Parameter estimation derivation of equations for lower bound in LDA with EP
$begingroup$
I am working on the derivations of EP for LDA, and I don't understand how the authors derived the last equations.
Basically, they get the following expression (the lower bound in eq 29):
$L=int(sum_iq_i(veclambda))log q_i(veclambda)dlambda + sum_{iw}n_{iw}int q_i(veclambda)log(sum_alambda_ap(w|a))dveclambda + C$
So, in order to solve the problem, they want to optimize $L$ by maximizing w.r.t. $alpha$ and $p(w|a)$. To do so, they need to get the derivative of $L$ and equal to zero.
For instance, when maximizing w.r.t. $p(w|a)$ ($frac{dL}{dp(w|a)}=0$), they get the following (eq 31):
$p(w|a)^{new} propto sum_in_{iw}int q_i(veclambda)frac{lambda_ap(w|a)}{sum_alambda_ap(w|a)}dveclambda$
However, I am getting something completely different when trying to evaluate $frac{dL}{dp(w|a)}$.
begin{align*}
frac{dL}{dp(w|a)} &= sum_i 0 + ldots + n_{iw}int q_i(veclambda)frac{d}{dp(w|a)}log(sum_alambda_ap(w|a))dveclambda +0+dots\
&=sum_i n_{iw}int q_i(veclambda)frac{lambda_a}{sum_alambda_ap(w|a)}dveclambda\
end{align*}
And, well as you can notice, by equating that expression to zero, I will get an inconsistency. Could you please provide me some hints to solve that optimization problem? Or point out my mistake...
calculus integration statistics derivatives statistical-inference
asked Dec 30 '18 at 22:49
c.uentc.uent
11
$endgroup$
add a comment |
$begingroup$
I am working on the derivations of EP for LDA, and I don't understand how the authors derived the last equations.
Basically, they get the following expression (the lower bound in eq 29):
$L=int(sum_iq_i(veclambda))log q_i(veclambda)dlambda + sum_{iw}n_{iw}int q_i(veclambda)log(sum_alambda_ap(w|a))dveclambda + C$
So, in order to solve the problem, they want to optimize $L$ by maximizing w.r.t. $alpha$ and $p(w|a)$. To do so, they need to get the derivative of $L$ and equal to zero.
For instance, when maximizing w.r.t. $p(w|a)$ ($frac{dL}{dp(w|a)}=0$), they get the following (eq 31):
$p(w|a)^{new} propto sum_in_{iw}int q_i(veclambda)frac{lambda_ap(w|a)}{sum_alambda_ap(w|a)}dveclambda$
However, I am getting something completely different when trying to evaluate $frac{dL}{dp(w|a)}$.
begin{align*}
frac{dL}{dp(w|a)} &= sum_i 0 + ldots + n_{iw}int q_i(veclambda)frac{d}{dp(w|a)}log(sum_alambda_ap(w|a))dveclambda +0+dots\
&=sum_i n_{iw}int q_i(veclambda)frac{lambda_a}{sum_alambda_ap(w|a)}dveclambda\
end{align*}
And, well as you can notice, by equating that expression to zero, I will get an inconsistency. Could you please provide me some hints to solve that optimization problem? Or point out my mistake...
calculus integration statistics derivatives statistical-inference
asked Dec 30 '18 at 22:49
c.uentc.uent
11
$endgroup$
add a comment |
$begingroup$
I am working on the derivations of EP for LDA, and I don't understand how the authors derived the last equations.
Basically, they get the following expression (the lower bound in eq 29):
$L=int(sum_iq_i(veclambda))log q_i(veclambda)dlambda + sum_{iw}n_{iw}int q_i(veclambda)log(sum_alambda_ap(w|a))dveclambda + C$
So, in order to solve the problem, they want to optimize $L$ by maximizing w.r.t. $alpha$ and $p(w|a)$. To do so, they need to get the derivative of $L$ and equal to zero.
For instance, when maximizing w.r.t. $p(w|a)$ ($frac{dL}{dp(w|a)}=0$), they get the following (eq 31):
$p(w|a)^{new} propto sum_in_{iw}int q_i(veclambda)frac{lambda_ap(w|a)}{sum_alambda_ap(w|a)}dveclambda$
However, I am getting something completely different when trying to evaluate $frac{dL}{dp(w|a)}$.
begin{align*}
frac{dL}{dp(w|a)} &= sum_i 0 + ldots + n_{iw}int q_i(veclambda)frac{d}{dp(w|a)}log(sum_alambda_ap(w|a))dveclambda +0+dots\
&=sum_i n_{iw}int q_i(veclambda)frac{lambda_a}{sum_alambda_ap(w|a)}dveclambda\
end{align*}
And, well as you can notice, by equating that expression to zero, I will get an inconsistency. Could you please provide me some hints to solve that optimization problem? Or point out my mistake...
calculus integration statistics derivatives statistical-inference
asked Dec 30 '18 at 22:49
c.uentc.uent
11
$endgroup$
I am working on the derivations of EP for LDA, and I don't understand how the authors derived the last equations.
Basically, they get the following expression (the lower bound in eq 29):
$L=int(sum_iq_i(veclambda))log q_i(veclambda)dlambda + sum_{iw}n_{iw}int q_i(veclambda)log(sum_alambda_ap(w|a))dveclambda + C$
So, in order to solve the problem, they want to optimize $L$ by maximizing w.r.t. $alpha$ and $p(w|a)$. To do so, they need to get the derivative of $L$ and equal to zero.
For instance, when maximizing w.r.t. $p(w|a)$ ($frac{dL}{dp(w|a)}=0$), they get the following (eq 31):
$p(w|a)^{new} propto sum_in_{iw}int q_i(veclambda)frac{lambda_ap(w|a)}{sum_alambda_ap(w|a)}dveclambda$
However, I am getting something completely different when trying to evaluate $frac{dL}{dp(w|a)}$.
begin{align*}
frac{dL}{dp(w|a)} &= sum_i 0 + ldots + n_{iw}int q_i(veclambda)frac{d}{dp(w|a)}log(sum_alambda_ap(w|a))dveclambda +0+dots\
&=sum_i n_{iw}int q_i(veclambda)frac{lambda_a}{sum_alambda_ap(w|a)}dveclambda\
end{align*}
And, well as you can notice, by equating that expression to zero, I will get an inconsistency. Could you please provide me some hints to solve that optimization problem? Or point out my mistake...
calculus integration statistics derivatives statistical-inference
calculus integration statistics derivatives statistical-inference
asked Dec 30 '18 at 22:49
c.uentc.uent
11
asked Dec 30 '18 at 22:49
c.uentc.uent
11
asked Dec 30 '18 at 22:49
c.uentc.uent
11
asked Dec 30 '18 at 22:49
c.uentc.uent
11
asked Dec 30 '18 at 22:49
c.uentc.uent
11
11
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