Chapter 2 bootstrap-based inference

For bayesian network inference, two methods can be used.

2.1 bootstrap-based inference

Original boot.strength() function from bnlearn follows parameterization of Imoto et al. (2002). Specify strType="normal" which is default.

Estimate the gene network $$T$$ times from randomly sampled $$X^*_{n} = (x^*_{1},...,x^*_{n})^T$$. Edge intensity is defined as $$(t1+t2)/T$$, and if $$t1>t2$$, edge confidence is defined as the confidence of direction of gene $$i$$ to gene $$j$$ is $$t1/(t1+t2)$$, where the $$t1$$ corresponds to the number of edges of gene $$i$$ to $$j$$ and $$t2$$ the number of edges of gene $$j$$ to $$i$$.

2.2 Multiscale boostrap-based inference

Additionally, the multiscale boostrap-based inference is implemented .
Specify strType="ms" for the multiscale version. The drawback is that it consumes more time. $$n'/n$$ was defined as the same parameter as the original paper.

$n'/n = (0.5, 0.6, 0.7, 0.8, 0.9, 1, 1.1, 1.2, 1.3, 1.4)$

For the arrays with altered numbers of multiple $$\tau=\sqrt{n/n'}$$, the bootstrap-based inference are performed. $$MS_{ij}$$ is defined as $$1-\Phi(d_{ij}-c_{ij})$$, using the geometric quantities $$d_{ij}$$ and $$c_{ij}$$. Fitting $$BP_{ij}(\tau) = 1-\Phi(d_{ij}/\tau+c_{ij}\tau)$$, we can determine the $$MS_{ij}$$. The fitting was performed by msfit() available in pvclust.

The confidence of direction was obtained by the same method as obtaining $$MS_{ij}$$. The resulting edges were filtered by the threshold determined by the function inclusion.threshold from bnlearn. For both approaches, the superposed network possibly does not hold the cyclic assumption .

References

Imoto, Seiya, Sun Yong Kim, Hidetoshi Shimodaira, Sachiyo Aburatani, Kousuke Tashiro, Satoru Kuhara, and Satoru Miyano. 2002. “Bootstrap Analysis of Gene Networks Based on Bayesian Networks and Nonparametric Regression.” Genome Inform. 13: 369–70.
Kamimura, Takeshi, Hidetoshi Shimodaira, Seiya Imoto, Sun Yong Kim, Kousuke Tashiro, Satoru Kuhara, and Satoru Miyano. 2003. “Multiscale Bootstrap Analysis of Gene Networks Based on Bayesian Networks and Nonparametric Regression.” Genome Inform. 14: 350–51.