# Contents

# Context

**Conditional Probability Distributions (CPD)** are typically represented by **Conditional Probability Tables (CPT)**, in which there is one probability distribution for each combination of the states of the parent nodes.

Even though in BayesiaLab the final internal model is in the form of a **CPT**, the definition of the *CPD* can be facilitated by using the * Deterministic* or

*modes in the*

**Equation****Node Editor.**

BayesiaLab 5.3 now offers the ability of using **Conditional Probability Trees (CPTr)** for compactly representing **CPDs **by exploiting

**Contextual Independencies,**e.g. when the state of one parent makes the other co-parent(s) independent of the child node.

**CPTr **are available in the **Node Editor, **so you can manually specify a **Tree.** Also, they are now integrated in BayesiaLab's machine learning algorithms:

- For estimating the parameters of a given network: this allows generalizing probability distributions, either if data is scarce for some combination of parents' states, or if the distributions are not significantly different.
- For estimating the parameters
*and*learning the structure of a graph: this returns a more complex structure as adding parents with**CPTr**can be less costly than with**CPT***.*

# New Feature: Tree Mode in Node Editor

#### Node Editor | Probability Distribution | Tree

**Example**

Let's define a logical OR between 3 nodes: * Tuberculosis*,

*and*

**Cancer**

**Bronchitis**The tree below represents this deterministic function by exploiting two contextual independencies:

- When
*Tuberculosis*is*True*, then*Cancer*and*Bronchitis*are independent of the node*OR.*

Right click on **No Selector** to select the **Parent** to add in the **Tree**

Define the probability distribution for this context, i.e. *Tuberculosis=True.*

Right click on **False** to select the **Parent** to add to the **Tree.**

Define the probability distribution for this context, i.e. *Tuberculosis=False *and* Cancer=True.*

Right click on **False** to select the **Parent** to add to the **Tree.**

Define the last two probability distributions.

Upon **Validation**, we obtain the following **Conditional Probability Table**:

# New Feature: Parameter Estimation with Trees

#### Edit | Parameter Estimation with Trees

If this option is checked, the parameters will no longer be estimated using the Maximum Likelihood Estimation for each combination of states. Rather, they will be estimated by machine learning a probabilistic tree that exploits any regularities in the data. This has the objective of compactly representing the relationship between the parent and child nodes.

- We define the
**MDL**score (**Minimum Description Length**) as a measure of the quality of candidate trees with respect to the available data. This score, which is derived from Information Theory, automatically takes into account the data likelihood with respect to the tree and its structural complexity. - Trees are available for
**P****arameter Learning, Unsupervised Structural Learning (Taboo**and**Taboo Order**only**)**and**Supervised Learning.** - The
**Structural Coefficient**and**Local Structural Coefficients**used for computing the MDL score of a Bayesian networks are also used for computing the MDL score of**Trees**. - In this context, the
**Smoothing Parameter**is also taken into account.

- If no regularities can be exploited, tables are used as usual.
If the (conditional) dependency of a parent with its child is not strong enough, that parent will not be included in the induced tree.

Even if the link is not physically removed, it is "soft-deleted" through the parameters.

In this example,

is not dependent on**Evoque Joy**given**Flowery****Fresh.**

Nodes with **CPTr** are displayed in pink.

Such nodes can be selected by using** Edit | Select Nodes | Generalized**

Whenever **Trees** are used instead of **Tables,** an icon is added in the lower right corner of the graph window:

**Example**

Let's take our textbook *Perfume* example to illustrate the impact of estimating parameters with **Trees**.

As expected, the structure is much more complex (+12 arcs), as adding parents does not have necessarily an exponential impact on the size of the **CPD** representation.

Let's focus on the node *Elegant*:

It's strongest relationship is with *Chic* (as learned with EQ and CPT).

*Pleasure* is only interesting if the evaluation of *Chic* is *Medium* (<=6.4)

*Easy to wear* is only interesting if the evaluation of *Chic* is *High* (<=8.2)