Therefore, it is important to ensure the date range of the data and model are appropriate for each other, and to exclude dates from the dataset that do not reasonably fall within the modelled range. We achieve this with our real datasets by only including a date if more than 50% of its probability falls within the modelled date range-i.e. it is more probable that its true date is internal than external. Similarly, we achieve this with our extremely small toy dataset (N = 6) by constraining the modelled date range to exclude the negligible tails outside the calibrated dates.
7. Search algorithm for parameters
The CPL model is a PMF such that the probability outside the date range equals 0, and the total probability within the date range equals 1. The exact shape of this PMF is defined by the (x, y) coordinates of the hinge points. Therefore, there are various constraints on parameters required to define such a curve. For example, if we consider a 2-CPL model, only the middle hinge has a free x-coordinate parameter, since the start and end date are already specified by the date range. Of the three y-coordinates (left, middle, right hinges), only two are free parameters, since the total probability must equal 1. Therefore, a 2-CPL model has three free parameters (one x-coordinate and two y-coordinates) and an n-phase CPL model has 2n?1 free parameters. Continue reading “8. Estimating credible intervals using Markov chain Monte Carlo”