Partial Order Learning

Context

Partial Order Learning is a Meta-Learning algorithm that can be compared to Bagging. It is based on Data Perturbation (Smoothed Bootstrapping) for learning a set of networks that are transformed into Essential Graphs to define a set of Partial Orderings.

In a Partial Ordering, the index of a node is incremented only when it has parents in the Essential Graph. The indices of a Partial Ordering are saved as Temporal Indices.

There are two options for summarizing the node indices that have been estimated on the bag of networks: the mean value (a real value) or a vote (an integer).

Once defined, the Temporal Indices are used for learning a network on the original unperturbed data set.

Diagram of the method: the data set is perturbed into many variants, a network is learned from each, ordering statistics are pooled into temporal indices, and a final network is learned with those indices

Example

This example reuses the one introduced for Structural Priors Learning.

We keep the structural priors that have been estimated and we compare the two options for summarizing the indices, Mean Value and Vote.

We uncheck Remove Current Temporal Indices to keep our initial temporal indices that define $\mathit{Age}$ and $\mathit{Gender}$ as root nodes.

Partial Order Learning window with Compute Index Mean checked

The network learned with mean temporal indices

Click to Zoom

Console output for that run, final MDL score 2,741.607

$Edit Temporal Indices with the computed means, Age and Gender at 0 and the rest fractional$

Below is the layout we get by using the Temporal Index Layout.

Temporal Index Layout: the mean indices spread every node along a single near-horizontal line

Click to Zoom

Temporal indices computed as mean values return information that can be useful, but it can also be misleading. It is possible to have mean indices that have never been found in any network (e.g. a mean index equal to 3 with two indices equal to 1 and 5). This is the reason why it is probably safer to use the Vote instead.

We uncheck Remove Current Temporal Indices to keep our initial temporal indices that define $\mathit{Age}$ and $\mathit{Gender}$ as root nodes.

The same window with Compute Index Mean unchecked

The network learned with integer temporal indices

Click to Zoom

Console output for that run, final MDL score 2,740.733

Edit Temporal Indices with whole-number indices, several nodes sharing each level

The graph below shows the layout we get by using the Alignment function that uses Temporal Indices.

Alignment layout: the integer indices stack the nodes into readable rows

Click to Zoom