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Maximum Likelihood Estimation with Priors

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BayesiaLab can also take into account Priors when estimating parameters using Maximum Likelihood Estimation.
Priors reflect any a priori knowledge of an analyst regarding the domain, in other words, expert knowledge. See also Prior Knowledge for Structural Learning.
These priors are expressed with an analyst-specified, initial Bayesian network (structure and parameters) plus analyst-specified Prior Samples.
Prior Samples represent the analyst's subjective degree of confidence in the Priors.
P^(X=xi∣Pa=pai)=N(X=xi,Pa=pai)+M0×P0(X=xi,Pa=pai)∑j(N(X=xj,Pa=paj)+M0×P0(X=xj,Pa=pai))\hat P(X = {x_i}|Pa = p{a_i}) = \frac{{N(X = {x_i},Pa = p{a_i}) + {M_0} \times {P_0}(X = {x_i},Pa = p{a_i})}}{{\sum\nolimits_j {\left( {N(X = {x_j},Pa = p{a_j}) + {M_0} \times {P_0}(X = {x_j},Pa = p{a_i})} \right)} }}P^(X=xi​∣Pa=pai​)=∑j​(N(X=xj​,Pa=paj​)+M0​×P0​(X=xj​,Pa=pai​))N(X=xi​,Pa=pai​)+M0​×P0​(X=xi​,Pa=pai​)​
where
​M0{M_0}M0​
 is the degree of confidence in the Prior.
​P0{P_0}P0​
 is the joint probability returned by the prior Bayesian network.
BayesiaLab uses these two terms to generate virtual samples that are subsequently combined with the observed samples from the dataset.
Virtual Database Generator
With your current Bayesian network, you can generate Priors 
Select  Main Menu > Data > Prior Samples > Generate.
You can specify M0{M_0}M0​
 by setting the number of Prior Samples.
​
 
BayesiaLab uses the current Bayesian network to compute P0{P_0}P0​
.
The existence of a new Virtual Database is indicated by an icon 
 in the lower right corner of the graph window, next to the "real dataset" icon 
.

Right-clicking on the Virtual Database icon 
 displays the structure of the prior knowledge that was used for generating the Virtual Samples.
These Virtual Samples will be combined with the observed "real" samples during the learning process.
​
Number of Uniform Prior Samples
Edit Number of Uniform Prior Samples allows you to define prior knowledge in such a way that all the variables are marginally independent (fully unconnected network), and the marginal probability distributions of all nodes are uniform.
For instance, if the number of Prior Samples is set to 1, one observation ("occurrence") would be "spread across" all states of each node, essentially assigning a "fraction of an observation" to each node's states.
To apply Smoothed Probability Estimation, select Main Menu > Edit > Edit Smoothed Probability Estimation
Specify the number of Prior Samples.
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Maximum Likelihood Estimation
Structural Coefficient
Last modified 4mo ago