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 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 is computed using the assigned State Values and the marginal probability distribution of the Node States:
where is the marginal probability of state and is its associated value.
The Monitor shows 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.
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As a result, BayesiaLab uses the numerical values of the Node States, as they appear in the States tab, as the State Values.
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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:
where is the numerical value of the Node State.
Note that Mean and Value are identical in this case.
State Values Specified
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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.
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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 is computed now using the assigned State Values 2 and the marginal probability distribution of the Node States:
where is the marginal probability of state and is its associated value.
- The Monitor shows 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 and 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:
The Expected Value is calculated analogously:
\
Associated Data
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If data is associated with the node, is defined as the arithmetic mean of the data points that are associated with each Node State.
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Furthermore, clicking on the Generate Values button in the Node Editor sets the values 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 before setting the modifying evidence, or
- the Expected Value 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 is computed from the subset of Node States that do have an associated value.
- If a node only has a single Node State with an associated value, the corresponding Monitor does not report the Expected Value .