Means and Values of Nodes

Context

• At the top of each Monitor, the items Mean, Dev, and Value are displayed.

• Mean refers to the Mean Value m and is only shown in the Monitors of numerical nodes.
• Dev stands for Standard Deviation and is shown alongside Mean.
• Value refers to the Expected Value $v$ and is shown in all Monitors, regardless of the node type, i.e., categorical or numerical.

Examples

• The calculations for Expected Value and Mean Value are shown in the context of the following examples:

Categorical Node

Let's take the Discrete Nodes and Continuous Nodes Age with three categorical Node States:

• Child
• Adult
• Senior

Categorial Node with Assigned State Values

• In the Node Editor, you can assign State Values to the Node States of Age.

• For each node, the Expected Value $v$ is computed using the assigned State Values and the marginal probability distribution of the Node States:

$v = \sum_{s \in S} p_s \times V_s$

where $p$ is the marginal probability of state $s$ and $V_s$ is its associated value.

$v = 0.23 \times 10 + 0.415 \times 40 + 0.355 \times 70 = 43.750$

The Monitor shows $v$ as the Value of Age.

Discrete Numerical Variable

• Let's suppose that the node Age has three numerical Node States instead of categorical Node States.\

• In this context, we need to consider two conditions, with and without State Values specified in the Node Editor:

No State Values Specified

• Here, State Values are not specified in the Values tab of the Node Editor. Note the empty Value column below.

• As a result, BayesiaLab uses the numerical values of the Node States, as they appear in the States tab, as the State Values.

• Furthermore, as Age is a numerical node, its Monitor will now display the Mean (Mean) and the Standard Deviation (Dev) in addition to the Expected Value (Value)

• The Mean m is computed using the numerical values of the Node States and the marginal probability distribution of the Node States:

$m = \sum_{s \in S} p_s \times c_s$

where $c_s$ is the numerical value of the Node State.

$m = 0.23 \times 10 + 0.415 \times 40 + 0.355 \times 70 = 43.750$

State Values Specified

• However, if State Values are separately specified in the Values tab of the Node Editor, they will be used for the calculation of Value in the Monitor.

• To highlight the distinction between the Node States 70 and the State Values, we assign unrelated arbitrary State Values of 0, 1, and 2.

• The Expected Value $v$ is computed now using the assigned State Values 2 and the marginal probability distribution of the Node States:

$v = \sum_{s \in S} p_s \times V_s$

where $p$ is the marginal probability of state $s$ and $V_s$ is its associated value.

$v = 0.23 \times 0 + 0.415 \times 1 + 0.355 \times 2 = 1.125$

• The Monitor shows $v$ as the Value of Age.

Continuous Numerical Variable

• Let's now consider a continuous variable Age defined in the domain [0; 99], discretized into three states:

• Child: [0 ; 18]
• Adult: ]18 ; 65]
• Senior: ]65 ; 99]

• Given Age is a numerical node, its Monitor shows the Mean (Mean), the Standard Deviation (Dev), plus the Expected Value (Value).

No Associated Data

• If no associated data is with the node, both $c_s$ and $V_s$ are defined as the mid-points of the minimum and maximum values of each Node State. For example, for the Node State Adult ]18; 65], the midpoint is 17.2225.

So, the Mean Value m is computed as follows:

$m = \sum_{s \in S} p_s \times c_s$

$m = 0.23 \times 9 + 0.415 \times 41.5 + 0.355 \times 82 = 48.4025$

The Expected Value $v$ is calculated analogously:

$v = \sum_{s \in S} p_s \times V_s$

$v = 0.23 \times 9 + 0.415 \times 41.5 + 0.355 \times 82 = 48.4025$\

Associated Data

• If data is associated with the node, $c_s$ is defined as the arithmetic mean of the data points that are associated with each Node State.

• Furthermore, clicking on the Generate Values button in the Node Editor sets the values $Vs$ to the current arithmetic means of each Node State.

Value Delta

• If you set a new piece of evidence on a node that modifies the distribution of the node, the Monitor displays a delta value in parentheses adjacent to Value.

• This delta is the difference between the current Expected Valuev and:
  • the Expected Value $v$ before setting the modifying evidence, or
  • the Expected Value $v$ that corresponds to the Reference Probability Distribution, which you can set with the icon in the toolbar.

Special Case: Some Node States Without Values

• If only some Node States have an associated value, the Expected Value $v$ is computed from the subset of Node States ${S^*}$ that do have an associated value.

$v = \sum_{s \in S^*} \frac{p_s}{P^*} \times V_s$

• If a node only has a single Node State with an associated value, the corresponding Monitor does not report the Expected Value $v$.