diff options
Diffstat (limited to 'posts')
| -rw-r--r-- | posts/2019-02-14-raise-your-elbo.md | 8 |
1 files changed, 4 insertions, 4 deletions
diff --git a/posts/2019-02-14-raise-your-elbo.md b/posts/2019-02-14-raise-your-elbo.md index 3b8c1a5..5700899 100644 --- a/posts/2019-02-14-raise-your-elbo.md +++ b/posts/2019-02-14-raise-your-elbo.md @@ -86,7 +86,7 @@ as variational Bayes (VB). **Definition**. Variational inference is concerned with maximising the ELBO $L(w, q)$. -There are mainly two version of VI, the half Bayesian and the fully +There are mainly two versions of VI, the half Bayesian and the fully Bayesian cases. Half Bayesian VI, to which expectation-maximisation algorithms (EM) apply, instantiates (1.3) with @@ -272,8 +272,8 @@ ocurrance of word $x$ in document $d$. For each datapoint $(d_{i}, x_{i})$, $$\begin{aligned} -p(d_i, x_i; \theta) &= \sum_z p(z; \theta) p(d_i | z; \theta) p(x_i | z; \theta) \qquad (2.91)\\ -&= p(d_i; \theta) \sum_z p(x_i | z; \theta) p (z | d_i; \theta) \qquad (2.92). +p(d_i, x_i; \theta) &= \sum_{z_i} p(z; \theta) p(d_i | z_i; \theta) p(x_i | z_i; \theta) \qquad (2.91)\\ +&= p(d_i; \theta) \sum_{z_i} p(x_i | z_i; \theta) p (z_i | d_i; \theta) \qquad (2.92). \end{aligned}$$ Of the two formulations, (2.91) corresponds to pLSA type 1, and (2.92) @@ -285,7 +285,7 @@ The pLSA1 model (Hoffman 2000) is basically SMM with $x_i$ substituted with $(d_i, x_i)$, which conditioned on $z$ 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: |