Change-split. Randomly pick a splitting node and assign it a new splitting variable and rule drawn from the corresponding priors. 

Change-rule. Randomly pick a splitting node and assign it a new rule drawn from a given prior. 

The first two moves, *birth* and *death*, are reversible and change the dimensionality of  as described in [10]. The remaining moves provide jumps within the current dimensionality of . Note that the *change-split* move is included to make "large" jumps which potentially increase the chance of sampling from a maximal posterior whilst the change-rule move does "local" jumps.

For the birth moves, the proposal ratio R is written

## , ( '| ) ( ) ( | ) ( ) Θ Θ Θ Θ Θ' Θ' Q P Q P R

where q(θ | θ') and q(θ'| θ) are the proposed distributions, θ' and θ are (k + 1) 
and k-dimensional vectors of DT parameters, respectively, and p(θ ) and p(θ') are the probabilities of the DT with parameters θ and θ' :

![0_image_0.png](0_image_0.png)

p
( )
where ( )
var i N s is the number of possible values of si var which can be assigned as a new splitting rule, Sk is the number of ways of constructing a DT with k terminal nodes, and K is the maximal number of terminal nodes, *K = n* - 1.

The proposal distributions are as follows

( | ') ,
$$\mathbf{\Phi}(\mathbf{\Phi})={\frac{d_{k+1}}{D_{o^{\prime}}}},$$
$\mathbf{M}$
q θ θ  
where DQ1 = DQ + 1 is the number of splitting nodes whose branches are both terminal nodes. 

Then the proposal ratio for a *birth* is given by

$\mathbf{M}$
.
$$={\frac{d_{k+1}}{b_{k}}}{\frac{k}{D_{Q1}}}{\frac{S_{k}}{S_{k+1}}}$$
The number DQ1 is dependent on the DT structure and it is clear that DQ1 < k  k = 
1, …, K. Analyzing the above equation, we can also assume dk+1 = bk. Then letting the DTs grow, i.e., k  K, and considering Sk+1 > Sk, we can see that the value of R  c, where c is a constant lying between 0 and 1. 

Alternatively, for the death moves the proposal ratio is written as