Bootstrap-based inference

The general case

A general-purpose bootstrap implementation, similar in scope to the boot() function in package boot, is provided by the bn.boot() function (documented here). bn.boot() takes a data set, a structure learning algorithm and an arbitrary function (whose first argument must be an object of class bn) and returns a list of the values returned by said function for the network structures learned from the bootstrapped samples.

For example, we may want to know how many arcs we can expect in a network learned with hill climbing from the learning.test data set (documented here). We can do it as follows:

> library(bnlearn)
> unlist(bn.boot(learning.test, statistic = narcs,
+   algorithm = "hc", R = 10))

 [1] 5 5 5 5 5 5 5 5 5 5

The R argument controls how many bootstrap replicates are performed. Or maybe we want to compare the computational complexity (measured with the numbers of test/score comparisons) between hill climbing and Grow-Shrink for a sample size of 500:

> unlist(bn.boot(learning.test, statistic = ntests,
+   algorithm = "hc", R = 10))

 [1] 40 40 40 40 40 40 40 40 40 40

> unlist(bn.boot(learning.test, statistic = ntests,
+   algorithm = "gs", R = 10))

 [1] 453 430 395 371 488 437 374 431 445 420

Many other questions can be answered with this approach; essentially any function of the network structure can be used for the statistic argument. We can also return the structures themselves using a dummy function as follows.

> bn.boot(learning.test, statistic = function(x) x,
+   algorithm = "hc", R = 2)

[[1]]

  Bayesian network learned via Score-based methods

  model:
   [A][C][F][B|A][D|A:C][E|B:F] 
  nodes:                                 6 
  arcs:                                  5 
    undirected arcs:                     0 
    directed arcs:                       5 
  average markov blanket size:           2.33 
  average neighbourhood size:            1.67 
  average branching factor:              0.83 

  learning algorithm:                    Hill-Climbing 
  score:                                 BIC (disc.) 
  penalization coefficient:              4.258597 
  tests used in the learning procedure:  40 
  optimized:                             TRUE 


[[2]]

  Bayesian network learned via Score-based methods

  model:
   [A][C][F][B|A][D|A:C][E|B:F] 
  nodes:                                 6 
  arcs:                                  5 
    undirected arcs:                     0 
    directed arcs:                       5 
  average markov blanket size:           2.33 
  average neighbourhood size:            1.67 
  average branching factor:              0.83 

  learning algorithm:                    Hill-Climbing 
  score:                                 BIC (disc.) 
  penalization coefficient:              4.258597 
  tests used in the learning procedure:  40 
  optimized:                             TRUE

In addition, we can control how many observations are included in each bootstrap sample with the m argument.

> unlist(bn.boot(learning.test, statistic = narcs,
+   algorithm = "hc", R = 10, m = 50))

 [1] 4 3 3 3 2 5 4 4 4 4

Measuring arc strength

Measuring the degree of confidence in a particular graphical feature of a Bayesian network is a key problem in the inference on the network structure. In the case of single arcs this quantity is called arc strength.

Friedman, Goldszmidt and Wyner (1999) introduced a very simple way of quantifying such a confidence: generating multiple network structures by applying nonparametric bootstrap to the data and estimating the relative frequency of the feature of interest. boot.strength() uses this approach to compute the strength of every possible arc, and has a syntax similar to that of bn.boot().

> library(bnlearn)
> boot.strength(learning.test, algorithm = "hc")

   from to strength direction
1     A  B    1.000     0.500
2     A  C    0.270     0.500
3     A  D    1.000     0.865
4     A  E    0.000     0.000
5     A  F    0.000     0.000
6     B  A    1.000     0.500
7     B  C    0.000     0.000
8     B  D    0.000     0.000
9     B  E    1.000     0.950
10    B  F    0.100     0.500
11    C  A    0.270     0.500
12    C  B    0.000     0.000
13    C  D    1.000     0.865
14    C  E    0.000     0.000
15    C  F    0.015     0.500
16    D  A    1.000     0.135
17    D  B    0.000     0.000
18    D  C    1.000     0.135
19    D  E    0.000     0.000
20    D  F    0.000     0.000
21    E  A    0.000     0.000
22    E  B    1.000     0.050
23    E  C    0.000     0.000
24    E  D    0.000     0.000
25    E  F    1.000     0.050
26    F  A    0.000     0.000
27    F  B    0.100     0.500
28    F  C    0.015     0.500
29    F  D    0.000     0.000
30    F  E    1.000     0.950

Note that this approach computes the joint strength of all the possible arcs; the estimates will not be independent. For each pair of nodes, the probability that there is an arc between them regardless of its direction is stored in the strength column, and the probability of each direction is in the direction column. This parameterization follows Imoto et al. (2002).

Last updated on Mon Aug 5 02:37:50 2024 with bnlearn 5.0 and R version 4.4.1 (2024-06-14).