class: center, middle, inverse, title-slide # Bayesian Decision Theory ## Statistical Summit at Lundbeck ### Claus Thorn Ekstrøm<br>(joint work with Corine Baayen)<br>UCPH Biostatistics<br>.small[<a href="mailto:ekstrom@sund.ku.dk" class="email">ekstrom@sund.ku.dk</a> ] ### Friday 22nd, 2021 --- class: animated, fadeIn layout: true --- background-image: url(pics/nvp.jpg) background-size: 112% --- # Formalizing decision making 1. Frame decision space - which decisions can I make? 2. Quantify information - what information do we have available to make the decision? 3. Quantify how good/bad a decision is given information Once we know 1-3: How do we make an optimal decision? --- background-image: url(pics/examples.jpg) background-size: 95% --- background-image: url(pics/choice2.png) background-size: 95% # Framing the decision space .pull-left[ Let `\({\cal A}\)` be the space where all possible decisions live. Can be discrete, continuous, multivariate, ... ] .pull-right[ ] --- # Examples - decision space `\({\cal A}\)` 1. **Used car**: Decision could be price `\([0, \infty)\)` 2. **Drug dosage**: Decide dosage `\([0, \infty)\)` 3. **Driverless car**: Decide route `\((X_t, Y_t)\)` for temporal `\(t\)`. --- background-image: url(pics/info.png) background-size: 95% # Quantify information .pull-left[ What will we use to choose between different decisions? Let `\({\cal \Theta}\)` be the space where all the information live. A piece of information will be `\(\theta \in \Theta\)`. ] .pull-right[ ] --- # Examples - information space 1. **Used car**: Run through database of old sales of roughly same make and model and price, and predict probability that it did sell. `\(\theta_\text{price} \in [0, 1]\)` 2. **Drug dosage**: For example use smooth function relating adverse event rate to drug dosage. `\(\theta \in {\cal C}^\infty\)`. From similar studies? From ethics? From risks? 3. **Driverless car**: Use knowledge of objects in 2D space. `\(\Theta\)` complicated. --- # What makes a good/bad decision? What do we want? Quantify loss `$${\cal L} : \Theta \times {\cal A} \rightarrow \mathbb{R}$$` `\(\cal L\)` ties together our decision space and our information space and quantifies how good/bad a decision is. --- # Examples - loss function 1. **Used car**: Maximize expected return: `\({\cal L}(\theta, a) = -\theta a\)`. 2. **Drug dosage**: Want to decrease adverse event rate. More loss with higher rate: `\({\cal L}(\theta, a) = \theta(a)\)` 3. **Driverless car**: Stay as far away as possible from the objects: `\({\cal L}(\theta, a) = - \|a - \theta \|_2^2\)` at any time `\(t\)` --- # Adding uncertainty Typically no perfect information, instead prior belief (distribution) of `\(\theta\)`. `$$P(\theta)$$` Or even better ... bring the old reverend in play and use the `$$P(\theta|X)$$` --- # Expected loss So how do we figure out the loss associated with individual decisions when we don’t even know the information we want to use to make a decision? `$$\text{Expected loss}(a) = \int_\Theta {\cal L}(\theta, a) p(\theta) d\theta$$` -- Simulate away ( `\(N\)` samples from `\(p(\theta|X)\)` ) `$$\text{Expected loss}(a) \approx \frac{1}{N}\sum_{i=1}^N {\cal L}(\theta^{(n)}, a)$$` --- # Optimal decision Pick the decision, `\(\hat a\)`, with lowest expected loss `$$\hat{a} \approx \arg\min_{a \in {\cal A}} \frac{1}{N}\sum_{i=1}^N {\cal L}(\theta^{(n)}, a)$$` Can get computationally demanding very quickly. --- # Example - used car Produce model to obtain `\(P(\text{sold} | \text{price}, x)\)`. `$$Y_i \sim \text{Bernoulli}(\pi_i)$$` `$$\pi_i = \text{Logit}^{-1}(\eta_i)$$` `$$\eta_i = \beta_0 + \beta_1 x_\text{price} + \beta_2 x_\text{mileage} + \beta_3 x_\text{year}$$` With priors on top, eg `\(N(-2, 1), N(-1, 1), N(1, 1)\)` --- # Used car <!-- --> --- background-image: url(pics/ibdt.png) background-size: 75% --- # Input information <img src="pics/R.png" width="80%" /> --- # Go / No go ? <!-- --> --- # Group workshop 1. Imagine that you want to do a phase II trial 2. Imagine that you want to do a pair of phase III trials 3. Imagine that you want to do a complete phase II + phase III trial Sketch out the decision space, the information space, and the loss function. --- # Future consideration How to construct the framework for pharmaceutical decision making? How to initiate discussions about the loss function? Is minimizing expected loss always desirable? How to we integrate this in a company-wide decision making process, where other products/fixed budgets influence decision making?