# Active Data Selection and Information Seeking
## 6 Clinical Trials  
In our final example, we demonstrate the potential utility of the ideas outlined above in the context of a more concrete example. Adaptive Bayesian clinical trials [40] have been increasingly popular in efficiently answering questions about medical interventions in rapidly evolving healthcare situations. A prominent example of this was the use of an adaptive Bayesian methodology to assess interventions during the 2014 West African Ebola outbreak [41]. A key argument made in favour of this design was that in a rapidly developing epidemic it is necessary to be able to efficiently compare several possible treatments and to be able to adapt as new information emerges [42]. For instance, we may wish to stop one treatment arm, or perhaps reallocate participants to another treatment arm, if there is early evidence of the inferiority or superiority of this treatment, respectively. These benefits are not restricted to the management of epidemics but may apply more broadly. See, for example [43; 44; 45; 46; 47; 48; 49]. We have two reasons for choosing a clinical trial design as our final example. The first is that they are common hypothesis-driven experimental designs with resource limitations. The second is that they offer us an opportunity to go beyond the illustrative linear models used above and force us to deal with more expressive model structures that require  
Figure 8: Trajectory of beliefs. This figure captures several snapshots of the time-evolving inference process summarised in Figure 7. The first row shows the inferences made following the first data sample. The second row shows the inferences made at time step 50. The third shows the inferences made by the end of the time period. Note the improvement in predictive precision in the second and third rows, corresponding to the use of prior information from previous time steps. pragmatic approximations to be able to apply the methods outlined above efficiently. Our hope is to demonstrate that use of expected information gain does not restrict us only to the simplest model classes but has broad applicability in real-world problems. The active sampling approach advocated in this paper offers two main opportunities to augment adaptive trial designs. First, it allows us to adapt the design (e.g., inclusion criteria and assignment to a treatment arm) to maximise the information we obtain about treatment efficacy. Second, it allows us to balance this information gain against various costs, namely, stopping the trial when the potential information gain no longer exceeds the cost of acquiring further data. Alternatively, it might include a preference for good clinical outcomes and aversion for bad outcomes--prompting adjustments of the allocations between treatment arms to maximise benefit as we infer more about the effect of the intervention. This blurs the line between clinical trial and public health intervention and can be seen as analogous to animal behaviour that is never fully exploitative or explorative but is a balance between the two. Our setup is as follows: for each new cohort of participants, we decide upon the randomisation ratio to adopt and the time(s) at which we follow them up. To simplify for illustration purposes, we assume each cohort is followed up only once, noting that the same principles can be applied to multiple follow-up points. Our generative model can be expressed as follows:  
\[ \begin{array}{l}p(y|\theta,v,d,\pi_{\mathrm{r}})=Cat([\rho,1-\rho])\\ p(d)=Cat(\left[\frac{1}{2},\frac{1}{2}\right])\\ p(v|\pi_{\mathrm{r}})=Cat(\pi_{\mathrm{r}})\\ p(\theta)=\mathcal{N}(\eta_{\mathrm{r}},\mathbb{E}_{\theta})\\ \rho=\prod\limits_{i}^{\pi_{\mathrm{r}}}\prod\limits_{1+e^{-X_{i}^{2}\theta}} \\ X_{i}=\left\{\phi_{i},d,v\right\}\end{array}  \tag{14}\]  
In this model, the data (\(y\)) are binary values reflecting survival up to the measurement time or death before this point. The probability of survival to a given time is the product of the survival probabilities for all previous time steps.