.

_bibliography/auto-int.bib

@@ -0,0 +1,25 @@
+---
+---
+References
+==========
+
+@misc{baydin2018automaticdifferentiationmachinelearning,
+      title={Automatic differentiation in machine learning: a survey}, 
+      author={Atilim Gunes Baydin and Barak A. Pearlmutter and Alexey Andreyevich Radul and Jeffrey Mark Siskind},
+      year={2018},
+      eprint={1502.05767},
+      archivePrefix={arXiv},
+      primaryClass={cs.SC},
+      url={https://arxiv.org/abs/1502.05767}, 
+}
+
+@article{JMLR:v21:19-346,
+  author  = {Shakir Mohamed and Mihaela Rosca and Michael Figurnov and Andriy Mnih},
+  title   = {Monte Carlo Gradient Estimation in Machine Learning},
+  journal = {Journal of Machine Learning Research},
+  year    = {2020},
+  volume  = {21},
+  number  = {132},
+  pages   = {1--62},
+  url     = {http://jmlr.org/papers/v21/19-346.html}
+}

_drafts/technical-posts/2020-05-31-learning-to-teach.md

@@ -1,6 +1,8 @@
 ---
 layout: post
 title: Helping others learn
+categories:
+    - "proposal"
 ---
 
 With our greater understanding of learning we can now train dogs to do X. (insert videos of clever dogs) [jenga](https://www.youtube.com/watch?v=1kl3Y82qRDg), [objects](https://www.youtube.com/watch?v=Ip_uVTWfXyI) ...

_drafts/technical-posts/2022-10-13-autoint.md

@@ -1,21 +1,76 @@
 ---
 title: "Automatic integration"
 subtitle: "Where is it?"
+layout: post
 categories:
     - "proposal"
 scholar:
   bibliography: "auto-int.bib"
 ---
 
-Within machine learning, automatic differentiation proves its awesome utility.
+Automatic differentiation has proven its awesome utility, especially within machine learning {% cite baydin2018automaticdifferentiationmachinelearning %}.
+Fasciliting the training of deep neural networks, it has become a staple of the field.
 
 However, as far as I know, there is no analog for automatic integration.
 Why not?
 
+***
+
+<!-- uses / motivation -->
+
+Firstly, what kind of problems could automatic integration solve?
+
+<!-- bayesian -->
+Two distributions.
+
+$$
+\begin{align}
+p(y| x) &= \frac{p(x| y) p(y)}{p(x)} \\
+p(x) &= \int p(x| y) p(y) dy
+\end{align}
+$$
+
+For example, we have a prior distribution over likely diseases $p(y)$ and a likelihood function $p(x| y)$ which tells us how likely the observed MRI $x$ is given the disease $y$.
+Where $p(y)$ is derived from the population statistics. And $p(x | y)$ is a generative model learning from data.
+<!-- y= discrete variable. 1000 different potential diseases -->
+<!-- y= continuous variable. how long are they likely to survive / how long until surgery is needed. measured in weeks. e.g 10.24 weeks. -->
+
+
+
+
+
+$$
+\int p(x; \theta) f(x; \phi) dx
+$$
+
+where $p(x; \theta)$ is a probability distribution and $f(x; \phi)$ is a function.
+{% cite JMLR:v21:19-346 %}
+
+or the Wave equation
+
+$$
+\frac{\partial^2 u}{\partial x^2} = c^2 \Delta u
+$$
+
+where $u$ is a function and $\Delta$ is the Laplacian operator.
+And in general, any PDE.
+
+
+
+
+<!-- what is AD? -->
+
 The key to automatic differentiation is the chain rule.
 It allows us to compute the derivative of a function by breaking it down into a sequence of elementary operations.
 Calculating their derivatives independently, we can then combine them to get the derivative of the original function.
 
+
+
+
+<!-- current state of integration tools -->
+
+
+
 ## Bibliography
 
 {% bibliography --cited %}