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Diffstat (limited to 'posts/2019-02-14-raise-your-elbo.md')
| -rw-r--r-- | posts/2019-02-14-raise-your-elbo.md | 15 |
1 files changed, 8 insertions, 7 deletions
diff --git a/posts/2019-02-14-raise-your-elbo.md b/posts/2019-02-14-raise-your-elbo.md index eeb115e..8e6cf8c 100644 --- a/posts/2019-02-14-raise-your-elbo.md +++ b/posts/2019-02-14-raise-your-elbo.md @@ -272,8 +272,8 @@ ocurrance of word $x$ in document $d$. For each datapoint $(d_{i}, x_{i})$, $$\begin{aligned} -p(d_i, x_i; \theta) &= \sum_z p(z; \theta) p(d_i | z; \theta) p(x_i | z; \theta) \qquad (2.91)\\ -&= p(d_i; \theta) \sum_z p(x_i | z; \theta) p (z | d_i; \theta) \qquad (2.92). +p(d_i, x_i; \theta) &= \sum_{z_i} p(z; \theta) p(d_i | z_i; \theta) p(x_i | z_i; \theta) \qquad (2.91)\\ +&= p(d_i; \theta) \sum_z p(x_i | z_i; \theta) p (z_i | d_i; \theta) \qquad (2.92). \end{aligned}$$ Of the two formulations, (2.91) corresponds to pLSA type 1, and (2.92) @@ -282,10 +282,10 @@ corresponds to type 2. #### pLSA1 The pLSA1 model (Hoffman 2000) is basically SMM with $x_i$ substituted -with $(d_i, x_i)$, which conditioned on $z$ are independently +with $(d_i, x_i)$, which conditioned on $z_i$ are independently categorically distributed: -$$p(d_i = u, x_i = w | z = k) = p(d_i | \xi_k) p(x_i; \eta_k) = \xi_{ku} \eta_{kw}.$$ +$$p(d_i = u, x_i = w | z_i = k) = p(d_i ; \xi_k) p(x_i; \eta_k) = \xi_{ku} \eta_{kw}.$$ The model can be illustrated in the plate notations: @@ -328,19 +328,20 @@ dimensional embeddings $D_{u, \cdot}$ and $X_{w, \cdot}$. Let us turn to pLSA2 (Hoffman 2004), corresponding to (2.92). We rewrite it as -$$p(x_i | d_i; \theta) = \sum_z p(x_i | z; \theta) p(z | d_i; \theta).$$ +$$p(x_i | d_i; \theta) = \sum_{z_i} p(x_i | z_i; \theta) p(z_i | d_i; \theta).$$ To simplify notations, we collect all the $x_i$s with the corresponding $d_i$ equal to 1 (suppose there are $m_1$ of them), and write them as $(x_{1, j})_{j = 1 : m_1}$. In the same fashion we construct $x_{2, 1 : m_2}, x_{3, 1 : m_3}, ... x_{n_d, 1 : m_{n_d}}$. +Similarly, we relabel the corresponding $d_i$ and $z_i$ accordingly. With almost no loss of generality, we assume all $m_\ell$s are equal and write them as $m$. Now the model becomes -$$p(x_{\ell, i} | d = \ell; \theta) = \sum_k p(x_{\ell, i} | z = k; \theta) p(z = k | d = \ell; \theta).$$ +$$p(x_{\ell, i} | d_{\ell, i} = \ell; \theta) = \sum_k p(x_{\ell, i} | z_{\ell, i} = k; \theta) p(z_{\ell, i} = k | d_{\ell, i} = \ell; \theta).$$ It is effectively a modification of SMM by making $n_d$ copies of $\pi$. More specifically the parameters are @@ -357,7 +358,7 @@ of SMM wherever applicable. The updates at the E-step is -$$r_{\ell i k} = p(z = k | x_{\ell i}, d = \ell) \propto \pi_{\ell k} \eta_{k, x_{\ell i}}.$$ +$$r_{\ell i k} = p(z_{\ell i} = k | x_{\ell i}, d_{\ell, i} = \ell) \propto \pi_{\ell k} \eta_{k, x_{\ell i}}.$$ And at the M-step |