The goal of this thesis is to implement and experiment with a Bayesian way of estimating a (smooth) monotone regression function by applying it to climate data. The method we use is proposed by Neelon and Dunson. This method uses a piece-wise linear model for the unknown regression function and enforces the monotonicity constraint by the specification of the prior distribution of the slopes. This thesis is also aimed at providing solutions to specific problems that we encounter during the process of applying this method. We encounter two main problems: a numerical problem and a boundary problem. The numerical problem concerns a fraction of very small numbers, which we can solve using an asymptotic approximation of the Mills Ratio. The boundary problem appears in the form of a steep upward-sloping curve at the left boundary. We provide a pragmatic solution to this boundary effect, in which we use an extension of the data set, to obtain a better curve estimate at this boundary.