# Adaptive information source selection during hypothesis testing

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3 1 Information Request Type Positive Evidence Probability of request type Negative Evidence Specific Instance Figure 3: Sample view from the same condition as Figure 2 after information requests. Requests that produced an instance within a hidden rectangle are colored red, while instances outside the hidden rectangles are colored blue. After each information request the participant had to move the rectangles into a position that is consistent with all of the information. This is one possible arrangement of rectangles consistent with all information requests. were instructed to try each information source at least once and familiarize themselves with the game. We analyze results from the second and third games only. Results Evidence requests Does the proportion of positive and negative evidence requests change as a function of the sparsity of the hypotheses? Figure 4 shows a significant effect of hypothesis sparsity on the proportion of requests for both positive evidence (F(1,499) = 21.95, p <.5) and negative evidence (F(1, 499) = 14.6, p <.5). As hypotheses became less sparse, positive evidence requests were increasingly less likely: the slope of the line of best fit was -.. The pattern was the opposite for negative evidence, with more negative evidence requests occurring when the hypotheses were less sparse (the slope of the line of best fit was.4). With the introduction of the possibility of instance requests, one might ask whether the relative proportion of positive to negative evidence requests parallels that found in Hendrickson et al. (in prep) and shown in Figure 1. Indeed, looking only among actions in which participants made evidence requests, shown in Figure 5, reveals a very similar pattern to what was found previously. The relationship between the proportion of positive to negative evidence requests is characterized by similar slopes of the line of best fit (.57 in this study,.6 in the previous one) and intercept at % sparsity (.71 in this study,.61 in the previous one). Instance requests The instance requests, shown by the dashed grey line in Figure 4, were made frequently in all spar- Figure 4: The proportion of information requests across hypothesis sparsity. Across all levels of sparsity participants made all three types of requests. The proportion of positive and negative evidence requests were strongly influenced by hypothesis sparsity, The effect of sparsity on on instance requests was much less strong. The solid line shows positive evidence requests, the dotted line shows negative evidence, and the dashed line shows instance requests. Error bars indicate standard error. sity conditions. Though people were very sensitive to sparsity when making evidence requests, they appeared to be less so when making instance requests. Although the effect of sparsity was significant (F(1, 499) = 6.4, p =.1), the slope of the line of best fit was less steep at -.1. People were slightly more likely to ask about specific instances when the ships were smaller. Were people selecting good instances when they requested a specific instance? As discussed by Navarro and Perfors (211), an instance request maximizes expected information gain if the proportion of hypotheses consistent with it is %. If the sparsity is 2%, a random instance will be within a rectangle in an average 2% of hypotheses: selecting an instance that is within a rectangle in closer to % of hypotheses improves information gain. Conversely, if the sparsity is 8%, to increase information gain the learner should select an instance with a lower than average probability of a being within a rectangle. To compute the proportion of hypotheses consistent an instance it is necessary to know what hypotheses have still not yet been ruled out; this is computationally intensive because this must be calculated for every request made by every participant on every game. At each of these points we therefore simulated a pseudo-random subset of all possible valid rectangle-position hypotheses. These were used to estimate, for each information request by each participant, the probability that it was located within a rectangle across all possible rectangle-position hypotheses games were simulated in most conditions due to computational complexity. Sample hypotheses were generated pseudo- 69

4 Probability of positive evidence 1 Probability within rectangle 1 Instances Chosen Random Instances Figure 5: People are sensitive to the sparsity of hypotheses when considering only evidence requests. Even if participants had the option of requesting instances, the ratio of positive to negative evidence requests matches closely the proportion of positive requests when instance requests are not available, as shown in Figure 1 from Hendrickson et al. (in prep). Error bars indicate standard error Figure 6: People do not select instances randomly. The y axis shows the probability than an instance selected by a participant is actually in a hidden rectangle. The dotted line reflects what would happen if people were choosing them randomly: it would parallel hypothesis sparsity. The probability of selecting a positive instance does depend on hypothesis sparsity (slope=.46). Error bars indicate standard error. The probability of a chosen instance being within a rectangle was strongly influenced by the sparsity of the hypothesis space (F(1, 378) = 17, p <.5, slope =.46) as shown by the solid line in Figure 6. Despite being influenced by the sparsity of the hypotheses, instances were not selected randomly if they were, one would expect that the probability of any instance being located within the rectangle would be the same as the sparsity of the condition (this is reflected in the dashed line in Figure 6). We find that the instances people requested were shifted towards being equally likely to be within or outside a rectangle. Expected utility of all information requests Why does the rate of positive and negative evidence requests depend on the sparsity of hypotheses, even when instance requests are taken into account? One possible explanation is that regardless of request type, people use a heuristic that picks information sources which are more likely to produce positive evidence when hypotheses are sparse. This heuristic predicts the linear changes seen in Figure 4 and Figure 5. An alternative explanation, first proposed by Oaksford and Chater (1994), is that people choose information sources based on how useful they expect those sources to be, and that this utility is sensitive to sparsity. One method to assess utility in this task is to look at the expected change in uncertainty about which hypothesis is correct for each information request option (for further discussion about different metrics of utility during hypothesis randomly by considering the set of possible hypotheses generated by permuting the locations of all rectangles to create a set of 12 base hypotheses then shifting each rectangle in each of those base hypotheses up to two grid cells in each direction to generate candidate hypotheses. testing see Markant and Gureckis (212)). Does the relative utility of evidence requests and instance requests drive information requests beyond hypothesis sparsity? We address this by using the results of the previous simulations. For each possible information request on each action we estimated the expected reduction in the set of remaining hypotheses. 2 We then compared the utilities of each kind of information request. If people were sensitive to relative utility, one would expect that they would be more likely to choose an evidence request when it had higher utility than an instance request, and to choose an instance request when it had higher utility than an evidence request. As Figure 7 shows, that is exactly what people do. The left panel shows actions in which the utility of evidence is higher than the utility of any instance; at all sparsity levels people are more likely to make evidence requests. 3 The right panel shows the opposite situation; people were more likely to make instance requests when an instance had higher utility than either evidence type. Across all levels of hypothesis sparsity, the most frequently chosen information response option was also the highest utility option. This suggests people were sensitive to the relative utility of each information request type and used this information appropriately. For most levels of sparsity 3 Figure 7 shows that at some times during a game instance requests had higher utility and at some times evidence requests did. How does utility depend on time, and why? Intuitively, positive evidence requests have higher utility than any instance when getting one 2 Following Austerweil and Griffiths (28) the expected reduction is weighted by the probability of obtaining each evidence type. 3 For sparsity levels 4 and, evidence requests were never more useful than instance requests. 61

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