterministic feature selection and isotropic covariance, beneficial overfitting occurs if we select features corresponding to the lowest weights. Without feature selection, beneficial overfitting occurs if the eigenvalues of the covariance matrix have a long, flat tail. In the second part of this thesis we check whether the same or similar results can be applied to other models as well. Specifically, we look at kernel regression, random Fourier features and a classification model. It seems that the linear regression results agree with the random Fourier features model, linear and quadratic kernel regression and classification model, but are not applicable for the Gaussian kernel regression case. Hence, more factors need to be considered, besides the eigenvalue behaviour of the covariance or kernel matrix and the way in which features are selected, to fully explain Double Descent and beneficial overfitting.

The goal of this thesis is to estimate parameters in a bidimensional Ornstein-Uhlenbeck process, namely a diffusion model which can be found in Favetto and Samson (2010), which considers plasma and interstitium concentrations. We first look at a general linear stochastic differential equation and some properties. Then we simulate possible paths and observations based on the diffusion model, and derive the underlying state space model. We consider state estimation, where we use the Kalman filter and smoother algorithm. It turns out that the smoother outperforms the Kalman filter. Next, we apply and derive the Liu and West filter for state estimation. We see that the performance for the Liu and West filter is close to the Kalman filter. Furthermore, we look at the parameter estimation. We again use and derive the Liu and West filter, but now for parameter estimation. We first apply this filter to a linear AR(1) model, which gives good results. Then we start with estimating one parameter in the diffusion model and finally we estimate all 7 parameters in the model, using priors concentrated around the true value and 10.000 particles. For one parameter, we obtain good results. For all 7 parameters, results are satisfactory, with for fully observed data better results than for partially observed data. We also look at the influence of the number of particles: for 5.000 particles estimation results are somewhat worse than for 10.000 particles. Furthermore, we change some priors such that they are no longer concentrated around the true value. From this, we see that the Liu and West filter does not seem to perform as well for certain choices of priors. Next, we look at state and parameter estimation in a non-linear AR(1) model using the Liu and West filter, which gives rather good results. Finally, we apply the Liu and West filter once more, but now with a real data set. In this case we are able to obtain parameter values that provide a reasonable estimate for the sum of the concentrations, but for the interstitium concentration it leaves much uncertainty.