# Active Data Selection and Information Seeking
## 1 Introduction  
When we look at the world around us, we are implicitly engaging in a form of active data sampling (also known as active sensing or active inference [1; 2; 3; 4; 5]). Despite the abundance of visual data available to us at any one moment, our visual systems are adapted to select (or foveate) only a small portion of these data. The advantage of this is that our brains can scale the processes that underwrite perceptual inference to large sensory datasets. All that is needed is an efficient means of sequentially selecting those data to best optimise our inferences about their causes [6; 7].  
This paper suggests that the same process of active sampling can be (and often is) used in situations in which there is a cost associated with collecting new data or analysing a very large dataset. Figure 1 illustrates the basic idea, akin to perception-action cycles in biology [8], of alternating between drawing inferences from the data we have available and selecting new data based on these inferences. This cycle is implicitly part of the scientific process, in which our pre-existing understanding is used to motivate experiments whose results update our understanding to motivate future experiments, and so on.  
Within the schematic shown in Figure 1, we highlight the key quantities in this cyclical process. They include the causes of data (which may include model parameters or hidden states) (\(\theta\)) and the data (\(y\)). Bayesian inference involves taking our prior beliefs about causes \(p(\theta)\), combining these with the likelihood (\(p(y\,|\,\theta)\)) of observed data, and arriving at a posterior belief (\(p(\theta\,|\,y)\)) as to the causes given the data. The priors and likelihoods together comprise what is known as a generative model, which can be represented in a number of ways, including as the Bayesian network shown in the centre of the graphic. Here, an arrow from one variable to another indicates a conditional probability distribution of the latter given the former.  
Careful data selection is especially important when we consider the problems associated with very large datasets of the sort that are now ubiquitous in machine learning and artificial intelligence settings. Computations with such datasets can be costly in terms of the hardware and computing power required, and their energy consumption raises important sustainability questions [9; 10; 11; 12]. Taking inspiration from the approach evinced by natural selection--that of sequentially selecting small amounts of sensory data--may help to alleviate the costs associated with big-data analysis.  
To optimise data selection, we first need to identify an appropriate optimality criterion. In what follows, we base this upon the idea of expected information gain--a measure used in optimisation of experimental design [13; 14], feature selection [15], and accounts of biological information-seeking and curiosity-driven behaviour [16; 17]. The idea behind this measure is that we form beliefs about hidden states or parameters in our model of how data are generated. Expected information gain is the degree to which we anticipate changing our beliefs under a given experimental design or data-sampling strategy. The implication is that optimisation of beliefs--and of data selection--work in tandem, as in Figure 1, and that both depend upon our model of how data are generated.  
In what follows, we consider the form this model, and therefore data-selection, might take in different settings, starting with abstract function-approximation and progressing to a more realistic example based upon the idea of an adaptive clinical trial. Before we move to these examples, we unpack the basic theory behind active data selection. We outline the basic principles behind Bayesian inference, the role of generative models, and the formulation of expected information gain. Our overall aim is to provide an intuitive overview of the principles that underwrite active data selection, and to illustrate this with some simple examples.