Likelihood factors that can be disregarded for inference are termed ignorable. We demonstrate that close ties exist between ignorability and identification of causal effects by covariate adjustment. A graphical condition, stability, plays a role analogous to that of missingness at random, but is applicable to general longitudinal data. Our formulation of ignorability does not depend on any notion of missing data, so is appealing in situations where missing data may not actually exist. Several examples illustrate how stability may be assessed. For each of a set of subjects, a sequence of observations is recorded, corresponding to the same property or feature of the subject evaluated at different times. Longitudinal data are common in scientific investigations, and their analysis has received much attention .Longitudinal studies typically propose a schedule of measurement times in advance. Nevertheless, even in carefully designed experiments, the precise number and timings of observations are rarely completely determined by the investigator. Most obviously, a subject who dies during the course of a study can give rise to no further observations, scheduled or otherwise. In panel studies, a subject may fail to present for evaluation at an intermediate wave, but then return for final assessment at a later date. More generally, measurements may be recorded on quite arbitrary occasions and determined by convenience to the particular subject, availability of the investigator, or external factors such as weather conditions and public holidays.In addition to external circumstances, observation times may be influenced by internal, subject-specific factors. Mood, medical intervention or indeed the biological processes underlying the longitudinal assessments can all affect the likelihood of recording an observation. An important special case is informative drop-out, where attrition relates to health .Dependence between timings and observations may still be ignorable, in the sense that certain likelihood-based analyses are equivalent under such dependence. Ignorability is ordinarily described within a missing data framework, which presupposes that the missing data actually exist , p. 8. EWe argue that this fiction is not needed. Instead, we provide an alternative characterization of ignorability for general longitudinal data that does not depend on any notion of missing data. We do so by applying the machinery of causal inference  to the cFor concreteness, consider the longitudinal measurement of foetal size. In early pregnancy, foetal growth is often monitored using the crown-rump length, assessed electronically from an ultrasound image. Crown-rump length could in principle be measured at any point after conception. We stress, however, that crown-rump length does not actually exist at every point after conception. This is most obvious following birth, when it is no longer meaningful to measure crown-rump length by ultrasound. However, our argument is more general: no unambiguous, unique definition of crown-rump length can be made except on those occasions when it is actually measured. Certainly, a healthy foetus has a shape and size that is complex and growing more or less continuously, but crown-rump length is not simply a one-dimensional slice of this high-dimensional, continuous-time process; it is an external procedure subject to many influences apart from foetal size, including the skill of the sonographer, the resolution of the ultrasound, and the cooperation of the foetus. Consequently, crown-rump length only exists in a meaningful way on precisely those occasions when it is measured. To fix ideas, we refer to this example throughout.2i, and let  be a marked point process  is a standard point process, and records the observation times for this subject. The sequence y =  contains the corresponding longitudinal measurements. We resist the temptation to define for each of the n subjects an underlying continuous or complete measurement process; the existence or otherwise of this complete measurement process, however it might be defined, is irrelevant in our subsequent development.We omit subject-specific subscripts  process , p. 10 rjt = ∞ if fewer than j events occur altogether; drop-outs require no special handling. If jt = ∞, we assign to the corresponding jy the irrelevant mark ∇. Defining m = max{j : jt < ∞}, we formally admit the possibility that m = 0 and no measurements are made on a particular subject, but this case is usually of little interest. In our crown-rump length example, an idealized realization  of the marked point process in which m = 2 might comprise the elements t =  weeks and y =  millimetres.Following y and the time-points t at which they are observed, for they arise together and neither can exist without the other. Dependence can also be associational and dynamic: writing jt̄ for  and jȳ for , a standard construction of marked point processes  and  sequentially for j ⩾ 1.There is a fundamental dependence between the marks rocesses , p. 22 st and y. σ, a parameter that governs the behaviour of the system. This σ indicates if the system is operating in its original, observed state (σ = o) or, when σ takes different values, whether particular interventions into the system are being considered. Under σ = o, probability functions are written pr or just pr(·). In general, arrows from σ into other nodes indicate that their stochastic behaviour may be altered under different regimes. We consider the specific case where arrows emanate from σ to every jt, and only to the jt. For a specific example, see σ is equivalent to the F used in the appendix of We use directed acyclic graphs to summarize possible dynamic dependencies between jt and jy for every j. Other nodes will usually appear; we shall write observed baseline covariates generically as x, with u denoting random variables whose realized values are not observed. Whatever the larger picture of dependence between u, x, t and y, the influence diagram should contain, for every j, the subgraphs jt → jy, in which a directed arrow from jt to jy formalizes the notion that jy comes into existence at time jt and that if jt is changed, so too is jy.Our influence diagrams will contain nodes representing 3t and y is indeed inherent in a marked point process, then it can never be dismissed entirely. Instead, we may ask when this dependence can safely be ignored. A popular approach to the analysis of longitudinal data , do(t2), … , or simply do(t). Circumstances in which the presence, σ = do(t), or absence, σ = o, of such intervention is irrelevant to inference might be deemed ignorable, and we make this our definition.If dependence between y design , p. 482.EFINITION 1. The point process t is ignorable for inference based on y if pr ∝ pr{y; do(t)} with respect to some parameter of interest. More generally, the point process t is ignorable for inference based on y, conditional on x, if pr ∝ pr{y | x; do(t)} with respect to some parameter of interest.DIf the two objects are indeed proportional, then they may be used interchangeably for likelihood-based inference. Establishing general conditions under which this proportionality holds is precisely the question addressed by Ignoring the process that causes missing data means proceeding by: (a) fixing the [missing data indicator] at the observed pattern of missing data […], and (b) assuming that the values of the observed data […] arose from the marginal density of the [observed data].y; σ = do(t)} as the likelihood upon which inference is to be based. An immediate advantage of the causal formulation is that it provides the informative label pr{y; σ = do(t)}, or more simply pr{y; do(t)}, for the nameless integral construction that y; do(t)} is indeed a marginal probability function, it is computed with respect to the interventional regime σ = do(t) and is quite distinct from both the conditional pr(y | t) = pr and the more complicated marginal pr(y) = pr .This refers to a derived, interventional, distribution. We interpret Rubin’s statement as an intent to employ pr{t, y) and pr{y; do(t)} may be inferred through his condition known as missingness at random. However, because within a marked point process formulation there are no missing data, we require an analogous condition that does not employ notions of complete, observed and missing data. Like missingness at random, our contender, stability, is a simple, general, sufficient condition under which ignorability may be shown to hold. However, also like missingness at random, stability is not a necessary condition for ignorability; we return to this point in our discussion.Rubin answers the question of when proportionality between pr jexhibits simple stability if y ⫫ σ | .DNITION 2 . The marσ only has directed edges into the jt allows the following equivalent graphical check where σ is omitted from the graph: the marked point process is stable if there are no unblocked back-door paths between jt and jy, i.e., paths between jt and jy, with an arrow into jt, that are not blocked by any of  in the sense of d-separation  is stable, thenTOROLLARY 1. If the marked point process  is stable, then maximizations over parameters occurring in only one of its two likelihood factors may equivalently be performed over either the full likelihood pr or the relevant factor. In particular, any parameters that occur only in pr{y | x; do(t)} may be maximized over this simpler likelihood.CProof. The standard decomposition for the likelihood of a marked point process is t, y) is stable, pr = pr{jy | x, jt̄, ȳj−1; σ = do(t)}, whence σ = do(t), the jt are deterministic, so jy | x, ȳj−1; do(t)} telescopes to yield the required factorization.t, y) is stable, likelihood inference may proceed solely on the basis of pr{y | x; do(t)}, provided that the parameters of ∏ pr are suitably distinct from the parameters of interest in pr{y | x; do(t)}. It is possible, though unusual, that stability may plausibly be assumed to hold but the parameters of ∏ pr and pr{y | x; do(t)} are not thought to be distinct. To see that this need not affect consistency of estimation, consider the following argument, due to Peter Diggle. Since parameterization is essentially a modelling decision, in this case we may consider the larger model in which any parameters in common are replaced by two variation-independent sets of parameters, one for each of the two likelihood factors. Providing parameters are replaced throughout any given conditional probability, the product of these conditional probabilities remains a probability measure that is consistent with the data-generating mechanism. It is then clear that estimation of the parameters of pr{y | x; do(t)} will still be consistent based only on maximizing over this component of the likelihood, albeit with some loss of efficiency relative to using the full likelihood pr.If the marked point process }, in particular with their identification via adjustment for previous observations through so-called inverse probability weighting. Philosophically, we find it preferable to frame problems of ignorability in terms of causal inference, without reference to counterfactual missing data. In the next section, we argue also for the practical importance of this approach, particularly in assessing the plausibility of assumptions required for ignorability.The likelihood factorization of 4σ = o and σ = do(t), perhaps by way of a causal graph or an influence diagram. This gives expert judgement a formal place in analysis, and here we give some examples to illustrate how an expert, loosely defined, might go about assessing the plausibility of stability within our working example of foetal crown-rump length.Like missingness at random, stability is not nonparametrically testable ; that isExample 1. Consider an antenatal clinic in which the next recommended ultrasound scan date jt is set on the basis of foetal length measurements ȳj−1 from previous scans. For instance, pregnancies falling within reference ranges for foetal length might follow a standard scan schedule, while those showing unusually slow or rapid foetal growth might be invited to attend more frequently. There may also be privately scheduled, parent-initiated scans, perhaps because of underlying anxiety or a particular concern, summarized by tu in jt depends structurally on previous observations times t̄j−1, previous observations ȳj−1 and underlying parental influences tu but, given these, is independent of other past factors or processes, and in particular of yu.Any lack of adherence to these appointments is then assumed to arise from external factors that play no discernible role in determining the foetal length measurements, for example school holidays or local traffic conditions. In this way an observation time y may be incorporated in the usual way, by means of shared random effects yu. This can be thought of as capturing the underlying physical characteristics of the unborn child; but, crucially, these underlying characteristics are only thought to influence scan dates indirectly, by way of their influence on the measurements y.Covariance between scan measurements tu and yu are marginally independent, it is straightforward to show from first principles that pr equals y; do(t)}, and its form is the usual mixture distribution arising when random effects are integrated out of a likelihood function, as in yu had an arrow directly into any tj.This is our canonical example of stability. Because Example 2. The assumption that parental influences on the timings of antenatal measurements are independent of their unborn child’s growth is arguably a strong one. In particular, it seems at least plausible that a mother’s health, u* say, could play some role in determining both her levels of antenatal anxiety tu and the growth of her child yu; such dependence is depicted in t, y) is now no longer stable: the likelihood function takes the form y; do(t)} is possible. The timings t are not ignorable for inference about y, and inferences based on pr{y; do(t)} will in general be biased. Stability may be re-established if tu can be replaced by a set of measurements x assessing antenatal anxiety, assuming for the sake of argument that this can be done without appreciable error in measurement. Conditioning on this x breaks the dependency between t and y, and allows us to write pr as x in the mixing distribution pr(u* | x). The second factor does, of course, reduce to pr{y | x; do(t)}.The marked point process  remains stable because the unobserved factor ju affecting jy influences only the future tj+1, not the current jt. Stability is immediately lost if equipment failures are not independent of one another.Despite unobserved common causes between jy. Stability would also be lost if this were replaced by the random effects yu of the previous two examples, a scenario that we find much more plausible.Also important for stability in this case is the assumed autoregressive dependence structure of the 5Biometrika] mentioned that he really wasn’t fond of the title of the already accepted [David Cox, then editor of  It seems to us that the missing data label might reasonably be applied if data were actually gathered but subsequently lost. An obvious example would be clinical data lost when a laptop computer disappears from a crowded train. In this rather specific sense, missing data do sometimes exist.We contend that missing data are not nearly so widespread as their prominence in the statistical literature would imply. It is, of course, sensible to formulate stochastic systems in terms of unobserved random variables. However, describing such unobserved variables as data seems to us appropriate only when they were at some time, by some means, given a specific value that could in principle have been observed, even if that value has subsequently become lost to us. This perspective can be traced back to the dawn of the missing data literature, as Formulations of ignorability that rely on missing data require the user to assign meaning to these quantities in order that the plausibility of assumptions concerning the missing data can be assessed. Little and Rubin observed that the ability to assign such meaning formally underpins the majority of their influential book  = pr(t | yobs). Since we have avoided defining complete data, we could not use this notation here. This is no bad thing, as the yobs, ymis notation is at best ambiguous and at worst confusing (yobs must at least encode the value of m and, in balanced monotone drop-out cases, actually determines t completely.In the context of longitudinal data, missingness at random has been defined in various ways and with varying degrees of formality. Many authors employ informal notation such as onfusing ; taken lMany variants of missingness at random, such as covariate-dependent missingness at random  or seque


In the field of neurodegenerative disorders (ND) such as Alzheimer’s disease, SG are currently employed both to support and improve the assessment of different functional and cognitive abilities, and to provide alternative solutions for patients’ treatment, stimulation, and rehabilitation. As the field is quite young, recommendations on the use of SG in people with ND are still rare. In 2014 we proposed some initial recommendations . The aim Recently, increasing attention is devoted to the field of neurodegenerative disorders (ND), such as Alzheimer’s disease (AD), where ICT is employed both to support and improve the assessment of different functional and cognitive abilities , and to ge e.g., . Howeverus group . Specifius group , SG traius group , and Serus group . A few mus group  and seveus group , but theND progress through several stages in several years, and ultimately lead to dementia, a decline in mental ability severe enough to interfere with activities of daily living. Dementia can result from different causes, the most common being AD. It is often preceded by a pre-dementia stage, known as mild cognitive impairment (MCI), characterized by a cognitive decline greater than expected for an individual’s age, which, however, does not interfere notably with activities of daily living . DependiN = 10) and healthy elderly controls (N = 8), and that during a 1-month trial participants experienced mainly positive emotions, improved their cardio-respiratory fitness, and were able to progress in the cognitive games scenarios. A summary of the training features of these three feasibility/pilot studies in reported in Table 1.Boosted by the publication of a Nature letter showing that video game training can enhance cognitive control in older adults , there iFigure 1) suggested that: (a) SG were considered between ‘adapted’ and ‘very adapted’ to people with MCI, and between ‘not very adapted’ and ‘adapted’ to people with AD and related disorders; (b) SG was rated between ‘adapted’ and ‘very adapted’ for assessment, stimulation and rehabilitation of people with AD and related disorders, to train family caregivers and healthcare professionals, with the best rated target being stimulation; clinical targets rated between ‘adapted’ and ‘very adapted’ included physical, cognitive and social stimulation, as well as apathy (while agitation and improvements in activities of daily living were considered between ‘not very adapted’ and ‘adapted’); (c) SG should be employed regularly ; (d) SG were rated between ‘adapted’ and ‘very adapted’ to be employed at home, in day centers and nursing homes, and between ‘not very adapted’ and ‘adapted’ to be employed in the hospital; (e) SG were rated between ‘adapted’ and ‘very adapted’ to be employed with someone , and between ‘not very adapted’ and ‘adapted’ to be employed alone (or with a robot).Data on the frequency of use employed in the three feasibility studies is converging with the recommendations reported by In the present paper we aimed to update and refine these initial recommendations thanks to a Delphi expert panel. A number of methodological changes were performed compared to the 2014 consensus panel. First of all, the 2014 recommendations were collected in a single round, without following the classical Delphi methodology. Indeed in the standard Delphi method  experts The recommendations reported in the present paper were collected and discussed during the workshop “Innovation Alzheimer 2016,” organized by the CoBTeK (Cognition – Behaviour – Technology) Research Unit of the Université Côte d’Azur, in Nice (France) on September 28th, 2016, in occasion of the 10th World Conference of Gerontechnology (ISG2016).N = 23) included researchers and health care professionals working on autism and other neurological and developmental disorders (n = 6), neurodegenerative disorders (n = 10), or both neurodevelopmental and neurodegenerative disorders (n = 2); ICT engineers (n = 2); and game developers (n = 3).The expert panel whom? Are SG adapted  to the following populations?SG for •Mild Cognitive Impairment (MCI);•Early to moderate dementia due neurodegenerative disorders such as AD (dementia)(b)What is the clinical target (in each condition)?•Assess •Train physical activity •Train cognitive functions •Improve autonomy (IADL)•Improve wellbeing •Favor social exchanges (Improve sociability and favor relations)•Teach contents(c)Where should SG be used, and with whom (in each condition)?•At home, alone•In a clinical facility , alone•At home, with a trained  caregiver•In a clinical facility, with a trained  caregiver(d)When (how frequently) should SG be used (in each condition)?•Once a week•Twice a week•3 times a week•4 times a week•5 times a week•EverydayParticipants were asked to rate each item on a 1–5 scale .N = 7), Bonferroni corrections were applied (α = 0.05/7 = 0.007). The third question  was analyzed by means of a repeated-measures ANOVA with Group (MCI vs. dementia), Where  and with Whom  as within-subject factors, in order to analyze the effect of the three factors and their interactions. Finally, the fourth question (“When should SG be used?”) was analyzed by means of a repeated-measures ANOVA with Group (MCI vs. dementia) and Frequency  as within-subject factors. As the methodology employed in the present study is not completely comparable to that employed in the 2014 recommendations paper, we compared the results of the two studies only at a descriptive level.Results were collected, and analyzed. During the workshop, a discussion session was organized with the objective to comment on the survey results, and to generate practical recommendations for the use of SG in MCI and dementia. Ratings from one participant were not considered for data analysis because more than 50% of responses were missing. Thus, reported data refer to 22 participants. For descriptive analysis purposes, we reported mean ratings. In order to compare ratings obtained for people with MCI and people with dementia in the first (“SG for whom?”) and second  questions, we performed separate repeated-measures ANOVAs on each response item with Group (MCI vs. dementia) as within subject factor. For the second question, in order to account for multiple comparisons  = 16.87, p = 0.001], suggesting that SG are considered as more adapted to people with initial cognitive decline than to people which are already loosing autonomy in activities of daily living. However, SG are considered to be adaptable also to people with dementia.Results are reported in Figure 2B. Participants reported that SG are between ‘very adapted’ and ‘completely adapted’ for assessment, to train physical and cognitive functions, improve wellbeing, and teach contents. Improving autonomy and favoring social exchanges were considered from ‘adapted’ to ‘very adapted.’ For people with dementia, SG were rated between ‘adapted’ and ‘very adapted’ for all the specified targets. This is in line with responses to the question ‘SG for whom,’ and it suggests that all these domains do represent useful targets for SG in this population. Repeated-measures ANOVAs (Bonferroni corrected) conducted to compare people with MCI and dementia for each category suggested that SG were considered as more adapted to people with MCI compared to people with dementia to train cognitive functions , to improve autonomy  and wellbeing  and to teach contents . All the other contrasts did not reach statistical significance (p > 0.007).Results are reported in Figure 2C) suggest that SG are mostly adapted (between ‘very adapted’ and ‘completely adapted’) to be employed with a trained caregiver, both in a home and in a clinical setting. The use of SG by patients alone was rated between ‘adapted’ and ‘very adapted’ for people with MCI, and between ‘not very adapted’ and ‘adapted’ for people with dementia. Repeated-measures ANOVA with Group (MCI vs. dementia), Where  and With whom  as within-subject factors confirmed a significant effect of With whom factor , with SG use with a caregiver as more adapted compared to SG use alone. The Where factor was not statistically significant , thus suggesting that SG are considered as equally adapted to be employed at home and in a clinical facility. A significant effect of Group was also found, with SG rated as more adapted to be employed with people with MCI compared to people with dementia in all the settings . Interestingly, a significant interaction between Group and With whom factor was also found , suggesting that employing SG with a trained caregiver is especially important for people with dementia. No other 2-way or 3-way interaction reached statistical significance .Results for all these patient populations  suggest that all the game frequencies were rated between ‘adapted’ and ‘completely adapted’ for all conditions, and the mean ratings of the different game frequencies can be visually described as skewed Gaussian distributions. Repeated-measures ANOVA with Group (MCI vs. dementia) and Frequency  as within-subject factors revealed no significant main effect of Group , and Frequency , and no significant interaction between Group and Frequency . Converging with descriptive data, the ANOVA’s contrast tests revealed an almost-significant quadratic contrast , suggesting that categories in the middle of the curve were rated as more adapted compared to extreme values (1 and 7 days per week). For participants with MCI and dementia, the highest scores were obtained for game frequencies from 2 to 4 days a week.Results . However, emerging SG are also targeting the social domain. Some of these SG are showing promising results . Training targeting the cognitive domain and teaching contents may be more suitable to people with MCI compared to people with dementia, as they require some intact learning abilities to be optimized. However, beyond the selection of a clinical target for each patient’s category, an important aspect to take into account is that the selection of a SG should be  patient . The ass patient , and dem patient . This do patient .The experts suggested that most of the SG are more effective when the patient is accompanied by a trained caregiver . This is consistent with the general recommendations on SG usability drafted by Concerning the Where question, most of the SG are considered as useful to be delivered both at home and in clinical facilities. Clinical facilities have the advantage to allow a complete standardization of the training, and to provide a secure, controlled environment. The main problem, however, is represented by the travel time and costs. In order to improve study adherence, Training frequencies between two and four times a week were rated as the most adapted for people with MCI and dementia. Evidence collected in recent studies on both cognitive and cognitive-physical training converges with this recommendation e.g., . HoweverAnother crucial aspect to take into account when designing training is the motivation aspect. In general, the longer and more intense the training, the better. However, ‘forcing’ employing the SG too often may result in lowered motivation, or even in addiction, as there is a fine line between motivational and addictive aspects . IdeallyThe present recommendations were gathered from a relatively small group of experts working in the domain of SG for health. In further work, it would be interesting to verify if these results hold for a wider expert population . Furthermore, it would be interesting to collect the opinion of healthcare professionals who do not work with SG and ICT in their practice, to verify the barriers for SG adoption in the healthcare domain. For instance, if clinicians non-expert in ICT consider SG as not adapted and not useful for people with ND, they will hardly suggest their use for training purposes. This means that more effort should be made to share the promising results of SG in these populations among the clinical community.VM analyzed the data. All the authors contributed to draft the paper.AG is co-founder, shareholder, BOD member, and advisor for Akili Interactive Labs, a company that produces therapeutic video games. PF works for Genious group, a company that produces serious games. All the other authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


The “entropic brain hypothesis” holds that the quality of conscious states depends on the system's entropy . However such a measure quantifies (the number of) Criticality of the psychedelic state is inferred through the following reasoning: Criticality maximizes the amount of metastable states  Criticality implies (a set of) scaling variables, e.g., avalanche size and duration. Nothing guarantees that the dynamics producing these metastable states in Carhart-Harris et al.  would alFurthermore, there is an essential contradiction in the suggestion that primary consciousness would be a “psychological atavism” and a “suboptimal mode” and, at the same time, “more critical” than normal states (p. 12). On the one hand, intuitively, the psychedelic state wouldn't appear to be functionally optimal. The hyper-connectivity which characterizes it (Tagliazucchi et al., per se, to assess criticality. Rather than just moving brain activity toward or away from criticality, context-specific demands can modulate the scaling properties of brain fluctuations (Popivanov et al., A true configurational entropy requires extensive phase space exploration, otherwise it merely reflects the random side of complexity. Under these conditions, quantifying the number of metastable states is not enough, The author confirms being the sole contributor of this work and approved it for publication.The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


The nervous system is a non-linear dynamical complex system with many feedback loops. A conventional wisdom is that in the brain the quantum fluctuations are self-averaging and thus functionally negligible. However, this intuition might be misleading in the case of non-linear complex systems. Because of an extreme sensitivity to initial conditions, in complex systems the microscopic fluctuations may be amplified and thereby affect the system’s behavior. In this way quantum dynamics might influence neuronal computations. Accumulating evidence in non-neuronal systems indicates that biological evolution is able to exploit quantum stochasticity. The recent rise of quantum biology as an emerging field at the border between quantum physics and the life sciences suggests that quantum events could play a non-trivial role also in neuronal cells. Direct experimental evidence for this is still missing but future research should address the possibility that quantum events contribute to an extremely high complexity, variability and computational power of neuronal dynamics. TherefoThere are two main arguments, which are usually raised against the quantum brain hypothesis:too large for quantum phenomena to play a significant role in their functioning. The conventional wisdom is that all quantum events are averaging out, so that fluctuations among quantum particles are not important. As expressed by Daniel Dennett:(1) Neuronal signaling molecules, neurons and neural networks are “Most biologists think that quantum effects all just cancel out in the brain, that there’s no reason to think they’re harnessed in any way. Of course they’re there; quantum effects are there in your car, your watch, and your computer. But most things — most macroscopic objects — are, as it were, oblivious to quantum effects. They don’t amplify them; they don’t hinge on them” .Christoph Koch and Klaus Hepp, too, identify the large size of neuronal objects and the huge number of particles involved in neuronal signaling as one of the critical weak points of quantum brain hypothesis:“Although brains obey quantum mechanics, they do not seem to exploit any of its special features. Molecular machines, such as the light-amplifying components of photoreceptors, pre- and post-synaptic receptors and the voltage- and ligand-gated channel proteins that span cellular membranes and underpin neuronal excitability, are so large that they can be treated as classical objects. … Two key biophysical operations underlie information processing in the brain: chemical transmission across the synaptic cleft, and the generation of action potentials. These both involve 1000s of ions and neurotransmitter molecules, coupled by diffusion or by the membrane potential that extends across 10s of micrometers. Both processes will destroy any coherent quantum states. Thus, spiking neurons can only receive and send classical, rather than quantum, information. It follows that a neuron either spikes at a particular point in time or it does not, but is not in a superposition of spike and non-spike states” .interaction of neuronal molecules, neurons, or neuronal networks with their noisy, wet and warm environment will destroy any non-trivial quantum states such as superpositions or entanglements. If this were true, only trivial quantum effects could be present in the nervous system. But what is the difference between trivial and non-trivial quantum effects? Trivial quantum effects provide the basis for the structure and chemical properties of molecules and they are ubiquitous . Hence, trivial quantum effects are crucial for the basic biochemistry of neuronal molecules, but when considering the neuronal function, these trivial quantum features can be ignored and the molecules important for neuronal signaling can, allegedly, be treated as essentially classical. All interesting coherent quantum states which are necessary for any non-trivial quantum computation ((2) The second important criticism is that the putation  can, allputation , decoherputation .non-trivial quantum effects are present in biological systems – and not just in spite of, but sometimes because of, the interaction with the noisy and warm environment. Furthermore, because the brain is a complex non-linear system with high sensitivity to small fluctuations, it is likely that it can amplify microscopic quantum effects. Specifically, there are two alternative but interrelated ways in which quantum events may influence the activity of the brain  Contrary to expectations, non-trivial quantum processes have been observed in living systems . Experime brain” .coupling to the noisy and warm environment is able to promote rather than hinder long-lasting quantum coherence in biological systems  If it turned out that quantum effects cannot be observed in living systems at the macroscopic level, would that mean that living systems can be fully described by classical physics? Or is there another plausible way in which small-scale quantum effects – there is evidence for their occurrence [see (1)] – might influence large-scale neuronal activity and behavior? Yes, there is. The common view that minuscule fluctuations, including quantum events, cancel out in larger systems need not be true in highly non-linear systems like our brain. The nervous system can be seen as a nested hierarchy of non-linear complex networks of molecules, cells, microcircuits, and brain regions. In iterative hierarchies with non-linear dynamics (at the edge of chaos), small  fluctuations are not averaged out, but can be amplified. Quantum fluctuations on the lowest level of scale may influence the initial state of the next level of scale, while the higher levels shape the boundary conditions of the lower ones. This hierarchy of nested networks with many feedback loops exploits rather than cancels out the quantum effects as proposed by Jeffrey Satinover:“[Q]quantum dynamics alters the final outcomes of computation at all levels – not by producing classically impossible solutions but by having a profound effect on which of many possible solutions are actually selected” .In his essay on free will and neuroscience, Haim Sompolinsky has also mentioned this possibility:“Chaos within the brain may amplify enormously the small quantum fluctuations … to a degree that will affect the timing of spikes in neurons” .Similarly, even Christof Koch, one of the major critics of quantum brain ideas, had to admit:“What cannot be ruled out is that tiny quantum fluctuations deep in the brain are amplified by deterministic chaos and will ultimately lead to behavioral choices” .The quantum amplification mechanism has been adopted also by Scott Aaronson whose recent “freebit” theory of free will “postulates that chaotic dynamics in the brain can have the effect of amplifying freebits to macroscopic scale” . A similneuronal avalanches. These avalanches have a characteristic distribution: each avalanche engages a variable number of neurons, but, on average, many more small avalanches are observed than large ones. This indicates that neuronal networks are poised near criticality  random/supracritical dynamics consisting of asynchronous irregular activity with low coordination and high variability, and (3) complex/critical dynamics with high coordination and high variability. Brain states exhibiting complex/critical dynamics are the most interesting ones because they support the most efficient information processing  world, surprising events do occur, but regularities, too, are present so that the brain needs to register but also to update the stored memories.What is the evidence for the proposal that the brain is a complex non-linear system, capable of chaotic dynamics ? Beggs aactivity . SimilarMRI data  and fromMRI data . In geneocessing . At the ocessing  is obserocessing : in a su“[B]rains seem “balanced on a knife-edge” between order and chaos: were they as orderly as a pendulum, they couldn’t support interesting behavior; were they as chaotic as the weather, they couldn’t support rationality” .Thus, it is plausible, that small quantum fluctuations can be amplified, since brain activity can develop to the critical point: the point of complex neuronal dynamics. Interestingly, recent calculations suggest that quantum coherence can become long-lived in complex systems which are in a critical state between chaos and regularity – at the edge of quantum chaos .In summary, the intricate interplay between quantum effects and non-linear complex dynamics might be able (a) to generate new persistent quantum-chaotic patterns at a microscopic scale, (b) to amplify quantum effects to a macroscopic scale . How exaHow could the two scales (the microscopic and the macroscopic one) be interconnected? Is there a biologically plausible way in which small-scale quantum effects could influence large-scale neuronal activity and behavior? In their book, Interestingly, recent research in psychology has provided another hint that the brain displays quantum-like behavior . This leLong-lasting quantum coherence phenomena have been first observed in non-neuronal systems. Therefore, non-neuronal organisms such as bacteria, algae or land plants can be useful model systems to study the relationship between biological information processing and quantum events. Many molecular mechanisms, which support complex information processing, including neurotransmitter molecules, ion channels, gap junctions, evolved before the emergence of the nervous system during evolution . Molecul“[Physicists] invented the deterministic-reductionistic philosophy and taught it to the biologists, only to walk from it themselves” (“anything can be reduced to simple, obvious mechanical interactions. The cell is a machine. The animal is a machine. Man is a machine” (Although new ideas and concepts are emerging , reductimselves” . The dommachine” . Direct The author confirms being the sole contributor of this work and approved it for publication.The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


We apply Intersectional Theory to examine how compounded disadvantage affects the odds of women having a cesarean in U.S.-Mexico border hospitals and in non-border hospitals. We define U.S. Latinas with compounded disadvantage as those who have neither a college education nor private health insurance.Analyzing quantitative and qualitative data from Childbirth Connection’s Listening to Mothers III Survey, we find that, consistent with the notion of the Latinx Health Paradox, compounded disadvantage serves as a protective buffer and decreases the odds of cesarean among women in non-border hospitals. However, the Latinx Health Paradox is absent on the border.higher odds of a cesarean compared to women without such disadvantage. Further, women with compounded disadvantage who give birth in border hospitals report receiving insufficient prenatal, pregnancy, and postpartum information, providing a direction for future research to explain the border disparity in cesareans.Our data show that women with compounded disadvantage who give birth on the border have significantly  The U.S.-Mexico border region is a politically significant geographical region, as evidenced by its centrality in the 2016 United States presidential campaign. In fact, this strip of land, which spans 2000 miles, 48 counties, and four states, is often used to highlight hot-button political issues like immigration, crime, and poverty t the population level, caesarean section rates higher than 10% are not associated with reductions in maternal and newborn mortality rates” . U.S. HeA few studies have analyzed cesarean rates on the U.S-Mexico border. An analysis of birth certificate data compared cesarean rates in Mexico to the cesarean rate of Hispanics in the United States and residents in border regions (both south and north of the border). This study found that U.S. border residents have a higher cesarean rate than the U.S. Hispanic cesarean rate. Both of these groups have a lower cesarean rate than the Mexican rate of 44.5% (in 2009). Additionally, a Texas-based study found that border residents, particularly Hispanic women, have a higher rate of cesareans than non-border residents . AlthougOur study fills a gap in this literature in two ways. First, we focus on how compounded disadvantage affects the odds of Latinas having a cesarean in border and non-border hospitals, which allows us to add nuance to the application of the Latinx Health Paradox on maternal outcomes. Second, we include an analysis of qualitative data on women’s birth experiences to better explore whether the Latinx paradox with regard to cesarean rates persists on the border.We used data from Childbirth Connection’s Listening to Mothers III survey: a nationally representative survey of 2400 women who gave birth in U.S. hospitals in 2011 and 2012 . This waThe dependent variable is mode of delivery, with attributes of cesarean or vaginal birth. The independent variable in this study is compounded disadvantage, measured as being Latina, without a college degree and without private insurance. We also ran analyses that examine the independent effect of ethnicity (Latina or non-Latina), educational attainment (four-year college degree and above or less than four-year college degree), and private health insurance status (absent or present). We further integrated border region into our analysis. We used the Centers for Disease Control and Prevention’s definition of the border, which is based on the La Paz Agreement and defined as “the area situated 100 kilometers on either side of the inland and maritime boundaries between [the United States and Mexico]”  needs and that my birth is special and important.” The majority of those who responded positively about their births were non-Latinas in the advantaged group on the border who had given birth vaginally (eleven out of twenty). Out of the three non-Latinas in the disadvantaged group on the border, two had vaginal births and gave positive responses. The remaining woman, Kristin, had a cesarean and did not give a positive response. When asked whether she would return to the same hospital, she responded: “I have no choice which hospital I give birth at. Our military medical makes us go to the [hospital] if we can make it. And, with c-sections with both living children, my third will most likely be a repeat c-section.” While Kristin’s answer is not inherently negative, it does reflect constrained hospital choice and a resignation with her history of repeat cesarean.nd c-section.” Another mother, also in the advantaged group, took her cesarean history and her hospital’s vaginal birth after cesarean (VBAC) rate into consideration when deciding whether she would return to her hospital for future births: “I might. I want to do a vbac and a midwife center is the best place to do one. The hospital I went to is not known for its great vbac success rate. I am afraid I would end up with another c-section if I went back to the same hospital.” Two women equated their cesareans with a loss of efficacy. Sharing a similar sentiment to Kristin’s response above, Melanie, a privately insured mother with an associate’s degree, did not have any advice for past self: “I don’t think there is anything I would tell myself. I had a c-section and did not push or anything, so I doubt anything I would tell myself would have an effect.”All five women who spoke about cesareans indicated that they either wished they had not had a cesarean in the past or that they did not want a cesarean in the future. For instance, Iliana, a college educated mother with private health insurance in her late twenties, simply stated that if she could go back in time and give herself advice, she “would have attempted a natural birth instead of a 2While most non-Latinas were happy with the staff attending their birth, one woman had a surprising comment about the staff at her hospital. When asked if she would return to this facility, Jessica, a college educated mother on government insurance, said, “No. The staff spoke very broken English and seemed to get upset with me when I didn’t understand them.”Similar to non-Latina women on the border, most Latinas on the border (nineteen out of twenty-three) had positive birth experiences. Cassandra, a teenage mother insured through Medicaid, gave a typical response: “I would give birth to my next child at the same hospital. The staff were so great with me. I enjoyed their presence in the room. I know many say hospital food is the worst but I loved it. I miss hospital food the most.” Like Cassandra, Latinas on the border in the disadvantaged group who had vaginal births had the highest number of positive responses (six out of nineteen). Latinas on the border in the disadvantaged group who had cesareans, however, had the highest number of non-positive responses (three out of four). Further, all four border Latinas who had responses that were non-positive had cesareans. Two women directly stated that they would not return to the hospital, one woman noted that she had a tubal ligation and thus would not be having children again , and one woman did not give a clear answer about returning to her hospital. All Latinas on the border who had vaginal births gave positive responses.While a few non-Latinas on the border mentioned that they were “not planning” on having any more children, none of them directly referred to tubal ligations. Two Latinas on the border who had cesareans, however, directly commented on having this procedure. Both of these mothers were in the disadvantaged group. Another border Latina, a mother in her early to mid-twenties without a college degree, wanted information about when she could get a tubal ligation.While no non-Latinas on the border indicated that they wanted more information from their healthcare providers or other resources, two Latinas on the border expressed a desire for more information during pregnancy, birth, and postpartum. Laura, a high school educated mother insured through Medicaid, mentioned communication with staff more generally: “i would ask nurses when im unsure.” Ida, a mother in her early to mid-twenties without a college degree, had a list of specific topics she wanted information about: “How many C-sections can you have at a certain age, information on preeclampsia, exercise tips after having a c-section, when can I get my tubes tied, etc.” Both Laura and Ida had cesareans. Briana, a mother without a college degree and insured through Medicaid, provided the most negative response: “The doctor’s attentiveness could have been a little better, and they didn’t have to wait so long. If a patient says that they don’t feel well, it is because they don’t feel well – not because they want to be stupid and wait for hours in the waiting room”The majority of Latinas in non-border regions (79.1%) provided positive comments about their births. Nina, a mother with a high school diploma and Medicaid, gave a characteristic response: “yes i would love too give birth in the same place because i love how the doctors and nurses were with me. they were so nice and respectful and helpful. cause being a first time mommy is kinda scare you dont know whats going to happen in the labor room but they sure cheered me up. god bless them.” Out of all non-border Latinas, women in the disadvantaged group who had vaginal births had the highest percentage of positive responses (35.8%). Interestingly, this same group of women had the highest percentage of neutral to negative responses (25.0%). Although some women cited reasons that they would not return to their hospital for another birth , the majority of them did not give a specific reason. Examples of responses coded as non-positive  are “I don’t know,” “no,” and “maybe.” However, one of these non-positive responses was from Kayla, a mother with her associate’s degree and insured through Medicaid, who stated, “The hospital i went to was in the poor part of town. i would like to go where the rich go, downtown [city].”Similar to the border non-Latinas, nine non-border Latinas stated that they were not planning on having any more children. When asked if she would return to her hospital, Rosa, a mother in her forties with a high school diploma and private insurance, responded, “I can’t have anymore children but if I could I would use the same hospital. They took excellent care of me and my baby.” Like Rosa, many women said they “can’t” or “won’t” be giving birth again without providing further explanation. We cannot say whether or not these women had a tubal ligation performed, but it is interesting that, in contrast, every border Latina explained that they were not having more babies specifically because they had a tubal ligation.Only two non-border Latinas mentioned cesareans at all in their responses. And unlike border Latinas, non-border Latinas did not give their past selves advice about avoiding the procedure. Rather, both women brought up their cesareans to highlight interactions with hospital staff. For instance, Lidia, a mother with an associate’s degree and Medicaid, said that she loved her hospital because “they informed me of the reasons why they recommended certain things, including the c-section.” Karla, a mother with an associate’s degree and private insurance, said she would not go back to her hospital partly because she had to “deal with a c-section all by [herself]” and because her “dr didn’t discharge [her] or visit [her] after [her] c section.” However, both of these women appear to have understood that cesareans are serious procedures. This explains why Lidia appreciated her hospital staff’s thorough information about the need for surgery and why Karla was dissatisfied with her doctor’s absence after the procedure.Five non-border Latina women spoke directly about issues regarding information. Two of these responses were negative, two were positive, and one was neutral. Similar to Lidia’s experience above, a woman in her early twenties with Medicaid said that she would return to her hospital because she was “given all the information.” Conversely, Olivia, a mother who completed some college and has Medicaid, gave herself the following advice: “Fight like the Mama bear you are. Do not let anyone tell you how to parent your child, especially doctors that try to bully you by giving misleading information. You can do it, because you have maternal instincts that will overcome so much in the future.” When asked whether she would return to her hospital, another woman who provided a negative response said she would not because she felt “rushed” and did not know who to ask for help. Both of these negative responses are not simply about information. Lack of information (and misleading information) resulted in these women feeling dissatisfied with their health care providers and hospital. The non-border Latina who provided a neutral response said that she wanted to take a natural birth class.Intersectional Theory posits that social identities, such as race, class, and gender, interact simultaneously to create multidimensional systems of oppression and lead to different experiences and outcomes . That isA strength of our study lies in our use of Intersectional Theory, which informed our creation of a “compounded disadvantage” variable. Using this variable, our analysis shows that intersecting statuses affect cesarean rates in particular geographic contexts for particular women. Research on variations in cesarean rates have found differences by hospital and geographic regions  that canThere are limitations to our study that should be considered in analyzing the results. First, we are limited to the LTMIII methodology, which required women to take the survey in English, a major consideration for a study focused on the border region where some women may be primarily Spanish speakers. New data from the 2017 Listening to Mothers California may solve this problem by oversampling Latinas and also by fielding the survey in both English and Spanish, though only in one state. While we cannot generalize our study to the U.S. population as a whole, the application of Intersectional Theory to investigate disparities in mode of birth provides an innovative way to explore non-medical factors that contribute to an overuse of cesareans. Second, the survey from which we draw our data has the limitation of selection bias, which is very difficult to control. However, all surveys, particularly those in which participation is voluntary, share this problem. In short, some women are more inclined than others to respond. However, the data we use are the only data from a representative sample of U.S. women who gave birth that also include the geographic location in which they gave birth.These findings deserve further exploration. First, it is important for clinicians working in the border region to understand the increased risk for disadvantaged Latinas to have cesareans. Hospital leaders already working to reduce cesarean rates may want to analyze their institutional rates by using these intersecting statuses to understand the processes and policies that produce these outcomes. Our qualitative analysis shows that Latinas who give birth in border hospitals desire more information throughout pregnancy and childbirth. Interestingly, this desire for information is less visible among non-Latinas who gave birth in a border hospital and among Latinas who gave birth in a non-border hospital, perhaps reflecting their lower rates of cesarean overall. Thus, clinicians and birth professionals should make an extra effort to get relevant information about childbirth and cesarean prevention to Latinas on the border. Second, Latinas on the border were the only group of women in our analysis to discuss tubal ligation. Because of the history of forced sterilization of Latinas in the United States, it is concerning to find a pattern of Latinas on the border discussing sterilization, whereas this topic is absent from the responses of other groups of women . We do n


Typically, a vast amount of experience and data is needed to successfully determine cancer prognosis in the face of (1) the inherent stochasticity of cell dynamics, (2) incomplete knowledge of healthy cell regulation, and (3) the inherent uncertain and evolving nature of cancer progression. There is hope that models of cell regulation could be used to predict disease progression and successful treatment strategies, but there has been little work focusing on the third source of uncertainty above. In this work, we investigate the impact of this kind of network uncertainty in predicting cancer prognosis. In particular, we focus on a scenario in which the precise aberrant regulatory relationships between genes in a patient are unknown, but the patient gene regulatory network is contained in an uncertainty class of possible mutations of some known healthy network. We optimistically assume that the probabilities of these abnormal networks are available, along with the best treatment for each network. Then, given a snapshot of the patient gene activity profile at a single moment in time, we study what can be said regarding the patient’s treatability and prognosis. Our methodology is based on recent developments on optimal control strategies for probabilistic Boolean networks and optimal Bayesian classification. We show that in some circumstances, prognosis prediction may be highly unreliable, even in this optimistic setting with perfect knowledge of healthy biological processes and ideal treatment decisions. NCI defines cancer prognosis as ‘...an estimate of the likely course and outcome of a disease. The prognosis of a patient diagnosed with cancer is often viewed as the chance that the disease will be treated successfully and that the patient will recover’ . A centrThere are a myriad of approaches to model both normal  and aberrant (cancerous) cell dynamics, including biological pathways, co-expression networks, Bayesian networks, Boolean networks (BNs), probabilistic BNs (PBNs), Petri nets, differential equation-based networks, etc. It is believed that these may be used to predict disease diagnosis, progression, and successful treatment strategies, which has led to much work on the identification and analysis of biological networks in genomics and biomedicine.There remain two questions regarding prognosis. First, even if the underlying network of a patient were perfectly known and the best drug to use for the patient were also known, would a patient necessarily be curable? Second, suppose the precise network of a patient were unknown, but probabilities of an uncertainty class of networks, for instance all possible mutations of some healthy network, were available along with the best drug to use for each abnormal network. Then based on available measurements, say genomic or proteomic profiles of the patient, what could be said regarding a patient’s treatability and prognosis? That is, might the very nature of cancer, with its uncertain progression and unique characteristics in each individual, make it impossible to predict prognosis, even given perfect knowledge of all biological processes and ideal treatment decisions? In this paper, we give quantitative answers to these questions, at least at a conceptual level in the context of optimal control strategies for PBNs, by studying intervention outcome in a framework of uncertain biology.PBNs are a class of dynamical models for functional gene regulatory networks (GRNs) . They caManaging uncertainty is especially important in modeling biological networks, where there is inherent uncertainty in the state of a network due to immeasurable latent variables, as well as uncertainty due to a lack of knowledge or partial knowledge of the relationships between observable variables even in a healthy network . Here, wWe assume that a patient’s network belongs to an uncertainty class of networks, each derived from a known healthy network that contains some structure essentially common to all networks. Each network in the uncertainty class possesses one or more ‘mutations’ of the healthy network, representing various possible subtypes or stages of cancer. Some networks in the uncertainty class may be very treatable (good prognosis), while others may be difficult or impossible to treat (bad prognosis). In fact, we will partition the space of networks into four classes based on the severity of disease with treatment and the benefit of treatment. We measure the severity of disease by the long-run probability that cancerous cells visit certain known undesirable states, or equivalently, the SSD mass of these undesirable states. We measure the benefit of treatment by the difference between steady-state mass in undesirable states before and after treatment, which we call the steady-state shift.Our objective is to optimally classify patients into our four prognosis categories and to study the impact of network uncertainty on predicting prognosis. Recent work on optimal Bayesian classification (OBC) furnishes an elegant framework for designing optimal classifiers and optimally estimating their error ,5. In thIn prior work, there have been several studies developing subnetwork markers extracted from protein or gene interaction networks to improve cancer diagnosis -9. Whileper se, since we are modeling uncertainty in the progression of cancer itself while assuming a perfect understanding of cell regulation, as opposed to modeling uncertainty due to ignorance of biological relationships between genes, where knowledge could be enriched with training examples. Figure In this work, we assume a single GAP is observed from the patient, which is essentially a snapshot of the state of the patient’s network at the moment the sample is drawn. The patient’s sampling distribution is thus equivalent to the steady-state distribution of their network without intervention, giving a correspondence between the uncertainty class of networks and the uncertainty class of sampling distributions. We impose a prior distribution over the uncertainty class of networks, with the interpretation that certain mutation events are more or less likely with known probabilities. We can therefore cast our classification problem in a discrete Bayesian setting and directly apply closed-form optimal Bayesian classification and Bayesian error analysis. Note there is no training data nnodes, vi∈{0,1} for i=1,…,n, representing the expression level of genes or their products, and a collection of n Boolean predictor functions, fi:{0,1}n→{0,1} for i=1,…,n, describing the functional relationships between genes. In this setting, 0 and 1 represent down- and upregulation of genes, respectively.Since PBNs fundamentally rely on the dynamics of constituent BNs, we shall define BNs first. A BN is characterized by a set of n binary vector, n genes at time k=0,1,…, where i at time k. The Boolean function fi determines the value of node i at time k+1 by fi takes as input the entire GAP, vk, in general it might depend on only r(i)predictor nodes for gene i. We assume all genes update synchronously. Several methods for constructing transition rules have been proposed, for instance the ‘majority vote’ rule  and −1 with probability 1−β. β reflects a bias toward what type of regulatory relationship (activation or suppression) is more likely to occur. Given the perturbation probability p, we calculate a TPM and its SSD for the network corresponding to the seed regulatory matrix [To construct synthetic uncertainty classes of networks, we begin by outlining a methodology to construct healthy networks that are calibrated to have low undesirable steady-state mass. We generate a y matrix . We thenΘ contains only networks with unfavorable steady-state distributions. Given γ, we then calculate the probability distribution Λ for elements of Θ.Let REM and ADD be two non-negative integers. We enumerate all regulatory matrices such that no greater than REM and ADD edges are removed from or added to n=7). For each network, we select at most three predictors for each gene (rmax=3), with both types of edges being equally likely (β=0.5) and set the BNp random gene perturbation probability p to 0.01. We define the set of undesirable states, , to be the set of all states in which the gene corresponding to the most significant bit (v1) in the binary representation of the state is downregulated. This results in half of the states being undesirable. We also set the number of edge removals to REM=1, the number of edge additions to ADD=1, and the mutation probability γ to 0.5. Each seed network corresponds to an uncertainty set Θ.We generate 250 random seed networks with seven genes  equal sizes. Given Λ, the prior probability of networks in Θ, we use Equation Θ and probability distribution Λ. We also estimate the error of this classifier using Equation Θ, however it will change the partitioning of Θ and classification results. Thus, we set the control gene, in turn, to every gene in the network excluding the target gene.In the next stage of our procedure, given a control gene, we design the optimal intervention policy for each Λ) undesirable mass of networks before and after optimal intervention, respectively. In each scatter plot, each point represents a specific seed network and its corresponding uncertainty class. In general, we observe smaller undesirable mass after control, which is not unexpected since, by definition of the objective function, the undesirable mass after applying the optimal control cannot exceed that of the uncontrolled network.Figures Θ. Furthermore, the size of Θ is affected by the steady-state criteria for including mutated networks in the set. We expect that as uncertainty regarding the underlying mechanism of cancer increases, i.e., as the size of Θ increases, classification becomes a harder task. This effect is observed in Figure Θ . As uncertainty sets grow in size, we observe a trend of increasing error rates.Since the seed networks are randomly generated, they differ in the total number of edges and thus produce different sized uncertainty sets, i, consider the probability of correct classification with respect to random observed states: For a fixed network having label Figures For each uncertainty set, class probabilities conditioned on each state may be found across all networks via Equation β∈{0.1,0.5,0.9}, p∈{0.1,0.5,0.9}, γ∈{0.005,0.01,0.1}, and rmax∈{2,3}, and a discussion on the effect of these parameters.See the supplementary materials for analogous results on 54 different settings, varying We now apply our methodology to a dynamical network modeling the behavior of normal mammalian cells during the cell cycle . The netp=0.01. This network will serve as the nominal healthy network in our analysis. Due to computational constraints, we only consider regulatory networks for which no more than one edge is removed from Θ, and since we only allow one mutation, the distribution Λ will be uniform. For every network in Θ, we take each gene in the network, excluding CycD, Rb, and p27 as control genes in turn, and find the optimal intervention, which maximally shifts the SSD away from undesirable states. Given Λ, Θ, a control gene and the controlled networks, we follow a similar procedure for partitioning Θ as used for the synthetic networks and design an OBC. The results are presented in Table Θ contains 21 mutated networks. Although the average improvement in the SSD of undesirable states is significant, the classification error rates are poor, which indicates that prognosis classification is difficult for this network. The best classification performance in achieved when E2F is the control gene, which is slightly better than random guessing.We construct a BNp following the majority-voting updating rule for this network and set entclass1pt{minimaWe next consider a p53 signaling pathway derived from the KEGG database . p53 is p=0.01. We then enumerate all the networks that have no greater than one edge removed from or added to the nominal p53 network and use the same criterion as in the mammalian cell cycle example to select networks for inclusion in the uncertainty set of mutated networks. We also calculate the distribution Λ assuming a mutation probability of γ=0.5. Each node in the network, except DNAdamage and p53, is allowed as a control gene. Given Λ, Θ, a control gene, and the controlled networks, we partition Θ and design an OBC. For this network, Θ consists of 829 mutated networks. The classification results are shown in Table We construct a BNp following the majority-voting updating rule with We have outlined a framework in which it is possible to utilize prior knowledge regarding cell regulation, for instance pathway information in healthy and aberrant networks, to optimally predict prognosis. That being said, there are several important generalizations of our model that merit further study: (1) integrating partial ignorance of the healthy network itself into our uncertainty class of networks, (2) allowing the network to change over time, thereby taking into account the progressive deterioration of cancer as mutations accumulate, (3) modeling uncertainty in the ideal drug regimen for each network, (4) integrating different types of observations into the analysis, and (5) combining optimal prognosis prediction with optimal treatment recommendations under network uncertainty.ψOBC makes optimal prognosis predictions under network uncertainty, obtaining the GAP or any other relevant information from a patient has the effect of reducing uncertainty. A key point in this work is that we study performance with respect to prognosis only. Although one must overcome network uncertainty, it is not necessary to be able to actually infer the network or any mutations, rather, for our purposes one only needs enough relevant data to make good predictions regarding prognosis. Thus, a second major question we address is whether it is possible to successfully predict prognosis with a relatively small amount of data and available biological knowledge.While The larger the uncertainty class, generally the more difficult prognosis becomes. This is an intuitive result: more uncertainty requires more information to draw accurate conclusions. Furthermore, prognosis performance depends on many factors, including the type of cancer , the individual patient’s network, and the particular sample drawn from the patient. Very often, prognosis prediction from a single GAP is highly unreliable, even in this optimistic setting with perfect knowledge of healthy biological processes and ideal treatment decisions. In this case, the remedy is to collect more data, for instance time-series GAP measurements, to help identify the patient’s network or at least ensure reliable prognosis prediction. One may be lucky and find that their condition is quite clear from a single measurement, but, at least in our examples, it is typical to find that very little is revealed about one’s condition, necessitating additional lab tests.