# 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 (Kamimura et al. 2003).

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 (Imoto et al. 2002).

### References

*Genome Inform.*13: 369–70.

*Genome Inform.*14: 350–51.