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Confidence drives a neural confirmation bias

Abstract

A prominent source of polarised and entrenched beliefs is confirmation bias, where evidence against one’s position is selectively disregarded. This effect is most starkly evident when opposing parties are highly confident in their decisions. Here we combine human magnetoencephalography (MEG) with behavioural and neural modelling to identify alterations in post-decisional processing that contribute to the phenomenon of confirmation bias. We show that holding high confidence in a decision leads to a striking modulation of post-decision neural processing, such that integration of confirmatory evidence is amplified while disconfirmatory evidence processing is abolished. We conclude that confidence shapes a selective neural gating for choice-consistent information, reducing the likelihood of changes of mind on the basis of new information. A central role for confidence in shaping the fidelity of evidence accumulation indicates that metacognitive interventions may help ameliorate this pervasive cognitive bias.

Introduction

The philosopher Bertrand Russell opined “The most savage controversies are about matters as to which there is no good evidence either way”. While this view applies in some situations, even more troubling are instances where polarization and entrenchment of opinion persists in the face of contrary evidence, exemplified by debates on climate change and vaccinations. This polarization is most evident when opposing parties are highly confident in their positions1,2. A psychological-level explanation for such entrenchment is the idea that people selectively incorporate evidence in line with their beliefs, known as confirmation bias3. Although an extensive literature has documented this bias in behaviour3,4, the underlying cognitive, computational and neuronal mechanisms are not understood.

So far, an investigation of confirmation bias has been restricted largely to scenarios involving complex real-world beliefs such as political attitudes4,5,6. However, the complexity of such higher-order beliefs makes it difficult to disentangle the various contributors to biased information processing. For instance, people may have a strong personal investment in their political opinions, leading to a significant motivation to discount new information that goes against their beliefs. Intriguingly, confirmation biases have recently been demonstrated in low-level perceptual tasks7,8,9, that are unlikely to evoke such motivated reasoning. These studies indicate a source of confirmation bias may be a generic shift in the way the brain incorporates new information. Here we adopt such a task to study the computational and neural basis of post-decisional shifts in sensitivity to choice-consistent information.

Perceptual decision-making is well-described using sequential sampling models which assume the brain accumulates noisy evidence for each choice option to a decision bound10. This accumulation process is thought to be supported by neuronal populations in parietal and prefrontal cortex11,12. Importantly, while perceptual tasks allow tight control over the processes involved, they also permit generalisation to more complex decisions13,14,15, and similar principles appear to underlie choice and confidence formation in both simple and more complex tasks16,17. However while the processes underlying perceptual decision-making have been studied in detail, little is known about the mechanisms governing accumulation of evidence after a choice has been made, or how such processing is shaped by pre-existing beliefs and confidence7,17,18,19,20,21,22,23.

Here we combine theoretical models and neural metrics to identify alterations in post-decisional processing that may contribute to the phenomenon of confirmation bias. Across all experiments, participants were presented with a sample of moving dots (pre-decision evidence) before indicating their initial decision (motion to the left or right) and confidence in their choice (see Fig. 1a). They were then presented with a second sample of moving dots (post-decision evidence) before making a final choice and providing a confidence estimate. Importantly, pre- and post-decision evidence always indicated the same direction of motion such that post-decision evidence was helpful. Accordingly, an ideal Bayesian observer should use post-decision evidence to change its mind after initial mistakes (see Supplementary Note 1 for analysis of the adaptive usage of post-decision evidence), whereas a confirmation bias would blunt this belief flexibility13,18.

Results

Effects of confidence on changes of mind

In a first experiment we hypothesised that a confirmation bias would occur more often when people are highly confident in their original choice24,25,26. In order to dissociate subjective confidence from objective performance we used a psychophysical manipulation (“positive evidence”27, see Methods) to selectively boost participants’ confidence (mean difference = 0.024, CI = [0.008, 0.04], Cohen’s d = 0.21, t(27) = 3.0, p = 0.005, Fig. 1c) while leaving performance (mean difference = 0.006, CI = [−0.022, 0.034], Cohen’s d = 0.02; Bayesian t-test indicating equality: BF01 = 4.61; Fig. 1b) and reaction times (mean difference = −0.005, CI = [−0.029, 0.018], Cohen’s d = −0.04; Bayesian t-test indicating equality: BF01 = 4.51, Supplementary Fig. 5a) unaffected.

We next set out to test whether this boost in confidence influenced changes of mind. There were notable individual differences in the degree to which our manipulation boosted participants’ confidence (see Fig. 1c, d). Importantly, subjects who experienced a stronger confidence boost through the positive evidence manipulation also showed a stronger reduction in changes of mind (r = −0.69, p < 0.0001, see Fig. 1d), an effect not explained by an impact of positive evidence on accuracy or reaction time (p = 0.005 when controlling for these effects). This supports a notion that confidence drives reductions in changes of mind (see Supplementary Notes 5 and 6 for additional behavioural and magnetoencephalography (MEG) analyses that further confirm confidence as a critical driver of changes of mind).

Confidence induces a selective gain for confirmatory evidence

We next reasoned that confidence may reduce changes of mind by promoting a bias towards processing of confirmatory post-decision evidence. We sought to test this hypothesis by revealing the process through which confidence affects accumulation of post-decision evidence, applying a combination of drift-diffusion modelling and recordings of post-decisional fluctuations in a neural decision variable (DV) using MEG. We considered two potential mechanisms through which confidence might reduce changes of mind. First, confidence might reflect a shift in the starting point of post-decision accumulation to be closer to the bound associated with an initial decision, consistent with a continuation of pre-decisional evidence accumulation (influence on starting point; Fig. 2a upper panel). Second, confidence may induce selective accumulation of evidence in line with an initial decision (influence on drift rate; Fig. 2a lower panel)—a clear instance of confirmation bias.

Critically, these two mechanisms make different predictions in terms of the distributions of response times for the final decision8,9. We compared 10 drift-diffusion models (DDMs) that embodied these different predictions (see Supplementary Note 2 for a full model comparison). We employed accuracy coding such that the bounds correspond to a correct versus an incorrect decision, such that a positive drift-rate represents stronger integration of the presented (correct) motion direction. Note, by design, confirmatory post-decision evidence was received when an initial decision was correct, and disconfirmatory evidence when an initial decision was incorrect (Fig. 2b–d). In addition, in light of suggestions that confidence might also affect the separation of decision bounds, and thus the trade-off between speed and accuracy of subsequent decisions28,29 we allowed for a dependency of boundary separation on initial confidence in all models.

The models differed as to whether the starting point and/or drift-rate were affected by confidence (models 2–4), accuracy of the initial decision (models 5–7; i.e. correct = 1 and incorrect = −1, capturing a general confirmation bias) and their interaction (models 8–10; i.e. capturing a confirmation bias that depends on confidence). The winning model (Model 10, as indicated by the Deviance Information Criterion score; see Supplementary Fig. 2A) incorporated dependencies of starting point and drift-rate on all factors (confidence, initial decision and the interaction) and provided a good fit to the data (Fig. 2b, c):

$${\mathrm{Starting}}\,{\mathrm{point}} \sim {\mathrm{1}} + {\mathrm{confidence}} + {\mathrm{initial}}\,{\mathrm{decision}} \\ \,\,\,\,\,\,+ {\mathrm{confidence}} \times {\mathrm{initial}}\,{\mathrm{decision}}$$
(1)
$${\mathrm{Drift}} \hbox{-} {\mathrm{rate}} \sim {\mathrm{1}} + {\mathrm{post}} \hbox{-} {\mathrm{decision}}\,{\mathrm{evidence}}\,{\mathrm{strength}} \\ \,\,\,\,\,\,+ {\mathrm{confidence}} + {\mathrm{initial}}\,{\mathrm{decision}}\ + {\mathrm{confidence}} \\ \,\,\,\,\,\,\,\times {\mathrm{initial}}\,{\mathrm{decision}}$$
(2)
$${\mathrm{Boundary}}\,{\mathrm{separation}} \sim {\mathrm{1}} + {\mathrm{confidence}}$$
(3)

After accounting for main effects, we observed a dependency of the starting point on the interaction between confidence and initial decision (95% equal-tailed interval = 0.08−0.18; Fig. 2d right hand panel), indicating participants started the accumulation process closer to the bound of the initial decision when highly confident in their choice. Even more striking was the discovery of a similar interaction effect on drift rate (95% equal-tailed interval = 0.11−0.26; Fig. 2d right hand panel) indicating participants selectively accumulated evidence supporting their initial choice, and were more likely to do so when they were more confident. While a confidence-related shift in starting point might reflect normative usage of pre-decision evidence (because high confidence in an initial decision might reflect greater pre-decision evidence accumulation, and thus be closer to a post-decisional bound), an influence of confidence on the drift rate is a clear instance of confirmation bias. Indeed, effects of the initial decision and confidence on the drift rate were more pronounced than those on the starting point (see Fig. 2 and Supplementary Note 3). Such a confirmation bias led to a boost in accumulation of the veridical motion direction following high-confidence correct decisions (as such information served to confirm the original choice), whereas it led to a reduction in evidence accumulation (manifest as a lowered drift rate) following high-confidence errors (as new information served to disconfirm an originally wrong decision).

Neural markers of post-decisional processing

While our DDM fits support a distinct influence of initial choice and confidence on post-decisional processing, they allow only indirect inference on how confidence affects evidence accumulation. To quantify this process more directly we used MEG to obtain a time-resolved neural metric of post-decision accumulation. Specifically, we trained a SVM classifier on brain activity (normalized amplitude of all MEG channels) at each time point (10 ms timebins) in the pre-decision time window (lasting 850 ms from stimulus onset to the presentation of choice options; note that the trial timeline for the MEG study differed slightly to the timeline presented in Fig. 1a, see “Methods” for details) to predict which choice (left or right) was made on each trial. We then applied the trained classifier to brain activity at the corresponding time point in the post-decision time window, enabling us to derive a probabilistic prediction of neural evidence favouring a leftward versus rightward decision (see Fig. 3a left panel). Positive values indicate prediction of a rightward decision and negative values indicate prediction of a leftward decision (see Fig. 3b). We next fitted a linear regression to the time series of classifier predictions within each trial (see Fig. 3a right panel) to obtain a trial-by-trial neural measure of the starting point (intercept) and drift rate (slope). These measures of neural evidence accumulation (slope) should be highly responsive to the presented motion direction during the post-decision period, and we show this was indeed the case (hierarchical regression: β = 0.07, t(8550) = 6.89, p < 10−11, Fig. 3b).

The slopes extracted from this analysis are signed, such that positive values indicate evidence for a rightward choice and negative values evidence for a leftward choice. In order to obtain an unsigned metric of evidence accumulation strength, we flipped the sign of slopes extracted from trials in which leftward motion was presented (we conducted the same flip for the intercept to obtain an unsigned metric of the starting point). This unsigned metric quantifies a propensity to correctly integrate the presented information, analogous to a drift rate in the accuracy coded DDM employed in Fig. 2.

A neural analogue of the drift-rate (or change in internal DV) should be related to characteristic features of the observer’s decision. Specifically, stronger internal evidence accumulation should be related to a higher likelihood of having made a correct decision12, faster response times10 and higher confidence11. In order to check whether our classifier predictions satisfied these criteria for metrics of internal evidence accumulation, we entered both the trial-by-trial slope and intercept of the post-decision accumulation process as simultaneous predictors in a hierarchical regression model to predict (a) reaction times, (b) choice accuracy and (c) confidence of the final decision (see Supplementary Note 4 for a similar analysis of the pre-decision period). Steeper slopes predicted faster reaction times (β = −0.007, t(8549) = −2.83, p = 0.005, see Fig. 3d), a higher likelihood of a correct decision (β = 0.16, t(8549) = 3.05, p = 0.002, see Fig. 3e) and higher confidence (β = 0.14, t(8549)=3.53, p = 0.0004, see Fig. 3f). We also observed significant effects of the intercept on accuracy (β = 0.1, t(8549)=2.0, p = 0.045, see Fig. 3e) and confidence (β = 0.12, t(8549) = 3.07, p = 0.002, see Fig. 3f) which is to be expected if participants maintain a representation of the evidence obtained in the pre-decision phase, and if the strength of this pre-decisional accumulation predicts the likelihood of being both correct and confident.

We next asked whether specific sensor clusters drive the classifier performance. Previous studies using EEG have identified a centro-parietal event-related potential (the centroparietal positivity or CPP) as a neural marker of internal evidence accumulation12,30,31. Accordingly, when identifying the features that contributed most strongly to classifier decoding accuracy (Fig. 3c) we also found that centro-parietal sensors make a disproportionate contribution to an ability to differentiate between left and right decisions.

Having identified a neural metric of evidence accumulation, we next turned to our central question of whether confidence induces a selective accumulation for choice-consistent information as measured using MEG. As hypothesized, we found that after high confidence (vs. low confidence) decisions, accumulation of neural evidence was facilitated if it was confirmatory, but largely abolished if it was disconfirmatory (Fig. 4a, b). In other words, our MEG analysis reveals that high confidence leads to post-decision accumulation becoming “blind” to disconfirmatory evidence. To formally quantify this effect, we entered the slope and starting point of neural evidence accumulation on each trial into hierarchical regression models with initial decision, high vs. low initial confidence and their interaction as predictors. We obtained a significant effect of initial decision (β = 0.042, t(8547) = 2.96, p = 0.003) and its interaction with confidence (β = 0.038, t(8547) = 2.64, p = 0.008, see Fig. 4c) on slope in the absence of effects on starting point (p > 0.05). Consistent with our DDM fits, these results indicate that a confidence-induced confirmation bias is predominantly driven by a selective accumulation of choice-consistent information.

We further reasoned that this approach may remain blind to changes in the starting point of post-decision evidence accumulation because of an asymmetry in evidence availability at the start of the pre- and post-decision phases. In other words, simply reapplying the (non-predictive) classifier weights obtained at the beginning of the pre-decision phase to the same time point in the post-decision phase could render the analysis pipeline blind to starting point offsets. To address this concern, we evaluated the extent to which the entire timecourse of classifier predictions obtained in the pre-decision phase generalised to the post-decision phase, without making assumptions about their relative timing32. This analysis provides insight into how putative processing stages identified in the pre-decision phase are reinstated in the post-decision phase, and crucially how this timecourse is affected by confidence. We found a cluster of time points in which a representation of the initial decision was activated earlier in the post- compared with the pre-decision phase when confidence was high (p = 0.01, corrected for multiple comparisons; Fig. 4d). Such early reinstatement of a later processing stage is consistent with confidence enhancing a representation of the initial decision (i.e. shifting a starting point towards the bound of the initial decision) or inducing an expectation for evidence supporting an initial decision at the beginning of the post-decision period. Together these results indicate that confidence changes both the neural representation of evidence for an initial decision at the beginning of the post-decision phase (analogous to a change in starting point) as well as enhancing the processing of evidence supporting an initial decision (analogous to a change in drift rate).

Discussion

By combining behavioural and neural modelling we provide experimental evidence that holding high confidence in a decision leads to a striking modulation of post-decision processing and the emergence of a behavioural confirmation bias. These findings are consistent with a neural representation of confidence acting as a top-down controller25 (see Supplementary Note 7 for further analysis) that selectively amplifies processing of choice-consistent information.

A confirmation bias in the current experiment was observed in low-level perceptual decisions with limited emotional or cognitive content, suggesting that choice-induced biases in evidence accumulation represent a core principle of neural information processing8,33. In most real-world decisions, additional motivational34 and social35 influences (e.g. not revising a decision in order to appear self-consistent) are presumably also in play. These additional influences may amplify, or add to, effects of confidence on post-decisional processing in complex ways. An advantage of starting with an investigation of confirmation biases within lower-level tasks is that the potential for such interactions can be minimized, allowing a focused investigation of the processes that drive post-decisional shifts in evidence accumulation.

Computational modelling of the evidence accumulation process enabled further arbitration between apparently optimal information usage and a confirmation bias, by separating the influence of confidence on post-decisional starting point and drift rate. A shift in starting point is potentially normative as it may reflect the contribution of stronger pre-decision evidence to higher confidence, indicating that participants incorporate both pre- and post-decision evidence when reaching a final decision. In contrast, the influence confidence on drift-rate represents a distortion in the integration of new evidence and thus a classic instance of confirmation bias.

In turn, our usage of MEG recordings in combination with machine learning classification revealed a neural marker of these shifts in post-decision evidence accumulation. This measure complemented our behavioural modelling results and yielded direct support for a hypothesis that confidence alters the way in which the brain accumulates new information, consistent with a selective gating of choice-consistent information.

In the current task, where new evidence is always helpful, this bias against incorporating conflicting post-decision evidence is normatively maladaptive. In other scenarios, however, where new evidence may be distracting and/or actively misleading, a confirmation bias might prove helpful. For instance, previous attempts to explain the value of selective evidence accumulation focused on its role in directing attention towards aspects of the environment with the highest potential for information gain36,37, or in increasing the robustness of decisions against the influence of noise26,38. However, the fact that confidence increases choice-consistent information processing goes against the idea that confirmation bias is itself driven by a need for certainty3,39. Instead, we observed the strongest confirmation bias when people were already confident in their decisions.

The study of cognitive biases has remained largely distinct from parallel efforts to understand the processes governing evidence accumulation in simple decisions. We suggest that extending models of evidence accumulation to post-decisional processing enables a unique window onto biases in higher-order cognition7. Intriguingly, recent evidence suggests that alterations in post-decision processing are predictive of higher-level attitudes such as beliefs about political issues13, suggesting that insights gained from the study of confirmation bias in simple decisions can be applied to understand the drivers of polarization and entrenchment across a range of societal issues. For instance, a central role for confidence in shaping the fidelity of evidence accumulation indicates that metacognitive interventions may be one route towards ameliorating this pervasive cognitive bias.

Methods

Participants

Each study contained a different group of participants. We analysed data from 28 participants in study 1 (Mage = 23.8; SDage = 6.3; 16 female) and 23 participants in study 2 (Mage = 25.7; SDage = 7; 12 female). Participants were excluded based on the following set of pre-defined criteria: using the same initial confidence rating more than 90% of time (N = 3 in study 1; N = 2 in study 2), performance below 55% or above 87.5% correct decisions in one of the pre-decision evidence conditions (see explanation of the experimental conditions below) indicating non-convergence of the staircase procedure (N = 3 in study 1; N = 2 in study 2).

For the MEG study 3, participants conducted an initial behavioural training session before being screened according to the same criteria reported above. MEG data of a final sample of 25 subjects was analysed (Mage = 24.6; SDage = 4.1; 16 female). Data of four subjects could not be analysed due to technical problems with recording triggers. As we applied machine learning classification algorithms to the neural data in order to decode decisions (left versus right) and confidence (high versus low) it was important that participants showed relatively balanced responses for these two categories. 2 subjects were excluded because they chose one response more than 80% of the time for either the decision or confidence.

In addition to a basic payment (£10 for behaviour and £20 for MEG) participants received a performance-based bonus (up to £5 for behaviour and £8 for MEG). All studies were approved by the Research Ethics Committee of University College London (#1260-003) and all subjects gave written informed consent.

Stimuli and experimental design

The psychophysical task was an adaptation of the task used by Fleming and colleagues18, and programmed in MATLAB 2012a (Mathworks Inc., USA) using Psychtoolbox- 3.0.14. Stimuli were random dot motion kinetograms (RDKs), viewed at a distance of approximately 45 cm. The RDKs were clouds of white dots (0.12° diameter) within a white circular aperture with a radius of 7° on a grey background that lasted for 350 ms. The direction of motion was rightward or leftward along the horizontal meridian. The speed of movement was 5° per second and the density of dots in the whole experiment was set to 60 dots per degree. Each set was replotted three apertures later in which a subset of dots, determined by the percent coherence, was offset from their previous location towards the target movement direction, and another subset was offset in the opposite direction, whereas the rest was replotted randomly.

Unlike in a classical RDK stimulus, dots moved coherently in both the target direction and the opposite direction. The remaining dots moved randomly (percentages described below). We used a psychophysical manipulation of positive evidence to dissociate subjective confidence from objective task performance27. In the high positive evidence (HPE) the proportion of dots moving in the incorrect direction was set to 15% and the proportion moving in the correct direction was a higher percentage, staircased to ensure the targeted performance level (see below). In the low positive evidence (LPE) condition the motion coherence of dots moving in the incorrect direction was set to 5%, whereas the dots moving in the correct direction was also staircased to ensure the same performance as in the HPE condition. The rationale for this manipulation was that accuracy and confidence are usually highly correlated, hindering specific claims about the unique role of confidence. The positive evidence manipulation enabled us to selectively increase confidence while keeping performance constant, thus making it possible to determine a direct effects of changes in confidence on post-decision processing.

All experiments adapted a full 2 (pre-decision positive evidence level) by 2 (post-decision evidence strength) factorial design yielding a total of 4 experimental conditions each corresponding to 90 trials. HPE and LPE stimuli were each followed by one of two post-decision evidence conditions (weak or strong). For the post-decision evidence a constant level of evidence in the incorrect direction was employed (i.e. we did not manipulate the overall amount of positive evidence in the post-decision phase). The post-decision coherence level in the incorrect direction was derived from the averaged staircased pre-decision values as [incorrect coherence LPE + incorrect coherence HPE]/2. Weak post-decision evidence stimuli were created by specifying correct-direction coherence as [staircased correct coherence LPE + staircased correct coherence HPE]/2. Strong post-decision evidence stimuli were then derived by multiplying this coherence level by a factor of 1.3.

In every study, participants first performed 180 trials of a calibration phase before performing the main task which consisted of 360 trials (behavioural studies) or 352 trials (MEG study).

In the calibration phase subjects judged whether the dots were moving to the left or to the right side of the screen, without rating their confidence or seeing additional post-decision evidence. The response had to be given within 1.5 s after stimulus offset. LPE and HPE stimuli were randomly interleaved. As described above, the coherence of the target direction was adapted with a staircase procedure to obtain a performance of 60% correct in study 1 and 71% correct in studies 2 and 340.

The main task had the same core structure for all studies with slight variations, explained below, to optimize each study for the specific research question and planned analysis. Participants were first presented with a moving dot stimulus before they indicated their initial decision (left or right) together with a confidence rating. In behavioural studies 1 and 2 the decision was indicated by pressing the left or right arrow key on the keyboard and was directly combined with a graded confidence rating (7-point sliding scale between 50% and 100%), where pressing the (same) arrow key again moved a slider along the confidence scale. In the MEG study, subjects first made a left versus right decision, before giving a binary high/low confidence rating. After this initial decision, participants received a second sample of moving dots (i.e. post-decision evidence) which was always in the same (correct) direction as the pre-decision evidence presentation, but of variable strength. Subjects were instructed that this evidence was bonus information that could be used to inform their final decision and confidence. After the post-decision evidence, participants were again asked to judge the motion direction and indicate their confidence.

Design alterations in behavioural study 2

In study 2 we optimized the experimental design to allow drift-diffusion modelling of the second/final decision. While in study 1 subjects had to withhold their final response for 300 ms after the offset of the post-decision evidence (i.e. responding was only possible after this delay), in study 2 participants were able to make their final response freely as soon as they had decided. This allowed us to use response times as a proxy for crossing a decision threshold, which would not have been possible if the response was delayed.

Design alterations in MEG study 3

In the MEG study, participants indicated their responses by pressing and up or down button on a keypad with their right thumb. We disentangled the participant’s decision (left/right and high/low confidence) from the motor response they had to perform (pressing the up or down key on a key pad), by randomising the mapping between decision options and key presses. Specifically, on any given trial leftward motion could be indicated by pressing the up key and on another trial by pressing the down key. Similarly, high confidence could be indicated in one trial by pressing the up key and in a different trial by pressing the down key. The mapping between decisions and motor responses was revealed once responding was possible, by presenting the letters L or R (and H or L for confidence ratings) above/below the horizontal plane. This approach ensured that decoding of motion direction was not trivially confounded by motor preparation signals. Additionally, we introduced delays of 500 ms after the presentation of each stimulus but before participants were informed about the response mappings to allow decoding analysis to be applied in a time window when subjects could form an abstract decision about motion direction but were not yet able to prepare a response.

Scoring and bonus payment

Participants were instructed to rate their confidence as a subjective probability of being correct and were rewarded according to the correspondence between their confidence and task accuracy. An incentive-compatible Quadratic Scoring Rule41 was applied equally to both the initial and final decisions:

$${\mathrm{Points}} = 100\, * \,\left[ {1 - \left( {{\mathrm{correct}}_{\mathrm{i}} - {\mathrm{confidence}}_{\mathrm{i}}} \right)^2} \right]$$
(4)

where correcti is equal to 1 on trial i if the choice was correct and 0 otherwise, and confi is the subject’s confidence rating on trial i. The Quadratic Scoring Rule is a proper scoring rule in that maximum earnings are obtained by jointly maximizing the accuracy of both choices and confidence ratings. This scoring rule also ensures that confidence is orthogonal to the reward the subject expects to receive for each trial: maximal reward is obtained both when one is maximally confident and right, and minimally confident and wrong. The points gained on each trial were summed and participants were given a £1 bonus payment for every 15,000 points earned. After each block participants were informed of their current total number of points. This was the only performance feedback that was given and subjects did not receive specific information regarding the correctness of their motion direction decisions.

Multilevel meditation analysis

A mediation analysis was carried out to examine whether the effect of positive evidence on changes of mind was mediated by a shift in confidence (see Supplementary Notes 5 and 6). We implemented a multilevel mediation model with subjects as random effects, using the Multilevel Mediation and Moderation (M3) Toolbox42. Mediation analysis assesses whether covariance between two variables (predictor and dependent variable) is explained by a third mediator variable. Significant mediation is obtained when inclusion of the mediator in the model significantly alters the slope of the predictor-dependent variable relationship (evaluated as the product of the predictor-mediator and mediator-dependent variable path coefficients). In a logistic regression model the two positive evidence conditions (i.e. coded as HPE = 2, LPE = 1) were entered as the predictor variable, changes of mind as the dependent variable (coded as change of mind = 1, no change of mind = 0) with confidence ratings as the mediator variable. We controlled for covariates that potentially could have had a confounding influence on these linkages such as accuracy, reaction time, post-decision evidence strength and the interaction between accuracy × post-decision evidence strength. The following effects of interest were simultaneously tested: the impact of positive evidence on confidence ratings (path a); the impact of confidence ratings on changes of mind, controlling for positive evidence (path b); and the formal mediation of positive evidence on changes of mind by confidence (path a × b). The direct effect of positive evidence on changes of mind before and after controlling for confidence was also estimated (paths c and c’, respectively). Parameter estimates for each path (a, b, c, a × b, c’) were obtained by bootstrapping 200,000 times with replacement, producing two-tailed p-values and 95% confidence intervals. In a control model in which the predictor and mediator variables were swapped, no mediation effect was found.

Drift-diffusion modelling

Drift-diffusion modelling was conducted in Python 2 using Jupyter Notebook (5.50). The model was fit using accuracy coding such that decision boundaries and reaction time distributions corresponded to those for correct and incorrect responses. However, by design, initially correct decisions led to confirming post-decision evidence (because the motion direction was always the same in the pre and post-decision periods) and initially incorrect decisions always led to disconfirming post-decision evidence.

Within the DDM there are two natural ways to account for biases in a decision process: by shifting the starting point towards one of the decision boundaries, or by altering the drift rate to induce a bias in the processing of information. We also considered the possibility that other factors (e.g. decision bound) could be altered, but in initial simulations such changes were unable to explain the observed behavioural patterns. Since it has been reported that confidence might affect boundary separation29, we included a dependency of the boundary separation on confidence in each of the models (note however that a symmetrical influence on boundary separation cannot explain any choice-dependent effects on changes of mind).

A hierarchical Bayesian variant of the DDM (hDDM) enabled us to investigate the dependencies of the model parameters on the initial decision and confidence on a trial-by-trial basis43. The hDDM simultaneously estimates individual parameters drawn from a group distribution using Markov-Chain Monte-Carlo methods. This procedure not only estimates the most likely value of the model parameters but also uncertainty in the estimate. The hDDM toolbox43 was used to compare 10 hDDMs. The best-fitting model was identified by comparing Deviance Information Criterion scores and ensuring that the wining model adequately fitted the qualitative data patterns (see Supplementary Note 2). A regression analysis was used to investigate the dependency of the starting point and drift-rate parameters on the initial decision (1 = correct decision leading to confirmatory post-decision evidence, −1 = incorrect decision leading to disconfirmatory post-decision evidence), initial confidence (parametrically ranging from −1 to 1) or their interaction.

In all models the drift rate, starting point, non-decision time and boundary separation were fitted hierarchically with individual parameter estimates for each participant, whereas dependencies of starting point and drift-rate on experimental factors were estimated as fixed group-level effects. In all model fits we incorporated an influence of post-decision evidence strength on the drift-rate. First a baseline model was estimated where none of the parameters depended on confidence or an initial decision. Subsequently, we created three model families that had dependencies of starting point and/or drift-rate on (i) initial confidence, (ii) initial decision or (iii) the interaction of initial confidence × initial decision (i.e. confidence was allowed to amplify or attenuate the influence of the initial decision on the starting point and/or drift-rate). Within each model family we created three different models with dependencies of these variables on starting point, drift-rate or both.

Baseline model (Model 1):

$${\mathrm{Starting}}\,{\mathrm{point}} \sim {\mathrm{1}}$$
(5)
$${\mathrm{Drift}} \hbox{-} {\mathrm{rate}} \sim {\mathrm{1}} + {\mathrm{post}} \hbox{-} {\mathrm{decision}}\,{\mathrm{evidence}}\,{\mathrm{strength}}$$
(6)
$${\mathrm{Boundary}}\,{\mathrm{separation}} \sim {\mathrm{1}} + {\mathrm{confidence}}$$
(7)

Confidence dependency (Model 4):

$${\mathrm{Starting}}\,{\mathrm{point}} \sim {\mathrm{1}} + {\mathrm{confidence}}$$
(8)
$${\mathrm{Drift}} \hbox{-} {\mathrm{rate}} \sim {\mathrm{1}} + {\mathrm{post}} \hbox{-} {\mathrm{decision}}\,{\mathrm{evidence}}\,{\mathrm{strength}} + {\mathrm{confidence}}$$
(9)
$${\mathrm{Boundary}}\,{\mathrm{separation}} \sim {\mathrm{1}} + {\mathrm{confidence}}$$
(10)

Initial decision dependency (Model 7):

$${\mathrm{Starting}}\,{\mathrm{point}} \sim {\mathrm{1}} + {\mathrm{initial}}\,{\mathrm{decision}}$$
(11)
$${\mathrm{Drift}} \hbox{-} {\mathrm{rate}} \sim {\mathrm{1}} + {\mathrm{post}} \hbox{-} {\mathrm{decision}}\,{\mathrm{evidence}}\,{\mathrm{strength}} + {\mathrm{initial}}\,{\mathrm{decision}}$$
(12)
$${\mathrm{Boundary}}\,{\mathrm{separation}} \sim {\mathrm{1}} + {\mathrm{confidence}}$$
(13)

Full model (Model 10):

$${\mathrm{Starting}}\,{\mathrm{point}} \sim {\mathrm{1}} + {\mathrm{confidence}} + {\mathrm{initial}}\,{\mathrm{decision}} + {\mathrm{confidence}} \times {\mathrm{initial}}\,{\mathrm{decision}}$$
(14)
$${\mathrm{Drift}} \hbox{-} {\mathrm{rate}} \sim {\mathrm{1}} + {\mathrm{post}} \hbox{-} {\mathrm{decision}}\,{\mathrm{evidence}}\,{\mathrm{strength}} + {\mathrm{confidence}} \\ \,\,\,\,\,\,+ {\mathrm{initial}}\,{\mathrm{decision}} + {\mathrm{confidence}} \times {\mathrm{initial}}\,{\mathrm{decision}}$$
(15)
$${\mathrm{Boundary}}\,{\mathrm{separation}} \sim {\mathrm{1}} + {\mathrm{confidence}}$$
(16)

RTs faster than 200 ms were discarded from the model fits and the outlier probability was set to 0.05, as recommended in previous literature43,44. The models were estimated with a Markov chain of 100,000 samples with 50,000 burn-in samples (i.e. discarding the first 50,000 iterations), and a thinning factor of 25, resulting in 2500 posterior samples. To ensure convergence, the posterior traces and their autocorrelation were inspected and the Gelman–Rubin statistic was calculated for each parameter (see Supplementary Table 1). The posterior distributions of the best-fitting model were interrogated to retrieve parameter estimates.

The winning model was characterized by a regression equation that incorporates effects of confidence, the initial decision and their interaction (i.e. the full model) on the starting point and drift-rate. The Deviance Information Criterion scores of all models are shown in Supplementary Fig. 3A. The model parameters of the best-fitting model are shown in Fig. 2d.

MEG pre-processing

MEG was recorded continuously at 600 samples/second using a whole-head 273-channel axial gradiometer system (CTF Omega, VSM MedTech), while participants sat upright inside the scanner. Data was segmented into 8200 ms segments from −200 ms to +8000 ms relative to trial onset, where each segment encompassed one trial. Each epoch was aligned to the onset of the trial or, for analysis of the post-decisional phase, was realigned to the onset of post-decision evidence (to minimize any presentation delays that may have occurred during the trial). The data were resampled from 600 to 100 Hz to conserve processing time and improve signal to noise ratio, resulting in data samples spaced every 10 ms. All data were then high-pass filtered at 0.5 Hz to remove slow drift. All analyses were performed directly on the filtered, cleaned MEG signal, consisting of a 273 channel × 821 sample matrix for each trial, in units of femtotesla.

Generalising a pre-decision classifier to the post-decision phase

We built a machine-learning classification algorithm to predict participants’ decisions on each trial (leftward vs. rightward motion) at each timepoint during the decision phase. Having trained such an algorithm we could then apply it to a distinct set of trials and use the probabilistic prediction of the classifier as a neural DV for leftward versus rightward motion45,46. Specifically, we used a support-vector machine (SVM) classifier trained on sensor-level whole-brain activity (normalized amplitude of all MEG channels). The classifier labels were the trial-by-trial choices made by participants (left or right) while the features encompassed a matrix of activity at each MEG sensor (z-scored for each time point) at a given time point (average activity over 100 ms window, shifted in steps of 10 ms). The classifier was trained on MEG activity in the pre-decision time phase (e.g. 250 ms after the onset of pre-decision evidence) and then reapplied to the corresponding time point in the post-decision phase (e.g. 250 ms after the onset of post-decision evidence). We computed the predictions of the classifier across an 850 ms time window, starting with post-decision stimulus onset and ending with the presentation of response options (i.e. when the mapping between choices and motor responses was revealed).

We used linear kernels and a default regularization parameter of C = 1 within the svmtrain/svmpredict routines of libsvm47. A leave-one-out procedure was used, training the classifier on all trials except one (using pre-decision data only) and testing it on the left-out trial (using post-decision data). Training the SVM results in a hyperplane that best separates the two classes of trials (see Fig. 3a) in a high-dimensional space. If a trial is far away from this hyperplane it is unlikely to be a misclassification, while trials that are close to the hyperplane might easily be misclassified. Thus, the distance to the hyperplane represents the decodable evidence for a decision and can thus be used as a graded measure of the neural DV45,46.

After reapplying the classifier to every trial and time point during the post-decision phase, we obtained a timeseries of neural evidence accumulation within each trial (see Fig. 3a, right panel). We focussed on the time from the onset of the post-decision stimulus to the timepoint of peak decodability at which the pre-decision classifier best generalized to the post-decision phase. The accumulation process can be summarized by fitting a linear regression to the time series (see Fig. 3a, right panel) on each trial, where the slope is analogous to the drift rate in a DDM, and the intercept analogous to the starting point. A positive slope corresponds to a change of the neural DV towards predicting rightward motion decisions while a negative slope corresponds to a change towards leftward motion (see Fig. 3b). By taking the absolute value of these slope values (i.e. reversing the sign on trials in which leftward motion was presented), we could derive a general index for the sensitivity of the neural DV to the motion direction presented on the screen (see Fig. 4a, b).

Based on our behavioural findings we expected that both the slope and the intercept would be influenced by the interaction of initial decision (confirmatory post-decision evidence = 1; disconfirmatory post-decision evidence = −1) × confidence (low confidence = −1; high confidence = 1). Thus, we entered the initial decision, confidence and their interaction as simultaneous predictors in a hierarchical regression model.

MEG topography contributing to classification accuracy

To explore which brain areas carried the information about evidence for a left versus a right decision (or high versus low confidence as reported in the Supplementary Note 6), we trained a SVM classifier for each participant at the time point of highest decodability (see Supplementary Fig. 6 for the whole timeline) using subsets of 30 randomly selected sensors and repeated this procedure 2500 times. The contribution of each sensor s was taken to be the mean of all prediction accuracies achieved using an ensemble of 30 sensors that included s48,49.

MEG temporal generalization

The extent to which a classifier trained on neural data obtained from one time point generalizes to other time points can provide insight how mental representations change over time32. We utilized this temporal generalization method to formally test whether the same processing steps (leading up to a decision) occur at similar times in the pre- and post-decision phases (see Supplementary Fig. 9 for temporal generalization restricted to the pre-decision phase). Most critically, we also investigated whether this processing cascade was altered by participants’ confidence in their choice.

For the temporal generalization analysis we trained our classifier on every timepoint in the pre-decision phase and tested it on every timepoint in the post-decision phase yielding a 2D matrix of decoding accuracy (see Fig. 3a top-left panel). A fourfold stratified cross-validation was implemented for each subject and repeated 100 times to account for potential random biases in assigning trials to folds. Through this stratification we obtained a balanced number of trials within each condition in each fold (left/right decision, high/low confidence, change/no change of mind, and all combinations of these factors). Classifiers were trained on three out of four folds and tested on the left-out fold. Decoding accuracy was determined by the area under a Receiver Operator Curve (AUC) that sought to predict the decision based on the continuous DV outputted by the classifier. Decoding accuracy was calculated separately for the four different conditions (low confidence and change of mind; high confidence and change of mind; low confidence and no change of mind; high confidence and no change of mind). Importantly, classification accuracy was based on how well the initial decision (rather than the final decision) could be predicted based on neural data. Since we are dealing with a two-class decoding problem one can directly infer the decoding accuracy of the alternative decision from the classification accuracy of the initial decision.

We estimated the main effect of confidence on decoding accuracy to isolate confidence-induced changes in temporal generalisation from the pre- to post-decision phase. We used a cluster-based permutation test50,51 to determine statistical significance (p < 0.05, corrected for multiple comparisons). We calculated the contrast of high > low confidence averaging over change/no change of mind trials [[high confidence and no change of mind − low confidence and no change of mind] + [high confidence and change of mind − low confidence and change of mind]]. We identified adjacent timepoints all individually exceeding t-values corresponding to p < 0.05 uncorrected, and stored the sum of t-values for each cluster. We then applied a sign-flip permutation test (randomly switching the contrast direction for a subset of subjects of the sample, i.e. low-high instead of high-low) and repeated this procedure 1000 times. The distribution of summed t-values over all permutations built the null distribution for our statistical test. If the observed sum of t-values within a cluster exceeded the 5% quantile of this distribution (separately calculated for negative and positive values) we labelled this cluster as showing a significant main effect of confidence in this portion of the temporal generalisation matrix.

Reporting summary

Further information on research design is available in the Nature Research Reporting Summary linked to this article.

Data availability

Anonymised data and code are available at a dedicated Github repository [https://github.com/MaxRollwage/NatureCommunications]. The source data underlying Figs. 1b–d, 2b–d, 3b, d–f, and 4a–d are provided as part of this repository. A reporting summary for this Article is available as a Supplementary Information file.

Code availability

Code supporting this study are available at a dedicated Github repository [https://github.com/MaxRollwage/NatureCommunications].

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Acknowledgements

We thank Dr Sam Ereira for help with implementing the machine learning MEG analysis. M.R. is a predoctoral Fellow of the International Max Planck Research School on Computational Methods in Psychiatry and Ageing Research. The participating institutions are the Max Planck Institute for Human Development and University College London (UCL). The Wellcome Centre for Human Neuroimaging is supported by core funding from the Wellcome Trust (203147/Z/16/Z). S.M.F. is supported by a Sir Henry Dale Fellowship jointly funded by the Wellcome Trust and the Royal Society (206648/Z/17/Z). T.U.H. is supported by a Wellcome/Royal Society Sir Henry Dale Fellowship (211155/Z/18/Z), a grant from the Jacobs Foundation (2017-1261-04), the Medical Research Foundation, and a 2018 NARSAD Young Investigator grant (27023) from the Brain & Behaviour Research Foundation.

Author information

Authors

Contributions

S.M.F. and M.R. conceptualized the study. M.R. developed the methodology under supervision of S.M.F. and with input from T.U.H. A.L. and M.R. conducted the experiments. M.R. analysed the data under supervision of S.M.F. and with input from R.M. and T.U.H. M.R. and S.M.F. wrote the first draft of the manuscript which was revised and edited by A.L., T.U.H., R.M. and R.J.D.

Corresponding author

Correspondence to Max Rollwage.

Ethics declarations

Competing interests

All authors declare no competing interests.

Peer review information Nature Communications thanks Redmond O’Connell, and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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Rollwage, M., Loosen, A., Hauser, T.U. et al. Confidence drives a neural confirmation bias. Nat Commun 11, 2634 (2020). https://doi.org/10.1038/s41467-020-16278-6


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# 'Round-the-horn site-directed mutagenesis

‘Round the Horn Site-directed mutagenesis

Submitted by Sean Moore

## Background

I was in graduate school and had to make several codon substitutions at the same position in a cloned gene. I came up with this protocol and called it "Around-the-Horn", a phrase used in baseball meaning to throw the ball through all of the base positions in a circle. The phrase originally referred to "Rounding Cape Horn of South America" a very difficult, but rewarding journey ("Round-the-horn" is not difficult). Subsequently, a group published a paper describing the process, but I was first, take full credit, and have used it in ways not described by others.

I was used to doing "Quickchange" (Stratagene) wherein two complementary oligos containing the desired mutation are used to prime extensions all the way around the plasmid. Quickchange is a linear amplification (not PCR) and each round of extensions is performed on the plasmid template. There are several problems and limitations with Quickchange:

• The size of the mutation is limited to a few base pairs.
• Each primer has to be long to provide sufficient annealing on both sides of the mutation (money).
• Non-optimal extension conditions will result in "end-filling" wherein the polymerase fills the recessed 3' end of the product instead of extending the primer on the template (this is especially a problem as the reaction progresses and is why only 12 or so cycles are recommended). The blunt-end products won't give you any transformants and also mislead the researcher into thinking their reaction went great because they see a product on a gel, but the product is a dead end.
• For each different mutation at a given position, a new primer pair has to be ordered (mo' money).

'Round-the-horn is a PCR-based mutagenesis. The mutations are contained in one or both of the primers. The primers are phosphorylated so that the PCR product can be ligated into a circle and used to transform cells. The procedure has the advantages that the primers are smaller, a visible product on a gel means your reaction worked and must contain the encoded mutations, only one primer needs to be changed to change the mutation at the same site. If you are concerned with mutations elsewhere in your plasmid, you should sub-clone the segment of interest into a "fresh" vector (this is the case for any site-directed protocol). Finally, as opposed to standard site-directed mutagenesis, this procedure exponentially amplifies the plasmid, leading to increased yields, increased transformation efficiencies, and the potential to create a large library with randomized residues.

## Citing

This protocol can be cited directly OWW, but I get frequently asked for the first paper to use it. When I was nearing the end of my graduate work, I made the first mutations in the P22 scaffolding protein; found in this reference:

PMID: 12239300

However, if my shoddy memory serves correctly, there is another publication that came out shortly before mine, also from my graduate lab, the technique was used to introduce point mutations in HIV CA. I recall advising our technician, but I may be wrong. It is simply listed as "PCR mutagenesis".

PMID: 12507478

## Primer design

In this example, the Forward primer (primer A in the image above) has intended mutation at its 5’ end. Design such that the annealing portion has Tm ~60-63º.

e.g.,

5’-NNNACGTACGTACGTACGTACG-3’

The reverse primer in this example resembles the complementary strand and abuts the site of mutation.

                       5'-NNNACGTACGTACGTACGTACG
…ACGTACGTACGTACGTACGTACGTACGTACGTACGTACGTACGTACGTACGT…
…TGCATGCATGCATGCATGCATGCATGCATGCATGCATGCATGCATGCATGCA…
CATGCATGCATGCATGCAT-5'

• If the change needed is large, both primers can have mutant 5’ ends. Alternately, two rounds of PCR can be performed to build the desired PCR product.
• Primers without mutations at their 5' ends can be spaced apart on the template with a gap between them to make deletions of any desired size.
• You can amplify a low-copy plasmid with primers that introduce cloning sites for receiving an insert. The PCR product becomes the "vector" component of the ligation reaction. In this case, you don't need to phosphorylate the primers because the restriction digest of the ends will leave 5' phosphates. An advantage using PCR to prepare your vector is that background "vector re-ligation" reactions are practically eliminated.
• Remember to transform bacteria that have no restriction system as the plasmid will be unmodified. This is not essential, but really helps.

## Primer Phosphorylation

1. Primer stock should be at 100 μM in water or TE.
2. For each primer, mix:
• 37 μL water
• 5 μL 10 Kinase reaction buffer (comes with PNK)
• 1 μL 50 mM MgSO4 (comes with most polymerases)
• 5 μL primer (10 μM final)
• 1 μL 100 mM ATP
• 1 μL PNK
3. Incubate at 37°C for 30-60 minutes.
4. Heat each reaction to kill the PNK (65°C for 20 minutes or 95°C for 5 min).

## PCR

### The Mixture

Set up the following 50μL mixture on ice. After mixing, put into pre-heated PCR block. Keep in mind that your particular volumes may be different, some newer fusion polymerases (recommended) use 5X and 2X buffers.

• 40 μL water
• 5 μL 10X polymerase buffer
• 1.5 μL forward primer (0.3 μM final)
• 1.5 μL reverse primer
• 0.5 μL dNTPs (20 mM each)
• 1 μL template (mini-prep’d plasmid)
• 0.5 μL polymerase (not Taq, use Vent, Deep Vent, 9°N, Pfu, or a faster fusion enzyme)

### PCR program

1. 98°C 1’
2. 96°C 30”
3. 55-58°C 30” (~5°C below Tm, higher for fusion enzymes)
4. 72°C 2’ per kb being amplified (much less if using a fusion enzyme)
5. Goto (2) 25-27 more times
6. cool to 18°C for a minute or so (there's no need to hold at 4, you just cooked the sample for an hour!)
7. end

## Product Preparation

Run 5 μL on a gel. I put EtBr in the gel so I can check to see if there is a product within a few minutes. If you see the band, digest the template in the PCR reactions by adding 1 μL of DpnI, mix, and incubate for at least 60 minutes at 37º.

Run a preparative gel of the PCR product, excise the band, and use a kit to extract the product.

### Product ligation

• 2.3 μL water*
• 2 μL PCR product
• 0.5 μL 10X ligation buffer
• 0.2 μL ligase
• 5 μL

If there is not much product, replace the water fraction with more PCR product.

Incubate overnight at room temp. These blunt ligations can be pretty slow. Some enzymes (e.g. 9° North and Taq) leave crap on the 3' ends. This greatly reduces the efficiency of the blunt-end ligations.

### Transformation

Works best with electroporation or fresh TSS cells. Transform 4 μL into 50 μL cells. Plate all of it. You should get 20 to 1000 colonies. Many times, I have forgotten to phosphorylate the primers before doing the PCR. You can successfully phosphorylate the purified PCR product before ligation, but the ligation/transformation will be less efficient. PNK prefers exposed 5' ends.

## Optimizations

We have been using NEB's Q5 enzyme almost exlusively now. A rough estimate of 0.5 ng of template, 300 nM primers, and 2-3 degrees below the NEB recommended anneal. Also, use NEB's GC enhancer and 30 sec/kb.

I have had reactions that just wouldn't produce product (for whatever reason). When doing PCR, it is difficult to predict the yield or quality of a product. The longer the product is supposed to be, the greater the chance of competing reactions taking over. Usually, people will adjust the Mg++ concentration to influence polymerase behavior or adjust the annealing temp to influence primer annealing and polymerase extension.

Keep in mind that when the sample temperature is raised from the annealing temperature (say 55) to the optimal extension temperature (say 72), the oligo would normally melt off of the template. The reason primer extension works is the polymerase slowly extends the oligo during annealing and while the temperature is rising so that when the 72 degree mark is hit, the extended product is long enough to keep the primer attached and the polymerase happy. Therefore, oligo composition, the type of polymerase, the annealing temp, and ramp time can have significant effects on the reaction.

Take the following example: Each reaction had 1 μL of the same mini-prep, and all reactions were done with the same program at the same time. Three different primer pairs were used and three different enzymes. I have found that the surest way to get a successful 'Horn reaction is to set up the reactions with different enzymes (historically Deep Vent and Turbo Pfu, but now an array of fusion enzymes) rather than to set up arrays of Mg++ concentration or a gradient of annealing temps. The desired product is marked with an arrow. Note that the different enzymes gave varied performance simply depending on the primer pair used. Consider the cost of your time repeating steps relative to the cost of the enzyme aliquot. We now use NEB's Q5 enzyme with excellent results, even on >10 kb plasmids.

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# kistaro

## Kistaro Windrider, Reptillian Situation Assessor

### Unfortunately, I Really Am That Nerdy

Victory
There was a TopCoder match tonight (for those of you not aware, my handle there is Windrider), and the NSA was sponsoring a $5,000 prize pool. 70% of the pool goes to Division 1, with 30% going to Division 2. To give everybody at least some chance (not just the people who consistently score highly for the whole round), the prize is further divided by room, so your position in your room- not your overall rank- determines your payout. (Rooms are filled randomly, 20 people each.) The room winner scores 50% of the pot (in Division 1- D2 uses a different distribution), second place gets 30%, and third gets 20%. There are no extra prizes for overall rank, but overall is what decides your rating. I came in 2nd in Room 13, with a score of 1107.33 points. I came in 27th overall. This is my highest score for a Division 1 problem set, my highest ranking for a Division 1 problem set, and the only time I've won money in an SRM. (I've won cash in previous tournaments- we'll see if I can do it again when the TCCC Round 1 begins this Saturday.) This is also the first time I've swept the problems in Division 1, solving all three of them correctly. I'm happy about the (approximately)$45, but getting every problem (including the Hard) in Division 1 is what really has me in a good mood. That's the first time I've done that in a real match. (Early tournament rounds are more like D2 in difficulty and don't count.) This match did play to my strengths- discrete mathematics- but still, I'll take what I can get.

My rating is now at 1891, the highest it's ever been. It'll probably drop across the next few matches, but it's fun to have a rating like that, at least for a while. But getting the Hard problem right with the first algorithm I thought of... that's the really cool part.

The punchline is that I coded up my 1000 and submited it as a joke. I just thought up what I thought was a crazy, impossible algorithm, which I wrote mostly so I could use the variable name "antimanhattan". I just had this guess as to what might work, what might exhibit some properties of the right answer- which points I could ignore and still get the right answer. (The problem: Out of 250,000 points generated by an algorithm with fairly messy output, find the pair with the longest Manhattan distance in two seconds or less.) My "only use the manhattanest, the antimanhattanest, the unmanhattanest, and the unantimanhattanest; return longest corner-to-corner" was just this crazy idea I thought of, but then it passed all the example cases. So I submitted it... and it accidentally worked.

A lot of good ideas are like that.

Did I get it by accident? Not really. I coded it right. And I know discrete mathematics well enough that apparently, even when I don't know why, my intution can be correct- and good intuition requires at least some level of knowing what I'm doing.

But anyway. 27th place of 470. SQUEE.

Very nice! Congratulations!

Awesome, congratulations! :D

Mmm, bacon userpic.

...now I'm hungry.

*giggle* Uh... sorry? ;D

Congratulations!

The scary part here is that I think I understand what you mean by

> "only use the manhattanest, the antimanhattanest, the unmanhattanest, and the unantimanhattanest; return longest corner-to-corner"

... and my own brain chipped away at the problem.

I'm unfamiliar offhand with this Manhattan thing. Does the idea of "manhattanness" and "unmanhattanness" map to a standard X/Y coordinate plane with "Manhattan distance" mapping to straight-line distance?

Because if so, I think the algorithm breaks. Not always -- it depends on the data set -- but let's say you have the following data set:

(-10, 0)
(10, 0)
(0, -10)
(0, 10)
(9, 9)
(-9, -9)

The four points your algorithm selects are the first four, with corner-to-corner distances of 20. But the longest linear distance is between the last two, 18 sqrt 2.

OK, so I think that refined my understanding. You're not mapping "Manhattan" to X and "unManhattan" to Y; you're mapping them respectively to [ M(x,y): x>=0, y>=0, M = x + y ] and [ M(x,y): x>=0, y<0, M = x + |y| ], yes? Because that specifically addresses my sample failure.

I don't have a ready proof that that catches all such gotcha cases, but it's a much stronger contender.

... Actually, wait, no, that does work. If you pick one point from each quadrant with the highest M-distance to (0,0), then every most-efficient corner-to-corner path between them has a path of the same length which passes through (0,0). Thus the corner-to-corner distance is equivalent to the corner-to-origin-to-corner distance and there can be no gotcha relying on selection of points such that the most efficient path doesn't go through the origin.

### Re: answering my own question

The problem constraints included a constraint functionally equivalent to "all points are in the range (0, 0) to (107, 107), lower inclusive", so there are no negative points.

That's not my calculation of the Manhattanest and Unmanhattanest points. The Manhattanest pointis the point with maximal (X+Y). The Antimanhattanest point is the point with maximal (Y-X), which is equivalent to mirroring the board left to right. The Unmanhattanest had minimal (X+Y), and the Unantimanhattanest had minimal (Y-X).

The only candidate pairs I needed to consider were the Manhattanest and the Unmanhattanest, and the Antimanhattanest and the Unantimanhattanest.

My solution stood through the Challenge Phase and the System Tests; it's mathematically correct. But yes, as you noticed, the Manhattan distance is very different from the Euclid distance!

Using the Manhattan distance is what makes the problem solvable within two seconds- it becomes an O(n)-complexity problem. The Euclidian-distance variant can be solved by brute force in O(n2) time, but when n=250K this would time out relative to TopCoder's two-second time limit. (Anything involving 250,000 points generally has to be O(n) to pass.) I know that for the Euclidian-distance variant you can discard any point not on the convex hull of the collection of points-so-far, but the first trick is figuring out whether it's on the chull or not, and the second is "this doesn't help you at all when the points are in a ring".

It's also the Manhattan distance that let me solve it! It converts the problem from traditional mathematics to combinitorics- from my most serious weakness (relative to computer programming) to my greatest strength. This problem took a lot more insight than ability to pound out a thousand lines of code in ten seconds, which is a pleasant change for Hard problems.

The Nasty Little Algorithms I still really need to learn to get and maintain a red-bracket rating in TopCoder are Bipartite Matching and Max Flow. I'd still need to improve my math intuition and mathematical programming skills to have a stable red rating (rating >= 2200 and your username shows up in red; the 1500-2199 bracket gives you a yellow username, which is still considered skilled- my rating is 94th %ile and of 788 active American TopCoder members where "active" is defined as "played a match for rating points within the last six months", I'm in 32nd place; 288 of 4876 worldwide), but I think I could keep a low red rating with just those two. They solve a lot of problems, and they're the only graph theory problems I consistently miss. Of course, both of the problems are a huge pain in the tail...

### Re: answering my own question

What you said convinces me we're talking about the same algorithm over different data sets. If you somehow felt the urge to add (-5x10e6, -5x10e6) to all of your data points (which is, if I remember my set theory terms correctly, trivially a homeomorphism), your method would map to the one I discussed.

As for the Euclidean variant -- this might sound unorthodox, but, I suspect that for integral coordinates, the pair with the greatest Euclidean distance will be in the subset of points with the greatest Manhattan distance to the center of your data set, i.e., (Xmax - Xmin, Ymax - Ymin). [A proof would involve showing that all points within a ring of radius r from that origin have Manhattan distance to it bounded by some F(r), and you can find a larger Euclidean distance by selecting at least one point outside the ring than you can by staying iside of it.] If N is very large, you can then use Manhattan distances to identify the top candidates, and then brute-force among a very small subset of N.

### Re: answering my own question

The algorithm you're describing seems to rely on the faulty assumption that there will always be points in all four quadrants. If you have at least four points (not a given) that are not colinear, this is possible by redefining the origin somewhere in the middle. Otherwise, the problem is isomorphic to any coordinate space- that's why I can get away with Y-X (or X-Y would work just as well) for antimanhattanism rather than needing to add a max-X offset to it; it just doesn't make any difference at all. So corner-to-origin-to-corner? Almost, for some arbitrary pseudo-origin that is "between" the points. Note that the points are not guaranteed to be distinct; one of the example cases had all the points actually on the same point. The correct answer for that case was the first two points- the two points are required to be distinct, but ties are to be broken lexicographically by index of point as generated by the generator function (250K is too large for a TopCoder input, so they gave a "random" number algorithm and the input is the seed).

I can disprove your suspicion: A=(-4,0), B=(1,4), C=(1,-4), D=(4, 0). The Manhattanest distance is AB or AC, but the Eulerest distance is AD, and D is not among any of the Manhattan extrema relative to the origin or to point A.

### Re: answering my own question

> The algorithm you're describing seems to rely on the faulty assumption that there will always be points in all four quadrants.

I realized this at work today. The whole quadrants thing was just me being sloppy with rigor in order to get the thought experiment working a little more quickly.

Obviously, it's not needed at all; there's nothing about "the point with the maximum value of X+Y" that implies either or both X and Y have to be positive. In terms of helping me track in to the "corners" idea, it was helpful, though. The more we talk about the origin, the more confusing it'll get, so I'll just concede the model has a limited utility and try to move on to your terminology.

Anyway.

Your example as given doesn't actually disprove my suspicion. By my algorithm, the two points with the greatest Manhattan distance from the origin are the subset I'm checking for greatest Euclidean distance in. So I measure between B and C: Eu.Dist. = 8. A and D's Eu.Dist also = 8 and thus isn't longer.

However, if you double all the distances and nudge A, you have your disproof: (-9, 0), (2, 8), (2, -8), (8, 0). AD = 17, BC = 16, and B and C are chosen because MDist from origin are both 10.

However however, this counterexample fails (i.e., the algorithm works) if you simply move the origin one positive unit along the X-axis, showing that A) the algorithm is very sensitive to origin position, and B) thus shouldn't be tied to an origin at all. So I ask you:

If we revise my idea to base it on your formulations, can you still find a counterexample? (In other words, "The pair with the greatest Euclidean distance will be in the subset of {Manhattanest, anti-Manhattanest, unManhattanest, anti-unManhattanest}.")

### Re: answering my own question

Every point on the blue circle is equidistant from the origin. Every point in the green and red quadrilaterals has the same Manhattan distance from the origin.

I don't have time to get an actual set of counterexample points, but those regions between the larger diamond and the circle are where my suspicions lie. With points at all four locations where the large diamond is tangent to the circle, put a pair of points on the X-axis such that the first is in the left-side region bounded by the circle and the outer Manhattan diamond and the other is in the right-side such region. That pair has the greatest Eucidian distance to each other, but not any of the Manhattan extrema.

### Re: answering my own question

The green square reminds me that there's still room for a weaker version of my conjecture. If you set the radius and center of the red square s.t. that your points with greatest Manhattan distance are on a pair of its opposite faces, any point-set within the green square is guaranteed to not be a greatest-Euclidean-distance solution. (Proof: The minimum EuDist between facing sides of the red square is where it's tangent to the circle and is 2r [r = circle radius]. The maximum EuDist between facing sides of the green square is between corners and is 2r. But you already have a solution of that length.)

So this wouldn't be much of a help in a TopCoder problem, but might have some weird, abstruse applications in the "real world". Assuming normal distribution of points (this is the key phrase, and why it's useless for TopCoder), this can whack off 25% of the work of brute-forcing a large dataset. t(.75n^2 + n) is better than t(n^2) (well, for n>4). You run the risk that you're just wasting t(n)'s worth of work, but programmers take that risk with compression algorithms all the time. :P

### Re: answering my own question

Actually, you can prove a stronger claim than the red and green diamonds. The only points you need to consider are on the convex hull of the set(the shape you'd get by making the points nails in a board and snapping a rubber band around the whole thing). It's useless in TopCoder because you can engineer an input such that all the points are on the convex hull (ring-shape, for example), but a real-world application would definitely find it a major optimization- for any real-world application that needs "longest pair", which is a questionable proposition to start with. (Shortest pair has a clean O(n log n) algorithm that was my first assignment in my Intro to Algorithms class.) Quickhull (which is at least as good as Jarvis' March in the worst case) performs well in scattered cases, and then it would take most of the points out of consideration. From there, your algorithm might as well just check all pairs, because anything more intricate would wind up having to iterate over the remaining points to check them for possibility and wind up taking the same amount of time, if not more, than just checking everything.

But would calculating the Manhattan extrema (to give us the red diamond from which the blue circle and green diamond can be calculated fairly easily) help us anyway? We know that calculating the Manhattan extrema is O(n), and then checking for "inside diamond" is O(1) per point so we'd simplify the problem drastically for O(n) work. However, the simplification would make quick-hull much less efficient because it would leave a data set shaped like the worst case! Graham-scan would work better than quick-hull in that case, and it would probably be faster than a quick-hull over all the data points would have been.

Simplifying the problem overall is likely to produce better results (with regard to time complexity) than saving multiplications by using Manhattan distance comparisons rather than Euclidian distance comparisons (remember, the algorithm can save a lot of time by not executing the square root and just working with the squares of the distances- it's good enough for greater than/less than comparision purposes), so here's something I'm not sure of: What can we prove about the blue circle? Can we prove that the only points of interest are between the blue circle and the red diamond (inclusive)?

I'm not actually a mathemetician, I just sort of look like one when I'm doing things with combinatorics...

### This discussion is taking on a life of its own

Quickhull, Jarvis and Graham are Greek to me (as I never actually took any formal computing classes -- just their underlying mathematical ones). So I don't really have any comment on that part of the post.

> Can we prove that the only points of interest are between the blue circle and the red diamond (inclusive)?

I think this is the best we can do without having special-case evaluation based on the data:

The first step would be noting that by our definitions, we already have a path of at least 2r EuDist. So any point that is less than Z EuDist from every point inside the red square (and thus from every other point in our data set) is not a point of interest. These points are inside a circle of radius (2-sqrt2)*r centered on the blue circle's center.

If the particular Manhattanest points have a EuDist of Xr (>2r), we can substitute that in instead and the radius increases to (X-sqrt2)r. But the subset of points we can throw out entirely via this method still seems small.

Hmm ... on the other hand.

I'm thinking there might be some potential to taking the ... um ... diago-Manhattanest points, as well? i.e., the points with the greatest Manhattan distance if the coordinate system were to be rotated by 45 degrees. This should be still only a t(N) problem, since the algorithm for diago-Manhattanism can, I'm pretty sure, be dervied just from combinations of X1, X2, Y1, and Y2 (plus a single multiplication by sqrt 2 once you've figured the biggest pairs). This would give us a second set of red/green squares at 45-degree angles to the ones here, and I'm wondering if a combination of those and the originals would be enough to catch the exceptions that Manhattan points alone let through.

(On further thought, I don't think this would be sufficient, since there would be smaller slivers at the 22.5/67.5/etc. degree marks, and so on and so forth. But I don't have a formal proof of that yet, just intuition.)

### Re: This discussion is taking on a life of its own

Quickhull, Jarvis and Graham are Greek to me (as I never actually took any formal computing classes -- just their underlying mathematical ones). So I don't really have any comment on that part of the post.

They're all different algorithms for the same thing: find the convex hull. They all have different time complexities and perform well in different situations.

The first step would be noting that by our definitions, we already have a path of at least 2r EuDist.

No we don't. Consider an equilateral triangle.

Iterating the algorithm by rotating the coordinate system is going to run into the law of diminishing returns after about the third attempt. Finding out what the overhead is and how much of a difference it makes is probably interesting.

I have more thoughts on this, but it's too late and I need sleep and I just wanted to say "Won't somebody please think of the triangles?" before I forgot.

### Re: This discussion is taking on a life of its own

>> The first step would be noting that by our definitions, we already have a path of at least 2r EuDist.

> No we don't. Consider an equilateral triangle.

um ...? Consider what about it? The red square isn't arbitrary; we select its center and size based on your algorithm that finds the two points with the greatest Manhattan distance. Therefore, the red square is guaranteed to either have a pair of points on sides 1 and 3 respectively, or on sides 2 and 4 respectively (our pair with the greatest Manhattan distance), depending on whether your manhattan-antimanhattan pair or your unmanhattan-antiunmanhattan pair won out.

Since the length of the red square's sides are 2r, points on opposite sides must be at least 2r apart in EuDist.

### Re: This discussion is taking on a life of its own

Let's say we have the following three points:
(1,0), (0,1), and (-.4,-.4)

Now, in the manhatten metric, the distance between the first two is two, and the other two pairings both have a distance of 1.8, so the 'red square' would go to the unit points. That puts (-.4,-.4) inside the Blue Circle.

I think that your conjecture is based on the notion of a well-defined 'manhatten center', when, in fact, for the points, (a,b), (c,d), any point along the line segment (a,d)-(c,b) is a midpoint of sorts.

If the manhatten diameter of the point set is d_m, and the number of dimensions is n, then the radius of the inner circle should be:
\frac{1}{2} d_m (1 - \frac{1}{\sqrt{n})
which is not all that large.


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# ATLAS Conference Contributions

Ultimi arrivi:
2016-08-24
15:33
 Physics performances with the new ATLAS Level-1 Topological trigger in Run 2 / Artz, Sebastian (Institut fuer Physik, Universitaet Mainz) The ATLAS trigger system aims at reducing the 40 MHz proton-proton collision event rate to a manageable event storage rate of 1 kHz, preserving events valuable for physics analysis. [...] ATL-DAQ-PROC-2016-012. - 2016. - 5 p. Original Communication (restricted to ATLAS) - Full text

2016-08-24
15:26
 ATLAS measurement of Electroweak Vector Boson production / Vittori, Camilla (INFN Bologna and Universita' di Bologna, Dipartimento di Fisica e Astronomia) (+) Measurements of the Drell-Yan production of W and Z/γ∗ bosons at the LHC provide a benchmark of our un- derstanding of perturbative QCD and probe the proton structure in a unique way. [...] ATL-PHYS-PROC-2016-109. - 2016. - 5 p. Original Communication (restricted to ATLAS) - Full text

2016-08-22
22:03
 ATLAS/CMS Upgrades / Horii, Yasuyuki (Graduate School of Science and Kobayashi-Maskawa Institute, Nagoya University) Precision studies of the Standard Model (SM) and the searches of the physics beyond the SM are ongoing at the ATLAS and CMS experiments at the Large Hadron Collider (LHC). [...] ATL-PHYS-PROC-2016-108. - 2016. - 12 p. Original Communication (restricted to ATLAS) - Full text

2016-08-22
16:06
 Production of exotic and conventional quarkonia and open beauty/open charm at ATLAS / ATLAS Collaboration The ATLAS experiment at LHC is carrying on a wide programme to study the production properties of conventional and exotic quarkonium, beauty, and charm bound states. [...] ATL-PHYS-PROC-2016-107. - 2016. - 6 p. Original Communication (restricted to ATLAS) - Full text

2016-08-22
15:41
 Search for the Standard Model Higgs boson decaying into $b\bar b$ produced in association with hadronically decaying top quarks in pp collisions at $\sqrt{s}=8$ TeV with the ATLAS detector / Bruscino, Nello (University of Bonn) A search for Higgs boson production in association with a pair of top quarks ($t\bar{t}H$) is presented, where the Higgs boson decays to $b\bar b$, and both top quarks decay hadronically. [...] ATL-PHYS-PROC-2016-106. - 2016. Original Communication (restricted to ATLAS) - Full text

2016-08-20
18:10
 Flavour tagging algorithms and performance at the ATLAS experiment / Calandri, Alessandro (Centre de Physique des Particules de Marseille (CPPM), IN2P3-CNRS) The identification of jets containing $b$-hadrons ($b$-tagging) is a fundamental ingredient in the physics program of the ATLAS experiment. [...] ATL-PHYS-PROC-2016-105. - 2016. Original Communication (restricted to ATLAS) - Full text

2016-08-19
11:59
 The upgraded Pixel detector and the commissioning of the Inner Detector tracking of the ATLAS experiment for Run-2 at the Large Hadron Collider / Potamianos, Karolos (Lawrence Berkeley National Laboratory and University of California, Berkeley) Run-2 of the Large Hadron Collider (LHC) will provide new challenges to track and vertex reconstruction with higher energies, denser jets and higher rates. [...] ATL-PHYS-PROC-2016-104. - 2016. Original Communication (restricted to ATLAS) - Full text

2016-08-15
18:33
 Recent electroweak results from ATLAS / Schmitz, Simon (Institut fuer Physik, Universitaet Mainz) Recent measurements and searches in the electroweak sector of the Standard Model by the ATLAS experiment at the LHC are reported. [...] ATL-PHYS-PROC-2016-103. - 2016. - 5 p. Original Communication (restricted to ATLAS) - Full text

2016-08-12
15:29
 Search for exotic production of top quarks decaying into same-sign leptons at 13 TeV with the ATLAS detector / Berlendis, Simon (Laboratoire de Physique Subatomique et de Cosmologie de Grenoble (LPSC)) The signature of same-sign leptons and b-jets is a golden channel in the search for exotic production of top quarks. [...] ATL-PHYS-PROC-2016-102. - 2016. - 5 p. Original Communication (restricted to ATLAS) - Full text

2016-08-10
12:36
 Search for top squarks in final states with one isolated lepton, jets, and missing transverse momentum in $\sqrt{s}=13$ TeV $pp$ collisions using 3.2 $\mathrm{fb}^{-1}$ of ATLAS data / Kuechler, Jan Thomas (Bergische Universitaet (Wuppertal), Fakultaet fuer Mathematik und Naturwissenschaften) The results of a search for the stop, the supersymmetric partner of the top quark, in final states with one isolated electron or muon, jets, and missing transverse momentum are reported. [...] ATL-PHYS-PROC-2016-101. - 2016. - 5 p. Original Communication (restricted to ATLAS) - Full text


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## User: swbarnes2

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... Is this really a software error? ...
written 7 days ago by swbarnes2200
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... For interactions, even though the column is labeled "logFold2Change", that's not what it is. Edit: It's the log of the **ratio** of the fold changes. I found this site helpful https://rpubs.com/ge600/deseq2 Look down to the header "The different response in genotypes (interaction term)" ...
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Answer: A: DESeq2 log fold change
... > 1) get normalized count (2) log transformation (3) calculate averages > per condition (4) divide two averages: "FC". Is this a right way? No. Take the averages of the normalized counts, divide one by the other, take the log base 2 of that. ...
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... You need to post the top few lines of your files for anyone to be able to help you. Have you looked at the file formats used in datasets from tutorials, to see how they differ from what you have? ...
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... Lets say I have an experiment like this (assume every sample has replicates) treatment control control Treatment1 Treatment1 Treatment2 Treatment2 If I wanted to compare Treatment 1 to control, I'd likely make the dds object with **all the samples**, and use contrast ...
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... > From what I understand the results are telling me that there is a gene > expression difference between species 1 and species 2 without > accounting for treatment state, a difference between treatment 1 and > untreated control and treatment2 and untreated control without > accounting ...
written 18 days ago by swbarnes2200
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... For interaction tests, the column is labeled "LogFoldChange", but it really is a ratio of fold changes. You can eyeball this yourself by using the normalized counts, and working out the average values of the different treatments, and comparing their ratios. The number will not quite match what DES ...
written 20 days ago by swbarnes2200
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... intgroup.df isn't doing anything there. I like to throw in pca_merged <- merge(colData(dds), pca\$x, by.x = 0, by.y = 0) to get all the info in one place. ...
written 25 days ago by swbarnes2200
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... edata = read.csv(file="C:\Users\Owner\Dropbox\LouiseShoweProjects\lung cancer mRNA and miRNA\c.prevsn.pre\c.prevsn.pre.csv", header=TRUE, sep=",") For starters...no spaces in folder names, ever. ...
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Comment: C: How to subset data
... You need to walk through the example from the vignette, because you aren't following the example at all. ...
written 29 days ago by swbarnes2200

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# Oscillation in spring mass system questions

1. Feb 3, 2014

### xtrubambinoxpr

1. The problem statement, all variables and given/known data

I am having an issue with answering number 4 in the attached image.

2. Relevant equations

Relevant equations are given in question 3.

3. The attempt at a solution

Squaring the equation would only make it easier to solve for any of the other variables, or the period squared. I cannot see the correlation with the square of the period. Would it just make it more accurate?

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2. Feb 4, 2014

### jackarms

It's an oddly worded question -- I think the only thing it's getting at is that, like you said, it would allow you to solve for m or k, and it would also allow you to see the relationships of the variables more clearly.

With the original equation you have:
$T = 2\pi\sqrt{\frac{m}{k}}$

So that means $T \propto \sqrt{\frac{1}{k}}$, so graphing $T$ vs. $\sqrt{\frac{1}{k}}$ would give you a linear relationship, and you could solve for k.

If you square both sides, you get:
$T^{2} = \frac{4\pi^{2}m}{k}$, so $T^{2} \propto \frac{1}{k}$, so graphing $T^{2}$ vs. $\frac{1}{k}$ would also give you a linear relationship, and you could solve for k that way too. So maybe that would be little easier? It really just depends on what data you have.

3. Feb 4, 2014

### xtrubambinoxpr

I see where you are coming from! just makes no sense to me to square it given I had all the variables in the experiment that we did in class. Maybe for future reference know this is something great to remember. Thanks Jack!