Last updated: 2020-05-19

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Model and implementation details:

ebpmf_bg

Individual data analysis results: 

kos_K20,
kos_K50,
kos_K100

some observations

  • \(g_L\) ’s components have large \(\phi_l\) (\(100\)), \(g_F\) has components with both large (\(100\)) and small (\(0.01\)) \(\phi_l\).
  • bg model can identify rare but important words for a topic. (although they are not very easy to understand sometimes… may need better dataset?)
  • The fitted background is quite different from the rank-1 fit, and the row-wise median from PMF fit.

More comparisons across different \(K\)

Note that

\[ELBO = E_q(\text{log-lik}) - KL(q || g)\].

    K     ELBO       KL     E_ll   ll_pmf
1  20 -1700057 151380.3 -1548677 -1504099
2  50 -1688549 232036.2 -1456512 -1379746
3 100 -1702420 313105.3 -1389315 -1264891
4 300 -1847818 608624.9 -1239193 -1038952

For a large \(K\), we should expect optimal soultion to have ELBO no less than small \(K\): we can just make \(q, g\) to be point-mass at 0 for extra topics to ensure equal ELBO. But how can we achieve them? If we can do that, the algorithm can choose automatically \(K\).

(say \(K = 50\) is optimal, and we use \(K = 60\). we might initialize the last 20 to be point mass at 0. but the bg structure make it hard…)

look at KL divergence for different \(K\)

Version Author Date
8d1d007 zihao12 2020-05-19

TODO:

As suggested by Matthew:

add \(w_k\) to the model

\[\begin{align} & X_{ij} \sim Pois(l_{i0} f_{j0} \sum_k w_k l_{ik} f_{jk})\\ & l_{ik} \sim g_{L, k}(.), f_{jk} \sim g_{F, k}(.) \\ & \sum_{i} l_{i0} = \sum_j f_{j0} = 1 \end{align}\]

where \(g\) are mixtures of gamma, with mean 1 and different variances.
We use \(w_k\) to account for the inbalance of topics.

think about rank-1 update and automatically choose \(K\)

Analogous to the greedy algorithm (rank-1 update) in flashr, we want to do similar things in our model.

how to pick key words for each topic

We already see that we can learn some rare and important key words from \(\bar{f}_{jk}\) that we can’t from \(f_{j0}\bar{f}_{jk}\). Peter has some ways to find ways to find key words so it wold be great to apply them here.

find a better dataset

Some issue with the kos dataset: some words are not truncated properly (particularly names); does not include common and non-informative words like “a”, “the” (which would allow us to compare with common but informative words)

overfitting?

IN \(g_F\), most topics have around \(80\) percent of weight on large \(\phi_l\), which shrinks the enrichment of most words to almost 0. For a very small number in \(f_{j0}\bar{f}_{jk}\), which is more suitable: a small background or very little enrichment?


R version 3.5.1 (2018-07-02)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS  10.14

Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRlapack.dylib

locale:
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attached base packages:
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