Sensitivity to Pain Expectations: A Bayesian Model of Individual Differences

The thoughts and feelings people have about pain (referred to as ‘Pain expectations’) are known to alter the perception of pain. However we know very little about the cognitive processes that underpin pain expectations, or what drives the differing effect that pain expectations have between individuals. This paper details the testing of a model of pain perception which formalises the response to pain in terms of a Bayesian prior-to-posterior updating process. Using data acquired from a short and deception-free predictive cue task, it was found that this Bayesian model predicted ratings of pain better than other, simpler models. At the group level, the results confirmed two core predictions of Hierarchical Predictive Processing; that expectation alters perception and that increased uncertainty in the expectation reduces its impact on perception. The addition to the model of parameters relating to trait differences in pain expectation, improved its fit, suggesting that such traits (which have been relatively neglected in experimental work on pain perception) play a significant role in perception beyond those expectations triggered by the pain cue. When model parameters were allowed to vary by participant, the model’s fit improved further. This final model produced a characterisation of each individual’s sensitivity to pain expectations. These findings are relevant for the understanding of the cognitive basis of pain expectations, and potentially useful for guiding patient stratification and clinical experimentation.


Introduction
The experience of pain, like all other sensory experience, is a result not only of the objective reality, such as the degree of tissue damage, but also of the sufferer's beliefs about pain. Conscious and unconscious thoughts and beliefs that people have about imminent pain are referred to as 'pain expectations' (Schrooten, Vlaeyen, & Morley, 2012). The effect of pain expectations on pain experience is evident in both laboratory and clinical experiments (e.g. Atlas & Wager, 2012;Bingel et al., 2011;Colloca, & Benedetti, 2006;Peerdeman et al., 2016;Tracey, 2010). In addition, neural correlates of the effect of expectation on pain perception have been established (Brown., et al 2008a;Ploghaus et al., 1999;Seymour et al., 2004;Tracey, 2010;Wager et al., 2004;Watson et al., 2009). Despite these advances there remains a significant gap in our knowledge regarding the specific cognitive processes that underpin pain expectations.
Hierarchical Predictive Processing (HPP) provides the dominant theoretical framework for understanding the effects of expectation on perception (Clark, 2013, Friston, 2003, including pain perception (Buchel, Geuter, Sprenger, & Eippert, 2014;Tabor, Thacker, Moseley, & Körding, 2017;Van den Bergh, Witthöft, Petersen, & Brown, 2017). HPP stipulates that perception is biased towards the expected level of pain, and that this bias will be stronger when the expectation is more certain, since expectation uncertainty causes the suppression of top-down, prior-driven signals, leading to greater importance being placed on the bottom-up sensory input. A number of studies confirm the specific predictions of HPP for pain perception, beyond the established biasing effect that pain expectations exert on pain experience. The effect of pain expectations is enhanced when expectations are more precise (Brown, Seymour, Boyle, El-Deredy, & Jones, 2008b;Colloca, Petrovic, Wager, Ingvar, & Benedetti, 2010); and formal Bayesian models have been shown to provide a good fit to pain reports in placebo analgesia studies (Anchisi, Zanon, Plaghki, Kirsch, & Claggett, 2015;Jung, Lee, Wallraven, & Chae, 2017) and to neural data from the anterior insula (Geuter, Boll, Eippert, & Büchel, 2017).
Although HPP guides much of the research on the effect of expectations on perception, some findings appear to contradict its prediction concerning the impact of the precision of expectations on pain experience. Watkinson et al., (2013) delivered two different distributions of pain stimuli, a unimodal and a 4 Bayesian model of pain expectations bimodal distribution. The two distributions had the same mean pain level, but the bimodal distribution had a larger variance. In direct contradiction with HPP, and in support of the alternative, Range Frequency Theory (Parducci., 1963), the effect of expectations was stronger in the bimodal condition, despite the greater uncertainty generated by the larger stimulation variance. Likewise, Yoshida et al., (2013) altered participant's expectations relating to upcoming pain stimulations by presenting fictitious pain ratings of the stimulus prior to their delivery. They found that increasing pain uncertainty contributed independently to increased pain experience, in contradiction with HPP. In addition, it is not clear from Yoshida et al. data that uncertainty significantly modulated the bias induced by the mean of the fictitious pain ratings, as predicted by HPP. These findings suggest that further work is needed before HPP can be accepted as a viable framework for understanding pain expectations.
Over and above the conflicting findings around the impact of expectation uncertainty, we also know very little about how specific processes that underlie pain expectations are integrated to inform the pain percept. To unpack the construct of pain expectations to its underlying information processing mechanisms requires a statistical model of the pain perception process to be formulated which accommodates parameters that describe different facets of pain expectations. Furthermore, in order for such a model to be useful in understanding an individual's response to both pain and placebo analgesia, it needs to be capable of predicting the effect of pain expectations not just at a group level, but also at an individual level.
For example such a model would need to be able to identify individuals that experience high levels of pain because they have a trait-like bias to expect high levels of pain (i.e. always expecting high pain, independent of context), and to distinguish such individual from those who experience high pain because they are highly pessimistic in assessing the information they are given about the placebo treatment, or because they are over-confident in their mildly-pessimistic expectations, so that they rely less on their sensory data. That level of understanding is necessary in order to predict an individual's response, providing a basis for a clinical tool that can support clinical decision making.
This paper details the construction and testing of a model of impact of pain expectations on pain perception using experimental data garnered from a novel predictive cue task. The purpose of this model 5 Bayesian model of pain expectations construction was threefold. Firstly, we wished to assess the effect of expectation uncertainty on pain perception, in light of the conflicting past results mentioned above. Our second objective was to identify, using the experimental data, a number of putative cognitive processes that give rise to the umbrella term 'pain expectations'. Finally, leading on from the second objective, we wished to assess whether a model could be constructed that would enable individuals to be distinguished based on the aforementioned cognitive processes.
We collected two sets of experimental data via two separate experiments using independent samples. A series of increasingly complex statistical models were constructed and their performance was compared using the data from the first experiment. The simplest model (Model 1) where the pain participants experienced was influenced only by the actual, delivered pain was effectively used as a baseline to compare to three other models, which successively included additional facets of pain expectations (Table 1). In the 'Mean-only Bayesian model' (Model 2), pain experience was additionally influenced by the mean of the expected pain, computed based on the information provided by the cue. The 'Mean-and-variance Bayesian model' (Model 3), inspired by HPP, took into consideration the variance of expected pain, a function of the discrepancy between the possible pain levels indicated by the cue. Like Model 2, Model 3 also allowed pain experience to be affected by the mean of the expected pain, but here the impact of mean expected pain was modulated by its uncertainty. Finally the 'Full Bayesian model' (Model 4) additionally included the effect of cue-independent pre-experimental pain expectations in addition to the cue-dependent expectations used in models 2 and 3. In this formulation, cue-dependent pain expectations relate to those triggered on each trial by the cue, while cue-independent expectations capture more stable, trait-like differences in the propensity to believe that pain will be greater or weaker, independent of the effect of specific local cues. Finally, to satisfy the objective of creating a model that can characterise pain expectations at the individual level, we compared the winning group-level model to an individual-level variant that included individual-level random effects (Model 5). We considered the winning model to be able to usefully characterise pain expectations at an individual level if the 'individual-level' variant of the model was significantly better at predicting pain perception than its equivalent group level variant. The 6 Bayesian model of pain expectations second experimental dataset was used to validate the findings from the first dataset, thus providing evidence that the winning model could predict outcomes in a dataset other than the one from which it was constructed (cf. Maloney & Zhang 2010).
We hypothesised that the aforementioned model comparison will provide support for the HPP framework. More specifically we predict that Model 3 will provide a better fit to the data than models 1 and 2 (thus showing that expectation uncertainty significantly affects pain perception) and that the value of model parameter describing the effect of expectation uncertainty will support the predict from HPP that increased uncertainty reduces the influence of pain expectations. Based on the vast clinical literature on individual differences in pain catastrophizing (Sullivan, Bishop, & Pivik, 1995) we hypothesised that Model 4 will provide a better fit for the data than Model 3, suggesting that both cue-dependent and cueindependent expectations have separate influences on pain perception. Finally, we hypothesised that when individual-level random effects are added to the winning model this will significantly improve the fit of the model to the data even when that model was penalised for the increased complexity, suggesting that it is possible to characterise sensitivity to pain expectations at an individual level.

General experimental design
Two experiments were conducted on independent samples. Experiment 2 was conducted for validation, with task delivery modifications introduced to enhance the translational impact of the approach. During each experiment participants performed a 'pain rating' task (Figure 1.). In each trial of the task, participants' expectations regarding the upcoming stimulation were manipulated. This manipulation was achieved by offering the participants a choice between two 'cues' expressing different probability distributions for the intensity of the upcoming stimulation in terms of the possible intensities and corresponding probabilities. After selecting one of the cues the participant then experienced a pain stimulation level generated in accordance with the probabilities represented by the selected cue. Finally, 7 Bayesian model of pain expectations the participant was required to rate the intensity of the stimulation they received. This choice mechanism was used (in contrast to just presenting participants with one cue) to ensure that the participants both attended to, and understood, the pain probability cues they were presented with. The choice mechanism also acted to give the participants a sense of control over the upcoming pain, so as to minimise the possible influence of learned helplessness on their pain experience (Bhat et al., 2010). The trials were designed in such a way that one cue (herein referred to as the 'target') was always preferable to the other (herein referred to as the 'lure') in terms of expected pain intensity. Trials where the participants chose the lure (or failed to make a choice at all) were discarded from the data analysis as in such instances we could not be sure that the participants had understood the cues. The targets were designed in accordance with a factorial design involving the systematic manipulation of the relevant aspects of pain expectation.
The task utilised here is novel in the context of pain research. However, it has been used extensively to examine how people value financial outcomes (Talmi, Atkinson, & El-Deredy, 2013) as well as taste and pain outcomes (Hird et al., 2017). It differs from placebo paradigms because the manipulated quantity is the individual's expectation about the impending pain, rather than about the efficacy of treatment. One particular feature of our task (in contrast with placebo and other cue-based paradigms) is the avoidance of deception. In placebo paradigms participants are told that an ineffective intervention (e.g. an inert cream) is in fact a proven treatment (e.g. an active analgesic). In current cue-based paradigms (e.g. Atlas et al., 2010), cues intentionally mislead participants, for example informing them that an impending pain stimulus will be high intensity, when it may in fact be of moderate intensity. We avoided deception firstly because the integrity of data resulting from deception paradigms is reliant on the participant remaining naïve to the deception throughout the experiment, and secondly because deception is problematic as far as translation into clinical settings is concerned. Files needed to run the experimental paradigm and model detailed in this paper are available to download from BioRxiv.  Once a cue was selected it appeared on its own for 2s (second panel). That cue was assumed to influence participants' pain expectations. Here the selected cue gives the participant 50% chance of receiving pain level '4' and 50% of receiving pain level '7'. With this cue selected the expected pain level for a participant who is not particularly anxious or overly optimistic is 5.5. Participants then received a painful electric stimulation corresponding to one of the levels depicted on the selected cue, e.g. a '4' (third panel). Finally the participant rates their pain experience (fourth panel). In Experiment 1 the size of the portion on which the number was displayed corresponded to the probability of receiving pain of that level. In Experiment 2 all probabilities were fixed to 50%. Figure 1 shows an example of a trial from Experiment 1. In this example the target cue presented a 50% chance of receiving level 4 pain and 50% chance of receiving level 7 pain. This target therefore provided an 9 Bayesian model of pain expectations expected pain intensity of 5.5, and thus it is superior to the lure cue, which presents a 50% chance of receiving level 5 pain and 50% chance of receiving level 7 pain (expected intensity = 6). The cues were designed to provide a factorial design involving the manipulation of the following variables across the sequence of trials experienced by each individual: the expected value of the pain (the mean: 4.5, 5.5), the number of different pain levels that could potentially be delivered (2, 4), the prediction error (the difference between expected level of pain and intensity of the delivered stimulation: 0.5,1.5) and its sign (positive or negative). Using the trial depicted in Figure 1 as an example, if the target cue was selected and the level 7 pain subsequently delivered, then the prediction error would be +1.5 (7 -5.5). The positive sign of the prediction error indicates that the pain level that was experienced was higher than the average pain level participants would expect on the basis of the cue they selected. In summary, the experimental corresponded to a 2 (mean expected pain: 4.5, 5.5) x 2 (number of cue options: 2, 4) x 2 (direction of the prediction error: positive, negative) x 2 (the size of the prediction error: 0.5, 1.5) design.

Participants
Sixteen undergraduates from the Manchester University School of Psychological Sciences (14 female, Mean age 19.6, σ = 1.36) participated in the study for course credit. Participants were excluded if they had a history of psychiatric or neurological disorders. All participants achieved criterion of 85% accurate performance (defined as choosing the target cue). The study received ethical approval from the North West 6 Research Ethics committee (Greater Manchester South).

Materials
Acute experimental pain was delivered to the participants using electric stimulation, shown to produce effects of expectations that are equivalent to those produced with the more typical laser stimulation (Hird, Jones, Talmi, & El-Deredy, 2018). Compared to laser pain, which is used more often in the laboratory, the electric stimulation method is safe to use repeatedly and easier to implement in clinical settings. The electrical stimulations were delivered to the back of the right hand via a ring electrode built in-house (Medical Physics, Salford Royal Hospital) attached to a Digitimer DS5 Isolated Bipolar Constant Current Stimulator (www.digitimer.com). To ensure adequate conductance between the electrode and the skin, the back of each participant's hand was prepared with Nuprep Skin Preparation Gel and Ten20 Conductive Paste prior to the electrode being attached. The experimental paradigm itself was delivered via a laptop using Cogent2000 on a Matlab platform (www.Mathworks.com). The inputs to the DS5 machine were sent from Matlab via the Spike2 software (www.ced.co.uk) and a 1401plus data acquisition interface (www.ced.co.uk).

Procedure
Participants were given an information sheet prior to the study informing them of the justification for the study and of the use of electrical stimulation. On arriving at the laboratory participants were first asked to sign a consent form, and then to confirm that they had read the information sheet. The electrode was then attached to the back of the participant's right hand. Once the electrode was attached the participants undertook a pain calibration procedure. This procedure was necessary firstly to ensure that the participant could tolerate the stimulations, and secondly to ensure that the stimulations were psychologically equivalent across participants. Once the pain calibration procedure was complete participants were given instructions relating to the pain rating task. Participants were then given 6 practice trials of that task before undertaking the task proper.

Bayesian model of pain expectations
During this procedure participants received a series of stimulations, starting from 0.2V, and incrementing by 0.2V at each step. Participants rated each stimulation on a scale from 0 -10 where a score of 0 reflected not being able to feel the stimulation, 3 reflected a stimulation level that was on the threshold of being painful, 7 related to a stimulation that was deemed 'painful but still tolerable' and 10 related to 'unbearable pain'. The scaling procedure was terminated once the participant reported the level of pain as being equivalent to '7' on the scale. This calibration procedure was performed twice to allow for initial habituation/sensitisation to the stimulation. The voltage levels rated as '3', '4', '5', '6', and '7' on the second run of the calibration procedure were used as the different pain levels to be delivered during the pain rating task. The average voltage used for each pain level are shown in Supplementary Table 1.

Pain rating task.
Each trial began with a 250ms fixation cross, before two cues, presented as two pie charts, appeared sideby-side on the screen (+/-160 pixels from the centre: Figure 1). Participants were required to choose between these cues using the mouse. Numbers on the 'slices' of each pie chart indicated the level of stimulation that the slice related to, while the size of the slice depicted the probability of the stimulation level being delivered if the chart was selected. Participants had up to 8 seconds to make their selection. The target cue was always clearly preferable to the lure in terms of expected pain value (see General Methods).
If the participant failed to make any selection within the 8s time limit then the lure chart was automatically selected. The selected cue was then presented alone in the centre of the screen for 2 seconds, at which point the cue disappeared and one of the pain levels depicted on the cue was delivered to the back of the hand according to the probabilities depicted on the cue. Finally a visual-analogue scale (Hjermstad et al., 2011) appeared 500ms after the offset of the stimulation. Participants were required to rate their experience of the stimulation on this scale using the mouse, with the trial only proceeding once a rating had been given. Participants were told that even though the stimulation they would receive would always be one of those predicted by the selected cue, they should rate the pain they actually felt. After the rating 12 Bayesian model of pain expectations was provided an inter-trial interval occurred of a length selected randomly from 1000, 1500, 2000, 2500 & 3000ms (average 2000ms).

Experimental task.
Participants undertook 5 blocks of trials, with 32 trials in each block, giving a total of 160 trials. Each block consisted of a randomised presentation of 24 types of trials, depicted in Supplementary Table 2. In 16 of these the target cue corresponded to one of the 16 experimental conditions according to the 2x2x2x2 experimental design described above. Four additional trial types were required to complete the design (see Supplementary Table 2). Four further trials types, involving target cues which depicted the delivery of pain levels 3, 4, 5 and 6 with 100% probability, were included for the purpose of a manipulation check. A selftimed break was given every 20 trials.

Experiment 2: Method
The design of Experiment 2 was similar to that of Experiment 1. The following changes were made with the potential clinical application of the paradigm in mind. Firstly sets of playing cards were used as cues instead of pie charts ( Figure 2). This allowed the results from Experiment 1 to be (potentially) replicated using cues that may be more readily understood by a clinical audience. Secondly only trial types involving 2 different intensities with equal (50%) probabilities were included in the factorial design. This meant that only 3 aspects of pain expectation were manipulated within participants; the expected pain level (4.5, 5, 5.5), the size of the prediction error (0.5, 1.5, 2), and the direction of the prediction error (positive or negative). This change allowed a reduction to the number of trials appearing in the experiment, and therefore a reduction in the length of the experiment. This was thought beneficial due to the limited time available in the clinic for diagnostic procedures.

Participants
Thirty-two undergraduates from the Manchester University Psychology Department (26 female, Mean age 19.3, σ = 1) participated in the study for course credit. Participants were excluded if they had a history of psychiatric or neurological disorders. The study received ethical approval from the North West 6 Research Ethics committee (Greater Manchester South). Data from three participants was excluded because they did not achieve criterion of 85% accurate performance (defined as choosing the target cue during the cue selection phase) leaving a sample of 29 (24 female, mean age 19.3, σ = 1).

Materials and procedure
The materials were the same as in Experiment 1, but instead of charts, the cues consisted of spade suit playing cards. Each cue took the form of a pair of playing cards (100x130 pixels in size), with each card corresponding to one slice of the charts used in Experiment 1. During the selection phase the two cards from each pair of cards were presented 225 and 100 pixels from the centre of the screen, with one pair being presented to the left of the screen and the other to the right. After selection, the two cards comprising the selected cue were presented +/-75 pixels from the centre of the screen.
Participants undertook 5 blocks of trials. Each block consisted of a randomised presentation of 16 types of trials depicted in Supplementary Table 3. In 8 of these the target cue corresponded to one of the 8 experimental conditions corresponding to a 2 (the expected pain level: 4.5, 5.5) x 2 (the size of the prediction error: 0.5, 1.5) x 2 (the direction of the prediction error: positive or negative) experimental design. In 4 trial types the target cue corresponded to one of the 4 experimental conditions corresponding to a 2 (the size of the prediction error: 0.5, 1.5) x 2 (the direction of the prediction error: positive or negative) experimental design. We employed these two factorial designs together in order to compare their efficacy, and decide which design was best for future experiments. Four additional trial types involving target cues which depicted the delivery of pain levels 3, 4, 5 and 6 with 100% probability were again included for the purpose of a manipulation check. In total, therefore, each participant undertook 80 trials,

Modelling approach
HPP suggests that our perceptual systems are organised in a hierarchical manner. Expectations (priors) held at higher levels of the perceptual hierarchy, are compared to incoming pain at lower levels. Errors in prediction are then projected back to the higher level, causing the resulting pain experience (posterior) to be updated, which in turn alters the basis on which subsequent priors are calculated (Buchel et al., 2014).
Our model is influenced by this framework, formalising response to pain in terms of a Bayesian prior-toposterior updating process.
For participant i during trial j we represent the pain stimulation that was delivered as and the pain they reported as , both expressed on a 0-100 scale. The ratings were linearly transformed ( → ′ = + × ) using the pain responses on the 100% probability trials to standardize the responses across participants so as to avoid issues of differential scale use. Thus and were selected to minimize the squared error between participants' ratings and the delivered intensity. Before the pain stimulation was delivered, participant i at the jth trial also saw a cue. Let the cue consist of information , involving the possible magnitudes ( 1 … ) of the incoming pain stimulus, also expressed on a 0-100 scale, and their corresponding probabilities of occurrence, ( 1 … ). This information provides a specification of a probability distribution with mean and standard deviation respectively given by On the jth trial, an individual i exposed to cue represents the predicted intensity of the upcoming pain as a probability distribution P( | ). Using the terminology of HPP, this is the 'prior distribution'.
We formulate ( | ) as a product of two normal distributions, one incorporating cue-independent information and the other incorporating the cue information (see appendix for derivation): where Normal(x, y) stands for the normal distribution with mean a and standard deviation b, occasionally truncated to make sure that random variables satisfy their range constraints, whenever appropriate. In this term, each predicted value of is compared with the external sensory information, associating each such value with a measure of 'surprise', which our model takes to increase with an increasing distance between and the actual intensity of the stimulation, . The value of β determines the extent that the perceptual mechanism will tolerate a given disparity between the predicted experience and the actual sensation, with smaller values of β implying greater sharpness of the likelihood function around and hence a reduced tolerance. Plugging Equation 1 into Equation 2 we get the full model: Here the prior is combined with the surprise in accord with the Bayesian formalization of the integration of multiple sources of evidence.

Model specification
Data from each experiment were analysed separately with the following models:

Inference
Data from each experiment were analysed separately. For each model and each experiment, we calculated the Deviance Information Criterion (DIC, Spiegelhalter et al., 2002), which penalises models for increasing complexity. Using data from Experiment 1, we first compared fixed-parameter models 1-4 where all participants shared the same values for the parameters. The baseline model (Model 1) serves as the null hypothesis, as it assumes that the cue exerts no effect on pain experience, namely, no effect of expectations of any kind. Next, the same data were analysed with an elaboration of the winning fixedparameters model, now allowing the parameters β, ρ, η, µ, and ν, and to vary across individuals (the 'Variable full Bayesian model', Model 5). We took the values of these parameters in individual i (respectively denoted by β i ρ i , η i, µ i , and ν i ,) to be independently drawn from corresponding population-18 Bayesian model of pain expectations level normal distributions, with hyperparameters independently drawn from flat hyperprior distributions, resulting in a fully-Bayesian, hierarchical (multilevel) model. This model allows us to describe the variability of the individual-specific posterior estimates for parameters, and therefore to infer how the parameters vary across the population. Finally, we fitted these models to data from Experiment 2, to provide validation that the findings from the first dataset would generalise to an independent dataset.
As regards parameter estimation, the 'Full Bayesian model', when combined with data, gives rise to a Bayesian posterior distribution for the unknown model parameters. We sampled this posterior by using the

Results
The average performance during the cue selection task (i.e. proportion of targets selected) was 96% in Experiment 1 and 97% in Experiment 2, suggesting that participants understood the task well. In both experiments the mean ratings given during the trials where the cue provided 100% chance of a particular stimulation intensity increased linearly as a function of increased stimulation levels, confirming that participants were able to distinguish between the stimulation levels (Supplementary Table 4). Bayesian model of pain expectations that the cue has some effect on the rating. The best-fitting fixed-parameter model was the 'Full Bayesian model' (Model 4). This finding supports the hypothesis that both cue-dependent and cue-independent biases influence pain perception. When the same models were tested using the data from Experiment 2, the results of the model comparisons were the same (Table1).       Figure 3 illustrates the way in which Model 5 explains the responses of two different participants in Experiment 1 to a trial where these participants were exposed to the same cue. The charts on the top row illustrate their cue-independent expectations before either the cue or stimulus is delivered. In the middle row, the The left participant's value for ρ is markedly higher than that of the right participant. This causes the 'prior distribution' in the second row to be markedly less precise. In the third row, this results in the posterior being closer to the actual intensity of the delivered stimulus. We might characterize the behaviour displayed in the left diagram in terms of the cue exerting little influence on the participant's expectation and rating. The right participant's value for ρ is markedly lower, which is reflected by the sharper 'prior distribution', centred around the cue mean, . This causes the strong pull of the posterior towards the cue mean.

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Bayesian model of pain expectations  respectively. We see that the cue-independent priors differ in location and strength leading to differences in how the cue shapes the 'Prior Distribution'.

Figure 4: Individual-level, cue-independent priors representing trait-like bias. The graphs represent the cue-independent priors for participants in Experiment 1 under the 'Variable full Bayesian model' (Model 5).
For example, the bold blue distribution represents a participant who is more pessimistic about pain than the dashed blue distribution in terms of the mean pain that they expect. The distributions vary in both their mean and variance with the variance determining how strong the posterior will be biased towards the mean. Bayesian model of pain expectations µ, ν, β and ρ, have narrower credible intervals, and less overlap across individuals, which suggests these parameters could provide a good basis for classifying individuals. This suggests that individuals can be accurately ranked on the basis of the latter set of parameters.

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Bayesian model of pain expectations

Discussion
Five models of pain perception were tested using data collected from two experiments that utilised a short, novel and deception-free experimental paradigm. A model comparison performed using data from the first experiment supported a number of hypotheses. Firstly, it was found that increased variance in pain expectations decreases the influence that those pain expectations exert on the perception of pain.
Secondly, it was shown that adding parameters that describe cue-independent biases in pain perception significantly improved model fit, suggesting that such biases produce an effect on pain perception that is independent of cued pain expectations. Thirdly, as the best fitting 'Full Bayesian model' was influenced by the Hierarchical Predictive Processing (HPP) framework, the results of this study lend support for the HPP model of perception. Finally, this winning model produced a significantly better fit of the data when its parameters were allowed to vary at an individual level, suggesting that there are substantial individual differences in sensitivity to aspects of pain expectations. We also show that this individual-level model can be used to distinguish individuals based on the characteristics of their pain expectations. Importantly, the above model comparison results were replicated when the analysis was repeated using data from a second experiment which involved an independent sample.
The comparison between the fits achieved by the models presented in this paper allowed an assessment of the viability of the HPP framework to be made. The fact that Model 2 outperformed Model 1 replicated the typical finding the expectation can alter pain perception (e.g. Atlas & Wager, 2012;Bingel et al., 2011;Colloca, & Benedetti, 2006;Peerdeman et al., 2016;Tracey, 2010). More interestingly, the fact that Model 3 outperformed Model 2 confirms that the uncertainty around the expected pain value modulates its influence on pain perception (Brown, Seymour, Boyle, El-Deredy, & Jones, 2008b;Colloca, Petrovic, Wager, Ingvar, & Benedetti, 2010). Given that the model parameter which captures the effect of expectation uncertainty (η) was found to have a positive value, it can be concluded that greater uncertainty in the pain expectation decreases its effect on perception. This finding aligns with the predictions of HPP, but is in contrast to previous work which has suggested that greater expectation uncertainty may increase the influence of expectation (Watkinson et al., 2013) or may increase the level of perceived pain independent of the expectation (Yoshida et al., 2013). The reason for these divergent findings needs to be investigated in future research. It is possible that the different types of pain stimuli used (pressure by Watkinson et al., and temperature by Yoshida et al.,) may generate different results to those achieved via the electrical pain stimulation used in the current study. Indeed, although expectations of electric pain and laser pain influence ERP markers of pain delivery similarly (Hird, Jones, Talmi, & El-Deredy, 2018), their effect on neural markers of pain anticipation differs (Babiloni et al., 2007;Hird et al., 2018). Alternatively, as Yoshida  (Clark et al., 2008;Krol & El-Deredy, 2011;Krol & El-Deredy, 2015).
Another finding of interest in relation to the cognitive aspects of pain expectation was that Model 4 outperformed Model 3, a result which suggests that cue-independent (i.e. trait-like) differences in pain expectation have a significant effect on pain perception over and above the effect of cue-dependent pain expectations. The way we modelled the effect of cue-independent expectations, as shaping the prior distribution that was informed by the cue, was inspired by work on cognitive expectations where optimism bias modulates participants' ability to learn from the information they are given (Garrett & Sharot, 2017).
Experimental studies of the pain perception process have often ignored the influence of expectations that exist outside of those generated by the experimental paradigm, which in clinical settings are known to contribute greatly to pain experience. In the case of chronic pain patients such expectations are likely to contribute more to perceived pain than more cue-dependent expectations. Because pessimistic expectations about procedure pain could prevent patients from taking up critical preventative treatments, such as colonoscopy (Trevisani, Zelante, & Sartori, 2014) future work on pain expectation therefore needs to consider the influence that individual differences in cue-independent pain expectations have on pain perception.
While much previous research has supported HPP's account of pain perception (e.g. Brown et al., 2008a;Brown et al., 2008b;Brown, El-Deredy, & Jones, 2014;Clark, 2013)   . In a clinical setting, such individuals might benefit most from psychological pain therapies that concentrate on information processing biases relating to pain, as well as from training methods to increase their somatosensory sensitivity (Huque, Poliakoff, & Brown, 2017). In contrast, parameters µ, and ν, capture trait-like differences in pain expectation, which are independent of contextual information (e.g. the cue information). The value of parameter µ specifies the mean pain expected before the effect of any contextual information is considered (i.e. the cue in the experimental paradigm used here) while parameter ν allows us to set apart individuals with exceptionally low or high confidence in this cueindependent expectation. These cue-independent parameters can be thought of as analogous to an optimism/pessimism trait relating to future pain. In a clinical setting, individuals with a high value of µ and a low value of ν may be more likely to benefit from psychological pain therapies that concentrate on general beliefs about pain.
The individual differences that the methodology presented in this paper quantifies go beyond what is available through existing self-report instruments, such as the well-established pain catastrophizing scale (Sullivan, Bishop, & Pivik, 1995). In chronic pain, high PCS scores are associated with reports of more severe pain and reduced efficacy of physical and psychological therapies (Keefe et al., 1989). PCS and other selfreport measures offer a fast, reasonably accurate, but ultimately gross and subjective, description of pain expectation and experience. They cannot explain what aspects of expectations drive changes in pain experience. Specifically, the same score on a pain scale, given by two patients, could arise through very different psychological processes. Arguably, the method described here to characterise individuals' sensitivity to expectation goes further to reveal the underlying cognitive mechanism which drive the impact of expectations on pain experience.

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Bayesian model of pain expectations The ability of the methodology described in this paper to characterise the effect of pain expectations at the individual level could provide useful information in assigning treatment plans if it were applied in a clinical setting. As patients are likely to be differentially sensitive to expectations (Brown, El-Deredy, & Jones, 2014) a method, such as the one presented in this paper, which can potentially identify which patients are most sensitive to pain expectations, and which facets of pain expectations they are sensitive to, could help inform clinical decisions. For example, pain expectations are a key target in nonpharmacological therapies for pain, such as Cognitive Behavioural Therapy (CBT). CBT is effective in reducing chronic pain and pain-related disability partly because it changes patients' pain expectations (Bingel et al., 2011). Unfortunately, CBT is not as effective as we would wish: more than half of patients with chronic pain do not benefit from CBT (Peerdeman et al., 2016;Smith et al., 2016;de Williams, Eccleston, & Morley, 2012). Put simply, if a patient can be identified as being biased to report stronger pain they may be a better candidate for CBT than a patient who reports the delivered pain more accurately; and their specific expectation sensitivity profile can further improve the specific therapeutic intervention. It is hoped that the method described in this paper may ultimately prove useful in selecting suitable candidates for pain management treatment, adding quantitative information that is less affected by introspective limitations to existing self-report questionnaires that assess pain expectations. This could facilitate the personalisation of treatment selection, an approach that is particularly important in the current climate when health services are stretched, and access to effective treatment is limited (Enck et al., 2013). In addition to providing input to clinical decisions, the methodology presented here could also be used in clinical trial research to attempt to control for placebo effects when selecting control and experimental groups. To further the goal of providing a clinical tool using our approach, the current research needs to be repeated in a variety of clinical pain populations. Such work would be required in order to establish the clinical relevance of the individual parameters of pain expectations that have been established in this study.
One limitation of our design is that the pain intensity participants received was, on average, not very different from the pain intensity that they had expected. This may be important, because HPP resembles the assimilative part of assimilation-contrast theory (Hoyland, Harvey, & Sherif, 1957), which predicts that the distance between prior expectations and the physical stimulus is crucial. Both theories agree that when the distance is small, namely, when the input is not too discrepant from our existing anchor, we will assimilate it and our experience will reflect a 'compromise' between the reality and our expectations.
However, when the reality is very different to our expectations assimilation-contrast theory predicts that we will experience the real, discrepant stimulus as even more discrepant compared to our expectations.
Our data does not currently allow us to decide whether pain experiences show such contrast effects. An additional limitation of our design is that we are currently not able to distinguish between participants who rated what they 'really felt' and participants who rated their pain as closer to the values of the cue because of demand characteristics. The possible influence of demand characteristics is prevalent in studies of pain experience across paradigms, including placebo designs and cue-based designs, so this caveat is not unique to our design. Future experiments using our experimental paradigm could explore this issue further, for example by manipulating the working memory load of participants, which reduce goal-directed behaviour (Otto, Gershman, Markman, & Daw, 2013).
In summary, model comparison and validation yielded a number of important findings. The HPP framework was shown to provide an accurate description of pain perception. In particular the framework's predictions relating to the effect of expectation uncertainty were supported. Trait-like pain expectations that exist independent of any cued pain information were shown to have an influence on pain perception that was separate and distinguishable from those provided by cued pain expectations. Finally the methodology presented in this paper was able to characterise an individual's sensitivity to pain expectations, potentially providing the basis for a clinical tool that could guide practitioners who deliver individual psychological pain management interventions.
Bayesian model of pain expectations