The second type of regularization adds a penalty to the statistic, so that the test statistic becomes the maximum (over all m s) of the statistic plus penalty. For the K -sample problem, Jiang et al. (2014) suggested assigning a prior on the partition scheme and they regularized the likelihood ratio score using this prior. Specifically, they assumed the partition size is Poisson and the conditional distribution on the m partition widths (normalized to sum to one) is Dirichlet (1 ,..., 1). This led to their penalty term − λ$_{0}$ (log N )( m − 1), where λ$_{0}$ > 0 has to be fixed. We assume that the marginal distribution on the partition size is π ( m ) (e.g., Poisson or Binomial), and that the prior probability of selecting I given m , π ( I| m ), is uniform. There is an important difference between our uniform discrete prior distribution on partitions of size m and the continuous Dirichlet uniform prior of Jiang et al. (2014). Our prior is uniform on all partitions that truly divide the sample space into m cells, i.e., we cannot have two partition lines between two consecutive samples, since this is actually an m − 1 partition. Using the continuous Dirichlet prior results in practice in at most m partitions, but the partition size may also be strictly smaller than m if two partition points lie between two sample points. Therefore, their conditional distribution given the partition size parameter is not necessarily the true size of the partition. Their penalty translates to a conditional probability given a true partition size m of ( m − 1)! ( N − 1) ( m − $_{1)}$, compared to our π ( I| m ) = 1 / ( N − 1 m − 1 $^{)}$. Therefore, their score penalizes more severely large m s, and their regularized test statistic has less power when the optimal m is large in our simulations. For aggregation by maximum in the K -sample problem, we consider the regularized statistic, where we use the likelihood ratio score per partition. Due to the computational complexity, we do not consider a regularized score for M$_{m}$$_{×}$$_{m}$ . For aggregation by max m ∈{ 2 ,...,m$_{max}$ $_{}}${ M$_{m}$ + log[ π ( I| m ) π ( m )] } , (2.6) 18