# Bayesian Optimal Experimental Design (BOED)

A model-based approach to choose some design $$d \in \mathcal{D}$$ which maximizes the information gained about some model parameters $$\theta$$ from the outcome $$y$$ after performing experiment $$d$$. Given a predictive model $$\mathbb{P}(y|\theta,d)$$ and prior $$\mathbb{P}(\theta)$$, the objective is to choose a design $$d$$ that yields the greatest reduction in our uncertainty - which as a function of $$d,y$$ can be formulated as $\text{IG}(y,d) = \mathbb{H}[\mathbb{P}(\theta)] - \mathbb{H}[\mathbb{P}(\theta|y,d)]$

Where $$\mathbb{H}$$ is the entropy operator*. This function is called the Information Gain, and integrating out $$y$$ gives us the Expected Information Gain (EIG). $\text{EIG}(d) = \mathbb{E}_{\mathbb{P}(y,\theta|d)}\big[\log \frac{\mathbb{P}(y|\theta,d)}{\mathbb{P}(y|d)}\big]$

Which is the same as the mutual information between $$\theta$$ and $$y$$ given $$d$$. The Bayes optimal design $$d^*$$ is then defined as: $d^* = argmax_{d\in \mathcal{D}} \text{EIG}(d)$

## Thoughts

• A bit vague here to be honest.

Created: 2022-03-13 Sun 21:44

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