taken to be a Markov chain; for continuous-valued traits (such as "brain size"), the process is frequently taken to be a Brownian motion or an Ornstein-Uhlenbeck process. Using this model as the basis for statistical inference, one can now use maximum likelihood methods or Bayesian inference to estimate the ancestral states. Discrete-state models Suppose the trait in question may fall into one of states, labelled . The typical means of modelling evolution of this trait is via a continuous-time Markov chain, which may be briefly described as follows. Each state has associated to it rates of transition to all of the other states. The trait is modelled as stepping between the states; when it reaches a given state, it starts an exponential "clock" for each of the other states that it can step to. It then "races" the clocks against each other, and it takes a step towards the state whose clock is the first to ring. In such a model, the parameters are the transition rates , which can be estimated using, for example, maximum likelihood methods, where one maximizes over the set of all possible configurations of states of the ancestral nodes. In order to recover the state of a given ancestral node in the phylogeny (call this node ) by maximum likelihood, the procedure is: find the maximum likelihood estimate of ; then compute the likelihood of each possible state for conditioning on ; finally, choose the ancestral state which maximizes this. One may also use this substitution model as the basis for a Bayesian inference procedure, which would consider the posterior belief in the state of an ancestral node given some user-chosen prior. Because such models may have as many as parameters, overfitting may be an issue. Some common choices that reduce the parameter space are: Markov -state 1 parameter model: this model is the reverse-in-time -state counterpart of the Jukes-Cantor model. In this model, all transitions have the same rate , regardless of their start and end states. Some transitions may be disallowed by declaring that their rates are simply 0; this may be the case, for example, if certain states cannot be reached from other states in a single transition. Asymmetrical Markov -state 2 parameter model: in this model, the state space is ordered (so that, for example, state 1 is smaller than state 2, which is smaller than state 3), and transitions may