fix the gd vert again

_posts/2024-10-17-hawkes-LL.md

@@ -28,7 +28,7 @@ We're interested in learning the parameters $$\Theta$$ --- first we'll derive th
 
 ## Log-likelihood (in general)
 
-Consider a point process where the probability density of a single event is conditional on the event history, that is $$f^*(t)=f(t|\mathcal{H}_t)$$.  We also have the cumulative density function (CDF) as $$F^*(t)=\int_{-\infty}^t f^*(s)ds$$.  Then the conditional intensity (aka hazard function) is
+Consider a point process where the probability density of a single event is conditional on the event history, that is $$f^*(t)=f(t\vert\mathcal{H}_t)$$.  We also have the cumulative density function (CDF) as $$F^*(t)=\int_{-\infty}^t f^*(s)ds$$.  Then the conditional intensity (aka hazard function) is
 
 $$
 \lambda(t) = \frac{f^*(t)}{1-F^*(t)}
@@ -99,7 +99,7 @@ $$
 T\sum_u \mu_u - \sum_u \sum_i H_{uu_i}(T-t_i)
 $$
 
-where $$H_{uu_i}(t) = \int_^t h_{uu_i}(s)ds$$.
+where $$H_{uu_i}(t) = \int_0^t h_{uu_i}(s)ds$$.
 
 
 
@@ -119,7 +119,7 @@ Several important insights here.  First, the indicator variables are turning on/
 Third, notice that for any event $$t_i$$, that is for every term in the sum $$\sum_i$$, we will only have a single indicator variable active, by definition.  Either that event is a background event ($$q_{ii}=1$$) or exactly one of the preceding events caused it ($$q_{ij}=1$$).  So we could take the indicator variables *out* of the log *and split the log*:
 
 $$
-\log(q_ii \mu + \sum_j q_{ij} h) = q_{ii} \log \mu + \sum_j q_{ij} \log h
+\log(q_{ii} \mu + \sum_j q_{ij} h) = q_{ii} \log \mu + \sum_j q_{ij} \log h
 $$
 
 This is a remarkable step, and I don't find it clearly explained in any paper, dissertation, or text.  Maybe I'm just thick-headed.