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 cs-677sp2010:map-inference [2014/12/09 09:53]ryancha created cs-677sp2010:map-inference [2014/12/12 11:19]ryancha 2014/12/12 11:19 ryancha 2014/12/12 11:19 ryancha 2014/12/09 09:53 ryancha created Next revision Previous revision 2014/12/12 11:19 ryancha 2014/12/12 11:19 ryancha 2014/12/09 09:53 ryancha created Line 2: Line 2: ==MAP Queries== ==MAP Queries== - Given a set of random variables, $\Chi$, and a set of evidence variables $E$ + Given a set of random variables, $\chi$, and a set of evidence variables $E$ - <​blockquote>​$W = \Chi - E$​ + <​blockquote>​$W = \chi - E$​ The MAP query finds the most likely assignment to the query variables $W$ given the evidence variables $E$. The MAP query finds the most likely assignment to the query variables $W$ given the evidence variables $E$. Line 50: Line 50: The Marginal MAP query is very similar to the MAP query, except that it loosens the constraint that $W$ must contain all non-evidence variables. The Marginal MAP query is very similar to the MAP query, except that it loosens the constraint that $W$ must contain all non-evidence variables. - Given a set of random variables, $\Chi$, a set of evidence variables $E$, and a set of query variables $W$ + Given a set of random variables, $\chi$, a set of evidence variables $E$, and a set of query variables $W$ - <​blockquote>​$Z = \Chi - W - E$​ + <​blockquote>​$Z = \chi - W - E$​ The MAP query finds the most likely assignment to the query variables $W$ given the evidence variables $E$. The MAP query finds the most likely assignment to the query variables $W$ given the evidence variables $E$. Line 72: Line 72: ==Solution Methods== ==Solution Methods== - This discussion is simplified by using the, equivalent, factor representation of a joint distribution. ​ Our task is to find $\xi^{MAP}$,​ the most likely assignment to the variables in $\Chi$.  $P_{\Phi}(\Chi)$ is a distribution defined via a set of factors $\Phi$ and an unnormalized density $\tilde{P_{\Phi}}$. ​ $Z$ is a normalizing constant. + This discussion is simplified by using the, equivalent, factor representation of a joint distribution. ​ Our task is to find $\xi^{MAP}$,​ the most likely assignment to the variables in $\chi$.  $P_{\Phi}(\chi)$ is a distribution defined via a set of factors $\Phi$ and an unnormalized density $\tilde{P_{\Phi}}$. ​ $Z$ is a normalizing constant. <​blockquote>​$\xi^{MAP} = \underset{\xi}{argmax}~P_{\Phi}(\xi)$ <​blockquote>​$\xi^{MAP} = \underset{\xi}{argmax}~P_{\Phi}(\xi)$