RSEG: Supplmentary Information

1.1 Negative binomial distribution In Fig. 1, we show the application of the Poisson distribution and the negative binomial distribution to H3K27me3 and H3K36me3 ChIP-Seq data. The negative binomial distribution has a smaller BIC (Bayesian Information Criterion) and therefore is more appropriate to model read counts data. The negative binomial distribution may be formulated in several ways. In this paper the negative binomial dis-tribution is formulated as the number of failures before r successes where the success probability is p. With this formulation, its support is {0, 1, 2,...,}. The probability mass function is f(n; r, p) = n+ r − 1 r − 1 pr(1 − p)n. (1) In this article we sometimes re-parameterize the negative binomial distribution with the mean µ and the dispersion parameter α, where µ and α are related to r and p as following µ = r 1 − p p, (2a) α = 1/r. (2b) With this parameterization, µ is the expected number of failures before r successes and α controls how dispersed the number is. 1.2 NBDiff distribution The NBDiff distribution is the discrete distribution of the difference between two independent negative binomial random variables. It has four parameters: r1 and p1 from the first negative binomial distribution and r2 and p2 from the second one. Probability mass function: The probability mass function of the NBDiff distribution is given in Eq. (3), f(k; r1, p1, r2, p2) = f(k; r1, p1)p r2 2 × 2F1 k + r1, r2, k + 1, (1 − p1)(1 − p2), k ≥ 0 f(|k|; r2, p2)p r1 1 × 2F1 k + r2, r1, |k|+ 1, (1 − p1)(1 − p2), k < 0 (3) where f(n; r, p) is the probability mass function of the negative binomial distribution and 2F1(a, b; c;x) is the Gaus-sian hypergeometric function (Barnes, 1908). ∗to whom correspondence should be addressed 1 H3K27me3 Foreground read count F re q