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Question

How does BayesiaLab calculate the Means and Values in the Monitors? What is the difference?

Answer


For each node that has values associated with its states, an Expected Value $v$ is computed by using the associated values and the marginal probability distribution of the node $$v = \sum_{s \in S} p_s \times V_s$$ where $p_s$ is the marginal probability of state $s$ and $V_s$ is its associated value.


This Expected Value is displayed in the monitor.

Example

Let's take a discrete node Age with three categorical states:

  • Young Adult
  • Adult
  • Senior

The Node Editor allows associating numerical values with these states.

v = 0.23 \times 25 + 0.415 \times 45 + 0.355 \times 80 = 52.825

Let's suppose now that the variable Age has three numerical states.

As it's a numerical node, its monitor will have a Mean value, a Standard Deviation and an Expected Value.

When the states do not have any associated values, $V_s$ is automatically set to the numerical value of the state.

Otherwise, the state values defined by the user are used:

The Mean value $m$ is computed with the following equation: $$m = \sum_{s \in S} p_s \times c_s$$ where $c_s$ is the numerical value of the state.

Let's consider now a continuous variable Age defined on the domain [15 ; 99], and discretized into three states:

  • Young Adult: [15 ; 30]
  • Adult: ]30 ; 60]
  • Senior: ]60 ; 99]

Again, as it's a numerical node, its monitor will have a Mean value, a Standard Deviation and an Expected Value.

The Mean value $m$ is computed with the following equation: $$m = \sum_{s \in S} p_s \times c_s$$ where $c_s$ is the central tendency of the state defined as: the mid-range of the state when no data is associated, the arithmetic mean of the data points that are associated with the state. When the states do not have any associated values, $V_s$ is automatically set to the central tendency of the state.
If a dataset is associated to a continuous variable, clicking on the Generate Values buttons sets the values $V_s$ to the current arithmetic means.

When new pieces of evidence are set, a the delta value is displayed in the monitor:

This delta is the difference between the current Expected Value v

 and:


  • the previous one,
  • the one corresponding to the reference probability distribution set with  in the toolbar.
When only some states have an associated value, the Expected Value is computed on the states $S^*$ that have associated values $$v = \sum_{s \in S^*} \frac{p_s}{P^*} \times V_s$$ where $S^*$ is only made of one state, the node is considered as not having any associated values.