only occur between adjacent states. This model contains two parameters and : one for the rate of increase of state (e.g. 0 to 1, 1 to 2, etc.), and one for the rate of decrease in state (e.g. from 2 to 1, 1 to 0, etc.). Example: Binary state speciation and extinction model The binary state speciation and extinction model (BiSSE) is a discrete-space model that does not directly follow the framework of those mentioned above. It allows estimation of ancestral binary character states jointly with diversification rates associated with different character states; it may also be straightforwardly extended to a more general multiple-discrete-state model. In its most basic form, this model involves six parameters: two speciation rates (one each for lineages in states 0 and 1); similarly, two extinction rates; and two rates of character change. This model allows for hypothesis testing on the rates of speciation/extinction/character change, at the cost of increasing the number of parameters. Continuous-state models In the case where the trait instead takes non-discrete values, one must instead turn to a model where the trait evolves as some continuous process. Inference of ancestral states by maximum likelihood (or by Bayesian methods) would proceed as above, but with the likelihoods of transitions in state between adjacent nodes given by some other continuous probability distribution. Brownian motion: in this case, if nodes and are adjacent in the phylogeny (say is the ancestor of ) and separated by a branch of length , the likelihood of a transition from being in state to being in state is given by a Gaussian density with mean and variance In this case, there is only one parameter (), and the model assumes that the trait evolves freely without a bias toward increase or decrease, and that the rate of change is constant throughout the branches of the phylogenetic tree. Ornstein-Uhlenbeck process: in brief, an Ornstein-Uhlenbeck process is a continuous stochastic process that behaves like a Brownian motion, but attracted toward some central value, where the strength of the attraction increases with the distance from that value. This is useful for modelling scenarios where the trait is subject to stabilizing selection around a certain value (say ). Under this model, the above-described transition of being in state to being in state would have a likelihood defined by the transition density of an Ornstein-Uhlenbeck process with two parameters: , which describes the variance