Take the following two equations...

\begin{align}
J_\alpha(p||q) = \frac{1}{\alpha(1-\alpha)} \int_x \alpha p(x) +(1-\alpha)q(x)-p(x)q(x)^{1-\alpha}dx
 && \text{KL}(q || p) = \int_x q(x) \log \frac{q(x)}{p(x)} dx
\end{align}

Let $q(x)$ be the pdf for a Gaussian,
$\mathcal{N}(\mu, \Sigma)$, and $p(x)$ be unknown. I have demonstrated numerically that both equations have the same extrema for $\mu$, for $-1 \lt \alpha \lt 0$. (If you would like to see the notebook for that, I can send it too. For now just assume it is true)

I would now like to show it analytically. I thought it would be relatiely straight forward to take the partials and show that the same $\mu^*$ minimizes both. 

\begin{align}
\frac{\partial J_\alpha}{\partial \mu} &= \frac{1}{\alpha} \int_x q_{\mu}(x)\bigg(1-\frac{p(x)}{q(x)^{\alpha}}\bigg)dx
 && \frac{\partial \text{KL}}{\partial \mu} = & \int_x q_{\mu}(x) \bigg(\log \frac{q(x)}{p(x)}+1 \bigg) dx \\
 \frac{\partial J_\alpha}{\partial \mu} &= \frac{1}{\alpha} \int_x q(x)\Sigma^{-1}(x-\mu)\bigg(1-\frac{p(x)}{q(x)^{\alpha}}\bigg)dx
 && \frac{\partial \text{KL}}{\partial \mu} = & \int_x q(x) \Sigma^{-1}(x-\mu) \bigg(\log \frac{q(x)}{p(x)}+1 \bigg) dx
\end{align}


After observing that $\int_x q_{\mu}(x) dx = 0$, via...


\begin{align}
\int_x q_{\mu}(x) dx &= \int_x q(x) \Sigma^{-1} (x-\mu)dx \\
&= \Sigma^{-1} \bigg(\int_x q(x) x dx - \mu \int_x q(x) dx \bigg)\\
&= \Sigma^{-1} \big(\mu - \mu \big) \\
&= 0\\
\end{align}


Both equations reduce to...


\begin{align}
\frac{\partial J_\alpha}{\partial \mu} = -\Sigma^{-1} \mathbb{E}_{q} \bigg[ \frac{p(x)}{\alpha q(x)^{\alpha}}(x-\mu) \bigg]
 && \frac{\partial \text{KL}}{\partial \mu} = -\Sigma^{-1} \mathbb{E}_{q} \bigg[ \log \frac{p(x)}{q(x)}(x-\mu) \bigg]
\end{align}


And from here, setting them both equal to zero, we can derive closed form solutions...

\begin{align}
\mu_J = \frac{ \mathbb{E}_{q} \bigg[ \frac{p(x)}{q(x)^{\alpha}}x \bigg]}{\mathbb{E}_{q} \bigg[ \frac{p(x)}{q(x)^{\alpha}} \bigg]}
&& \mu_{KL} = \frac{ \mathbb{E}_{q} \bigg[ \log\big(\frac{p(x)}{q(x)}\big)x \bigg]}{\mathbb{E}_{q} \bigg[ \log\big(\frac{p(x)}{q(x)}\big) \bigg]}
\end{align}


```python

```

3196e409990c0c8e23c3cddcba7a7b48533e5cf1
ipynb
Jupyter Notebook
dissertation/.ipynb_checkpoints/Minimizing J finds the peaks of P-checkpoint.ipynb
mathnathan/notebooks
63ae2f17fd8e1cd8d80fef8ee3b0d3d11d45cd28
MIT
2019-12-04T11:04:45.000Z
2019-12-04T11:04:45.000Z
dissertation/.ipynb_checkpoints/Minimizing J finds the peaks of P-checkpoint.ipynb
mathnathan/notebooks
63ae2f17fd8e1cd8d80fef8ee3b0d3d11d45cd28
MIT
dissertation/.ipynb_checkpoints/Minimizing J finds the peaks of P-checkpoint.ipynb
mathnathan/notebooks
63ae2f17fd8e1cd8d80fef8ee3b0d3d11d45cd28
MIT
Qwen/Qwen-72B
1. YES
2. YES