Difference between revisions of "Relational Topic Models"
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In modeling sparsity, we assume that we draw another hidden variable say y_ij ~ Bernoulli(\eta_{z_ij, z_ji}). And then draw r_ij ~ Bernoulli(\rho) if y_ij = 1 and r_ij ~ \delta_0 otherwise. \rho represents how often we expect to observe a link between nodes that are actually linked in the latent space. | In modeling sparsity, we assume that we draw another hidden variable say y_ij ~ Bernoulli(\eta_{z_ij, z_ji}). And then draw r_ij ~ Bernoulli(\rho) if y_ij = 1 and r_ij ~ \delta_0 otherwise. \rho represents how often we expect to observe a link between nodes that are actually linked in the latent space. | ||
| − | We can thus rewrite r_ij ~ y_ij Bernoulli(\rho) + (1 - y_ij) delta_0 = | + | We can thus rewrite r_ij ~ y_ij Bernoulli(\rho) + (1 - y_ij) delta_0. We can integrate out y_ij by noting that p(r_ij = 1 | z_ij, z_ji) = \rho \eta_{z_ij, z_ji}. And p(r_ij = 0 | z_ij, z_ji) = 1 - \eta_{z_ij, z_ji} + (1 - \rho) \eta_{z_ij, z_ji} = 1 - \rho \eta_{z_ij, z_ji}. In other words, r_ij ~ Bernoulli(\rho \eta_{z_ij, z_ji}). |
== Choosing the sparsity parameter == | == Choosing the sparsity parameter == | ||
Latest revision as of 18:08, 7 April 2008
Modeling Sparsity
In an undirected setting, let us consider having chosen z_ij and z_ji and then selecting the response according to r_ij ~ Bernoulli(\eta_{z_ij, z_ji}). In modeling sparsity, we assume that we draw another hidden variable say y_ij ~ Bernoulli(\eta_{z_ij, z_ji}). And then draw r_ij ~ Bernoulli(\rho) if y_ij = 1 and r_ij ~ \delta_0 otherwise. \rho represents how often we expect to observe a link between nodes that are actually linked in the latent space.
We can thus rewrite r_ij ~ y_ij Bernoulli(\rho) + (1 - y_ij) delta_0. We can integrate out y_ij by noting that p(r_ij = 1 | z_ij, z_ji) = \rho \eta_{z_ij, z_ji}. And p(r_ij = 0 | z_ij, z_ji) = 1 - \eta_{z_ij, z_ji} + (1 - \rho) \eta_{z_ij, z_ji} = 1 - \rho \eta_{z_ij, z_ji}. In other words, r_ij ~ Bernoulli(\rho \eta_{z_ij, z_ji}).
Choosing the sparsity parameter
On the senate dataset, running spectral clustering for various values of K gives the following:
| K | False positives | False negatives |
|---|---|---|
| 5 | .606 | .058 |
| 10 | .354 | .078 |
| 15 | .126 | .078 |
| 20 | .193 | .094 |
| 25 | .157 | .107 |
| 30 | .135 | .114 |
Even with 30 topics, this would imply that we're not seeing at least around 15% of true links. Since spectral clustering is likely to be overfitting in this case, a reasonable compromise between all the K might be 25%. Although, since for this dataset we'd expect the true K to be small, 50% might be a better estimate.
--Jcone 18:27, 7 April 2008 (EDT)
