BayesiaLab
Means and Values of Nodes

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 vv 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 vv is computed using the assigned State Values and the marginal probability distribution of the Node States:

v=sSps×Vsv = \sum_{s \in S} p_s \times V_s

where pp is the marginal probability of state ss and VsV_s is its associated value.

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

The Monitor shows vv as the Value of Age.

A Monitor of a categorical node does not show a Mean value.

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=sSps×csm = \sum_{s \in S} p_s \times c_s

where csc_s is the numerical value of the Node State.

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

Note that Mean and Value are identical in this case.

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 vv is computed now using the assigned State Values 2 and the marginal probability distribution of the Node States:

v=sSps×Vsv = \sum_{s \in S} p_s \times V_s

where pp is the marginal probability of state ss and VsV_s is its associated value.

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

  • The Monitor shows vv as the Value of Age.

Note that Mean and Value are not identical in this case.

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 csc_s and VsV_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=sSps×csm = \sum_{s \in S} p_s \times c_s

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

The Expected Value vv is calculated analogously:

v=sSps×Vsv = \sum_{s \in S} p_s \times V_s

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

Associated Data

  • If data is associated with the node, csc_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 VsVs 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 vv before setting the modifying evidence, or
    • the Expected Value vv 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 vv is computed from the subset of Node States S{S^*} that do have an associated value.

v=sSpsP×Vsv = \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 vv.

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