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How does BayesiaLab calculate the Means and Values in the Monitors? What is the difference?
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For each node that has values associated with its states, an Expected Value [latex:25jc8k4h]v[/latex:25jc8k4h] is computed by using the associated values and the marginal probability distribution of the node.[latex:25jc8k4h]v = \sum_{s \in S} p_s \times V_s[/latex:25jc8k4h]where [latex:25jc8k4h]p_s[/latex:25jc8k4h] is the marginal probability of state [latex:25jc8k4h]s[/latex:25jc8k4h] and [latex:25jc8k4h]V_s[/latex:25jc8k4h] is its associated value.This Expected Value is displayed in the monitor.Example:Categorical Variable:Let's take a discrete node Age with three categorical states:[*:25jc8k4h]Young Adult[/*:m:25jc8k4h][*:25jc8k4h]Adult[/*:m:25jc8k4h][*:25jc8k4h]Senior[/*:m:25jc8k4h]The Node Editor allows associating numerical values with these states.categoricalVariable.jpeg categoricalVariable1.png [latex:25jc8k4h]v = 0.23 \times 25 + 0.415 \times 45 + 0.355 \times 80 = 52.825[/latex:25jc8k4h]Discrete Numerical Variable:Let's suppose now that the variable Age has three numerical states.discreteNumericalVariable.jpeg As it's a numerical node, its monitor will have a Mean value, a Standard Deviation and an Expected Value.discreteNumericalVariable1.png When the states do not have any associated values, [latex:25jc8k4h]V_s[/latex:25jc8k4h] is automatically set to the numerical value of the state.Otherwise, the state values defined by the user are used:discreteNumericalVariable2.jpeg discreteNumericalVariable3.png The Mean value m is computed with the following equation:[latex:25jc8k4h]m = \sum_{s \in S} p_s \times c_s[/latex:25jc8k4h] where: [latex:25jc8k4h]c_s[/latex:25jc8k4h] is the numerical value of the state.Continuous Numerical VariableLet's consider now a continuous variable Age defined on the domain [15 ; 99], and discretized into three states:[*:25jc8k4h]Young Adult: [15 ; 30][/*:m:25jc8k4h][*:25jc8k4h]Adult: ]30 ; 60][/*:m:25jc8k4h][*:25jc8k4h]Senior: ]60 ; 99][/*:m:25jc8k4h]continuousNumericalVariable.jpeg Again, as it's a numerical node, its monitor will have a Mean value, a Standard Deviation and an Expected Value.continuousNumericalVariable1.png The Mean value m is computed with the following equation:[latex:25jc8k4h]m = \sum_{s \in S} p_s \times c_s[/latex:25jc8k4h] where [latex:25jc8k4h]c_s[/latex:25jc8k4h] is the central tendency of the state defined as:[*:25jc8k4h]the mid-range of the state when no data is associated,[/*:m:25jc8k4h][*:25jc8k4h]the arithmetic mean of the data points that are associated with the state.[/*:m:25jc8k4h]When the states do not have any associated values, [latex:25jc8k4h]V_s[/latex:25jc8k4h] is automatically set to the central tendency of the state.When new pieces of evidence are set, a the delta value is displayed in the monitor:meansValueNode.png This delta is the difference between the current Expected Value [latex:25jc8k4h]v[/latex:25jc8k4h] and:[*:25jc8k4h]the previous one,[/*:m:25jc8k4h][*:25jc8k4h]the one corresponding to the reference probability distribution set with meansValueNode1.png  in the toolbar.[/*:m:25jc8k4h]Information: When only some states have an associated value, the Expected Value is computed on the states [latex:25jc8k4h]S^*[/latex:25jc8k4h] that have associated values[latex:25jc8k4h]v = \sum_{s \in S^*} \frac{p_s}{P^*} \times V_s[/latex:25jc8k4h] where [latex:25jc8k4h]P^* = \sum_{s \in S^*} {p_s}[/latex:25jc8k4h]When [latex:25jc8k4h]S^*[/latex:25jc8k4h] is only made of one state, the node is considered as not having any associated values.
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