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Unformatted text preview: Probablistic Graphical Models, Spring 2007 Homework 4 solutions 1 Importance Sampling Solution due to Steve Gardiner 1. Why is computing the probability of a complete instantiation of the variables in an MRF computationally intractible? The probability of a complete instantiation of the variables in an MRF is simply a quotient: the numerator, which is trivial to compute, is the product of the potential functions evaluated at the instantiated values ; the denominator, however is the partition function over all possible instantiations. Computing the partition function involves enumerating all possible assignments of the MRF, which is exponential in the number of nodes in the network. 2. Given a chordal graph, describe how to compute the likelihood weighting for an importance sampler. Given a chordal graph H C we know there is a clique tree for it (KF Thm 5.7.17), which we can construct using the methods of KF 10.4.2. We can compute a set of calibrated potentials for the graph, and then use the clique tree measure of Definition 10.2.10 to compute the distribution (up to a multiplicative constant, via KF Thm 10.2.11). Thus the likelihood weighting will be given by r m = π T ( x ( m ) ) /Q ( x ( m ) ). Since the clique tree measure is a measure and not necessarily a distribution, we need to use normalized likelihood weighting. 3. Given a non-chordal graph, describe how to compute the likelihood weighting for an importance sam- pler. Given a non-chordal graph we have to use just the unnormalized measure producttext φ , i.e. the likelihood weighting is given by r m = [ Q ( x ( m ) )] − 1 producttext φ ( x ( m ) ). Again, we will use normalized likelihood weighting. 4. Briefly comment on why it is not useful to use importance sampling for approximate inferences on MRFs Generally, we would be trying to evaluate the expectation of a function of the variables in the MRF over the distribution of the MRF, i.e.over the distribution of the MRF, i....
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- Fall '07
- Normal Distribution, Yi, likelihood weighting, Chordal graph, tree decomposition, Steve Gardiner