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# Maximum Likelihood Estimation with Priors

- 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.

$\hat P(X = {x_i}|Pa = p{a_i}) = \frac{{N(X = {x_i},Pa = p{a_i}) + {M_0} \times {P_0}(X = {x_i},Pa = p{a_i})}}{{\sum\nolimits_j {\left( {N(X = {x_j},Pa = p{a_j}) + {M_0} \times {P_0}(X = {x_j},Pa = p{a_i})} \right)} }}$

where

- ${M_0}$is the degree of confidence in the Prior.
- ${P_0}$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.

- With your current Bayesian network, you can generate Priors
- Select
`Main Menu > Data > Prior Samples > Generate`

. - You can specify${M_0}$by setting the number of Prior Samples.

- BayesiaLab uses the current Bayesian network to compute${P_0}$.
- 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.

**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.

Last modified 4mo ago