## Abstract

Understanding the causal relations that drive markets is integral to both explaining past events and predicting future developments. Although there are various theories of economic causality, there has not yet been a wide adoption of machine learning–inspired causal frameworks within economics and finance. Causal networks are among the latter and provide one of the most transparent and practical models for representing the relationships between economic causes and effects that are relevant for making investment decisions. Among the various approaches to causal networks, the noisy-or model stands out because it provides the formal means to simplify the calculation of the cumulative impact of more than one cause on an effect. One of the implicit assumptions of the noisy-or model is that the causal probability values posited by model builders are completely reliable. This assumption is unrealistic, however, especially in financial applications in which beliefs about market events are generally supported by significantly less data relative to beliefs about natural phenomena. Moreover, aside from the need to evaluate the reliability of individual beliefs, portfolio managers presumably also need to assess the investment processes that produce those beliefs. To address the foregoing challenges, in this article, a formal, logic-based framework to produce robust, uncertainty-adjusted causal probability assignments within the noisy-or model is proposed. The robust noisy-or framework described provides both a technical enhancement of the basic noisy-or model and a practical solution for addressing the challenge posed by the varying quality of evidence that supports most investment decisions.

**TOPICS:** Big data/machine learning, quantitative methods, statistical methods

**Key Findings**

▪ Causal networks provide an efficient framework for assisting with investment decisions that are supported by both quantitative and qualitative evidence.

▪ Within the formal setting of causal networks, the noisy-or model simplifies the processing of large sets of evidence that support investment views. It does this by assuming that causes act independently of one another in terms of their ability to influence the actualization of effects.

▪ Evidence-based subjective logic is a many-valued logic that provides a way for investors to precisely account for model uncertainty, with regard to both the reliability of their individual causal probability assignments and their overall conviction in their investment process. The logic also provides the means to combine different causal probability assignments on a single belief, each driven by a distinct set of evidential support.

Causal relations between actions, events, and processes have been objects of study for philosophers and scientists for centuries. Contemporary discussions of causality typically begin with the definition of causation provided by the philosopher David Hume (1777). Hume defines^{1} cause in the following manner:

Hume’s theory of causation is based on the observation of the “constant conjunction” of events and is thus a purely subjective one that assumes that human psychology is the source of the seeming necessity of causes leading to effects.^{2}

Although Hume’s theory does not leave causality on a very sound footing, it has nevertheless been very influential in framing discussions of causal relations over the years. That said, today there exist several prominent theories of causality that are used by researchers in the natural and social sciences. In artificial intelligence (AI), the use of graph theoretical methods has become popular as a way to represent causal relations. For example, a *causal network* is a type of Bayesian network (BN) that is used to model the conditional dependencies of variables in a directed acyclic graph and to draw inferences based on their probabilistic relations.^{3} In many real-life problems, we are confronted with situations in which there is likely to be more than one possible cause for an observed effect. In such cases, researchers require a framework to efficiently model the interaction of several causes and their aggregate impact on the realization of effects.

Although basic versions of BNs can model several causes simultaneously, the question of efficiency is of prime importance in more expansive models because, in BNs, each node with a parent has a conditional probability table (CPT) associated with it. A CPT contains a set of discrete probability distributions corresponding to all possible combinations of parent states. For example, in a binary model (with two possible states for each parent) in which a variable *x* has five parents, the CPT associated with *x* will have 2^{5} = 32 probability distributions. Thus, the number of parameters in a CPT increases exponentially as the number of parents increases. The latter characteristic of BNs can render them computationally expensive when a model is of sufficient complexity.

As a response to the foregoing challenge, the* noisy-or* (NOR) model has been introduced.^{4} NOR is a probabilistic variant of the disjunction connective ∨ from deductive logic. To simplify the computational operations in BNs, the NOR model gives a causal interpretation of the relationship between parent and child nodes and assumes that causes act independently of one another in terms of their ability to influence the actualization of effects. This allows the compound effect of multiple causes to be modeled in a linear fashion. Both positive (promoting) and negative (inhibiting) causes can be evaluated using the NOR model, allowing for an assessment of the net conditional probability of an effect.^{5}

The NOR model has proven popular in many applications, especially those in which probability assessments are elicited from domain experts, as in the case of medical diagnosis. In its basic form, however, the NOR model is an incomplete tool for investment applications. One of its major shortcomings is that it does not inherently possess a way to formally address model misspecification. The latter is a practical concern with most economic or financial models; significantly misspecified input parameters may render them unreliable guides to decision making. In economics and finance, model misspecification has been addressed by a class of models known as *robust models* that take explicit account of the degree of confidence that a model builder has in the formal framework they are using. Such models are of particular interest to investment professionals who want to calibrate their models to worst-case scenarios, have misgivings about the reliability of their parameter estimates, or both.^{6}

To that end, in this article the author describes a robust NOR framework that uses a many-valued logic known as *evidence-based subjective logic* (EBSL) that can be used to evaluate the reliability of both individual beliefs and the epistemic framework that produces those beliefs.^{7} In a portfolio management context, the relevant epistemic framework is the investment process used to form market views. Accordingly, the framework described here allows investors to express robustness along two dimensions. What we call *first-order robustness* applies to the output of an individual model and bases reliability assessments on the size of the sample used. *Second-order robustness*, on the other hand, applies to an individual’s confidence in their overall investment process. In this way, our framework gives investors the ability to calibrate their causal beliefs within the NOR model in a manner that is consistent with the total strength of their evidence and the reliability of their evidence-gathering process.

## THE MANY FACES OF CAUSALITY

Hume’s definition of causation is empirically rooted and based on the observation that some events or actions follow other actions and events more often than not. On the surface, Hume’s definition of cause seems to leave open the possibility that *C* may be a cause of *E* even if *E* does not always follow *C* (i.e., that *C* may not be a necessary condition for *E*). As such, subsequent definitions of cause have attempted to strengthen the definitional relationship between cause and effect in a number of ways. For example, a counterfactual definition of cause could be stated in the following manner:

Other definitions of cause have focused on particular aspects of the cause and effect relationship. For example, much of the causal language employed in economic policymaking seems to subscribe to the manipulation, or ‘wiggle’, conception of causality. The manipulation approach to causality is motivated in part by the causal inferences that are often made in a laboratory setting, where causal relations are attributed to different variables of an experiment when manipulating one set of variables causes changes in another set of variables.^{8} We may define the latter type of causality as follows:

The applicability of definition 3 is clearly seen in monetary policy, in which the federal funds rate is considered a variable that can be manipulated to produce changes in employment and inflation.^{9} Application of the manipulation account is not limited to economic policy, however. Indeed, once we accept that all economic activity is the result of human intentionality and action, we realize that the manipulation account can be applied to most economic and market phenomena.^{10}

The manipulation account is consistent with another popular approach to causal modeling, the *probabilistic causation* account. We lay out the basic formalism of probabilistic causality using a well-known framework developed by Suppes (1970).^{11} We define a *positive cause* in the following manner:

The preceding definitions state that (1) causes must precede effects, (2) the probability of a cause is greater than zero, and (3) the probability of an effect given a cause is greater than the probability of the effect by itself. We define a *negative cause* in an analogous fashion:

The probabilistic approach to causality may be combined with the manipulative account to produce the following definition:

10Whatever underlying conceptual framework one subscribes to, there are two basic uses of any causal model. The first is to formally depict available causal knowledge, and the second is to discover hitherto unknown causal relations. Representative of the former are models for knowledge representation that are prevalent in AI. These models focus on the study of how beliefs can be formally represented in a way that is amenable to automated reasoning. Causal networks, for example, are models for representing beliefs regarding causal relationships. In contrast, traditional econometric models are examples of models designed to discover causal relationships within sets of data.

Whatever the goal of a particular model, developing a framework for understanding causal relations between economic entities is arguably more challenging than doing so in the natural sciences given the open nature of economic systems, which makes them generally inaccessible to the type of experimental analysis we find in other domains.^{12} This is so because in open systems, a multitude of potential common causes may bring about a certain economic effect. Moreover, as noted by Hicks (1979), economics is characterized by a lack of natural constants, timeless facts that can be discovered and measured—unlike physics, which presupposes the existence of the latter kind of phenomena. Furthermore, in the realm of physical science, cause and effect are assumed to occur in (more or less) immediate succession. In contrast, economic phenomena often exhibit significant lags between perceived causes and effects.

Despite its epistemic opacity, economic causality has nevertheless been subject to a considerable degree of theorizing. Perhaps the most well-known characterization of economic causality is that provided by Granger (1969), which states that for a given predictor *x* and target variable *y*, *x* causes *y* if

One way of formally expressing Granger causality is *P*(*Y*_{t+1}|*X _{t}*,

*Y*) >

_{t}*P*(

*Y*

_{t+1}|

*Y*), which can be interpreted as saying that, given a target variable

_{t}*Y*, the probability of predicting its one-step-ahead value

*Y*

_{t+1}is higher given temporally prior information about a predictor variable

*X*in addition to temporally prior information about

*Y*alone. Granger causality may also be articulated in a more general form:

Granger’s approach to causality is of course but one approach to economic causality. In Exhibit 1, we see a classification of economic causality provided by Hoover (2008). The first distinction Hoover makes is between *structural* and *process* approaches to economic causality. The latter categories can be viewed as orientated, respectively, toward description and prediction. Structural approaches to economic causality are primarily concerned with explaining and formally characterizing economic phenomena. The *a priori* structural approaches, such as that taken by the Cowles Commission, look to economic theory as the primary tool to identify economic causes. *Inferential* structural approaches, on the other hand, such as the approach presented by Simon (1953), accept that empirical data can be used to identify economic causes.

In contrast to structural approaches, process approaches to economic causality are primarily concerned with uncovering economic phenomena that can aid in the production of accurate forecasts. The *a priori* process approach of Zellner views theory as a means to produce successful predictions and as a way to distinguish genuine economic laws from accidental regularities and false generalizations. In contrast, the inferential process approach, best exemplified by Granger causality, more or less dispenses with any supporting background theory and, instead, looks to data to provide insight into predictively useful causal associations. Of course, not all approaches to causality in economics fit neatly into Hoover’s classification scheme. Hoover’s scheme is nevertheless useful in providing a picture of the various ways to approach causal analysis in economics.

Inferential process approaches are arguably the closest in spirit to the practice of financial data science (FDS). Process approaches are consistent with FDS because the latter is heavily oriented toward prediction and is less concerned with explanation.^{13} Moreover, because FDS is inherently data-driven and less reliant on theory compared to traditional econometrics, it would also seem to fall squarely in the inferential camp. Indeed, contra Zellner, it would seem that, for FDS, accidental regularities will suffice as causal explanans as long as they help produce successful forecasts and have a long (enough) lifespan. That said, we also recognize that theories (even informal folk theories) may provide some assistance in producing successful forecasts. Given this, the framework presented in this article can be considered a hybrid approach that does not fit neatly into one of the four boxes in Exhibit 1 but is nevertheless most sympathetic to the inferential process approach.

## ROBUST NOISY-OR

In Exhibit 2, we provide a visual depiction of the causal structure assumed by the NOR model. In the exhibit, the values on the edges of the graph indicate the causal probabilities of each individual cause before it interacts with the other causes in the model.

The NOR model encodes and quantifies causal beliefs in graphical form and assumes that causal mechanisms work independently of one another. In Equation 14, we show the basic formal machinery for calculating the conditional probability for several causes of a single type, either positive or negative. In Equation 15, we show how the probabilities for each type of cause can be aggregated to produce an all-things-considered probability of an effect materializing given our respective probability assignments to our sets of positive and negative causes. Positive and negative causes are, respectively, denoted by plus and minus sign superscripts.

14 15As an example of how NOR works, consider the case in which we are considering two sets of causes that we believe respectively promote or inhibit an appreciation in equity prices in the United States (the effect). The set of prospective promoting causes includes (1) a cut in the federal funds rate following the next Federal Open Market Committee meeting, (2) a positive gross domestic product (GDP) print for the most recent quarter, and (3) a drop in the unemployment rate. The set of inhibiting causes includes (1) a proposed across-the-board income tax hike by some members of Congress, (2) the potential for escalated trade tensions with China, and (3) a new set of proposed regulations by the Environmental Protection Agency that may have a negative impact on several industries. In Exhibit 3, we show the NOR operations for each set of causes given specific probability assignments for each cause.

With the values from Exhibit 3 in hand, we can proceed to calculate the value of Equation 15. If we assign True to the truth values of all three positive and negative causes, then the value of Equation 15 is

16One notable feature of the NOR model is that the mere presence of an additional cause in a disjunction is often enough to materially raise the probability of an effect actualizing, even when low probability values have been assigned to each individual cause. In Exhibit 3, low probability values were deliberately chosen for each of the causes to illustrate this point. As the exhibit shows, in the bottom row of each panel, where each cause takes a truth value of True, the resulting probability value is above 0.50 even though, in each case, all three constituent causes have a probability assignment below 0.50.

In many applications of NOR, the source of the probability assignments is assumed to be some type of expert. For example, imagine a healthcare setting in which medical professionals assign probability values based on their belief in the cause and effect relationship between various diseases and a particular symptom that has been observed in a patient. In this case, the probability assignments posited are purely subjective, and our confidence in them is based solely on the perceived reliability of the expert in question. Although the latter type of evaluative exercise is also common among portfolio managers with fundamental, qualitative investment processes, quantitatively oriented investors generally prefer to base their probability assignments on more clearly measurable information. However, the NOR model is not equipped with the formal means to calibrate probability assessments based on the sample sizes used to produce them. A further shortcoming of the NOR model is that it does not by itself provide a way to adjust input parameter values so that they reflect the model builder’s confidence in their methodology or the expert they are using as a source of knowledge.

Given the foregoing, we believe that the NOR model needs to be enriched to accommodate investment processes that use quantitative information as their basic inputs. To do so, we use EBSL, a many-valued probabilistic logic. Many-valued logics are logics that are characterized by their abandonment of the principle of bi-valence, the assumption that a given proposition is either completely true or completely false. They accordingly allow propositions to be assigned truth values in the range [0,1] rather than binary assignments of 1 (True) or 0 (False). One interpretation of many-valued semantics, and one that motivates our methodology, is that propositions can be assigned values that fall between complete truth and complete falsity based on a consideration of evidence for or against their correspondence to facts in the world.^{14}

EBSL begins with a formal definition of an opinion. An opinion *x* of some proposition *P*, (e.g., “lower interest rates will lead to higher equity prices”) is defined as a tuple *x* = (*x _{b}*,

*x*,

_{d}*x*), where

_{u}*x*

_{b}represents the degree to which a belief in

*P*is supported by the evidence,

*x*

_{d}represents the degree to which disbelief in

*P*is supported by the evidence, and

*x*

_{u}represents the degree of uncertainty in the available evidence. Accordingly, an opinion space is defined as Ω = {(

*b*,

*d*,

*u*) ∈ [0,1]

^{3}|

*b*+

*d*+

*u*= 1}. We further introduce a fourth parameter, the base rate

*a*∈ [0,1], to derive an expected probability value

*E*(

*x*) =

*x*+

_{b}*x*. The base rate is simply the prior probability of

_{u}a*P*before the introduction of data relevant to the decision at hand. The latter formulation lends itself to how investment decisions are made in practice. For example, consider again the proposition “lower interest rates will lead to higher equity prices.” Assume that this proposition is generally believed by an investor to the extent that it is assigned a prior probability

*a*of 0.8. If we further assume that the available evidence supports an assignment of 0.4 for

*x*

_{b}and 0.2 for

*x*

_{u}, then the expected probability value

*E*(

*x*) is 0.56.

We use a variant of EBSL that gives us the ability to consider time-series data as the evidentiary basis for our beliefs, in a way that explicitly accounts for the impact of sample size on our probability assignments. Its governing equations, derived from the pdf of the beta distribution, are as follows:

17The constituent elements of Equation 17 accordingly allow us to produce an alternative form of the beta probability density function (pdf):

18where 0 ≤ *p* ≤ 1, (*r* + *Wa*) > 0, (*s* + *W*(1 − *a*)) > 0 and where *r* denotes the number of observations of *x*, and *s* denotes the number of observations of ¬*x*. The parameters of the beta distribution α and β are thus expressed as a function of the observations *r* and *s*, respectively, in addition to the base rate (prior) *a*. The weight *W* is the non-informative prior weight and is typically assumed to have a value of 2 to ensure that the prior beta pdf is a uniform pdf when the default base rate *a* = 0.5.

The formalism in Equations 17 and 18 allows us to apply what we call *first-order robustness* to our assessment of causal probability. We see that the range of the distribution will be a function of the number of observations used to generate the probability values for a given belief. As the number of observations increases, the distance between the upper- and lower-bound values of the distribution’s confidence interval will narrow. If we take some lower bound of the confidence interval as our robust causal probability value, then the assumed causal strength will weaken (strengthen) as the size of our sample decreases (increases). Thus, the variant of the beta distribution described in Equations 17 and 18 allows us to directly link a robust lower-bound probability value to the number of observations in a sample. When the number of observations is large, as is often the case in natural science experiments and high-frequency trading, there will be minimal difference between the upper- and lower-bound values of the confidence interval. However, in many investment contexts, the size of the sample will be such that there will be a meaningful difference between the upper- and lower-bound values.

The foregoing methodology provides the formal means to generate causal probabilities for a single source of evidence (e.g., a particular model). In practice, however, portfolio managers often combine output from more than one model to support their investment views. Thus, there is a need for a mechanism to combine two or more independently generated opinions on the same proposition. EBSL provides such a mechanism, which is formally described in the following manner:

19where *x* and *y* represent distinct opinions of some proposition *P*.

The formal machinery of EBSL also provides a way to discount a causal probability value based on the portfolio manager’s confidence in the epistemic framework being used to form investment views. We call this discounting *second-order robustness*. As in the case of an individual opinion, the reliability of an opinion is also expressed as a tuple *x* = (*x _{b}*,

*x*,

_{d}*x*

_{u}). In turn, the value for the reliability-discounted opinion is produced through a combination of the values in the tuples corresponding to the original opinion and the reliability of the opinion as follows:

where *x* ⊗ *y* denotes the discounting of opinion *y* using reliability assessment *x*. As in the case of the primary opinion (investment view), the values that populate the tuple in Equation 20 correspond to the belief, disbelief, and uncertainty that an individual attaches to an opinion’s reliability. It is important to note that investors’ assessments of the reliability of their investment processes may be subjective but may also be linked to their track record in various ways (e.g., by employing batting-average measures to various aspects of their performance).

With the formal description of EBSL in place, we now provide a numerical example. Let us consider two independently formed beliefs regarding the causal relationship between the Fed funds rate and equity prices from Exhibit 3, with values of 0.40 and 0.45, respectively. For our first belief, we assume that it is described by a distribution with an α value of 40, a β value of 60, a prior value *a* of 0.50, and a value of 2 for *W*. For our second belief, we assume that it is described by a distribution with an α value of 45, a β value of 55, a prior value *a* of 0.50, and a value of 2 for *W*. If we further assume that the manager’s confidence in the first belief is 25%, then the first-order robust value for the belief is 0.37. For the second belief, we assume that the manager expresses a confidence level of 20%. This gives us a first-order robust belief of 0.41. Next, we proceed and combine our beliefs according to Equation 19. For the purposes of the latter calculation, we assume that the tuples corresponding to each individual opinion are composed of a belief *b*, which is the same as the first-order robust belief; a belief *d*, which is 1 minus the original belief (e.g., 0.40); and a degree of uncertainty *u*, which is equal to 1 − (b + d). The latter operation gives us a tuple with the values (0.37, 0.60, 0.03) for our first opinion and a tuple with the values (0.41, 0.55, 0.04) for our second. Combining them according to Equation 19 yields the opinion tuple with the values (0.39, 0.59, 0.02). Finally, we apply Equation 20 to discount the portfolio manager’s hybrid first-order robust opinion and produce a second-order causal probability value based on the manager’s assessment of the reliability of their overall investment process. If the manager’s view of the reliability of their investment process is represented by a tuple with values (0.75, 0.20, 0.05), then a discounted second-order robust opinion is generated with values (0.295, 0.44, 0.265). The first element in the tuple, 0.295, is the value we then proceed to use as a causal belief within the NOR framework.

## CONCLUSION

Building a view of causal relations among economic entities is important when formulating an investment process. The latter endeavor can be challenging, however, given the complex and opaque nature of many market interactions. Over the years, various models of economic causality have been advanced to both explain the past and assist with decision making. Among the latter, causal networks present among the most transparent and tractable models of causality. That said, when building causal networks, as more causes are added, the issue of computational complexity arises from the need to calculate the combined effect of larger and larger sets of causes. One formal remedy to the latter challenge is provided by the noisy-or model, which provides the means to calculate the aggregate effect of causes in a linear manner. Although useful, the basic noisy-or framework does not admit consideration of uncertainty with regard to individual models or sets of evidence or with regard to the model builder’s overall epistemic process. The latter shortcoming is particularly acute in financial applications of the noisy-or model, in which the availability of financial data is relatively limited compared to that available in the natural sciences. To address the question of uncertainty, in this article, the author provides a robust, uncertainty-adjusted noisy-or framework that uses evidence-based subjective logic, a many-valued logic explicitly designed to assess the reliability of the evidence supporting an individual’s beliefs. We find that the latter logic allows us to augment the noisy-or model so that it is sensitive to both individual model uncertainty and the uncertainty relating to a portfolio manager’s overall investment process. In this way, the robust noisy-or framework presented in the article gives investors the ability to apply the noisy-or model in a manner that is consistent with the strength of both their evidence and overall research methodology.

## ENDNOTES

↵

^{1}The various definitions of causality provided in this article are adapted from Illari and Russo (2014).↵

^{2}It is thus generally considered a skeptical solution to the problem of causality. Acceptance of Hume’s theory of causality also gives rise to the*problem of induction*, in which inductive inferences are impossible to justify. Because inductive inferences are based on observation, they are contingent and thus cannot be proven deductively. However, they also cannot be justified inductively because that would be assuming the very truth (that inductive reasoning can lead to justified belief) we are trying to prove (i.e., begging the question).↵

^{3}Bayesian networks are also known as*belief networks*.↵

^{4}The NOR model originates in the work of Good (1961), Peng and Reggia (1986), and Pearl (1988).↵

^{5}The basic NOR model has been extended in various ways. Henrion (1987) introduced the notion of a*leak probability*, which is a nonzero probability that the effect materializes even if all of the causes considered are not actualized. Díez (1993) and Srinivas (1993) extended the model to non-Boolean variables. Lemmer and Gossnik (2004) proposed a recursive version of the model that includes synergies among the causes.↵

^{6}The mathematical inspiration for the development of robust methods can be found in the drive to extend the static maxmin expected utility theory of Gilboa and Schmeidler (1989) to a dynamic environment that can be exploited in various contexts, including portfolio selection. For its part, Gilboa and Schmeidler’s theory provides a formal avenue for modeling decision making that is sensitive to individuals’ aversion to uncertainty and ambiguity in the sense described by Ellsberg (1961). Examples of the wide variety of applications of robust methods in economics and finance include work by Hansen and Sargent (2001, 2007); Rustem and Howe (2002); Garlappi, Uppal, and Wang (2007); Ben-Tal, El Ghaoui, and Nemirovski (2009); Fabozzi et al. (2007), Simonian and Davis (2010, 2011); and Simonian (2011).↵

^{8}The manipulation account of causation was developed by Collingwood (1940), Gasking (1955), von Wright (1971), Menzies and Price (1993), and Woodward (2003). It has had significant influence in experimental design (Cook and Campbell 1979) and AI (Pearl 2009).↵

^{9}The federal funds rate would thus be considered a common cause to changes in employment and inflation.↵

^{10}The anthropocentric orientation of the manipulative account has been criticized by some because it does not seem to admit non-human drivers of causality (Hausman 1986, 1998). However, the latter criticism seems*prima facie*impotent for most accounts of causality in the social sciences, including economics. That said, we recognize that not all economically relevant causal relations are due to human manipulation. For example, different weather patterns can cause changes in crop yields. Nevertheless although the latter changes are not a result of human manipulation, the change in commodities prices that follow are.↵

^{11}Good (1961) provides an earlier account of probabilistic causation.↵

^{12}This point has been previously discussed by Simonian, López de Prado, and Fabozzi (2018).↵

^{13}See Simonian and Fabozzi (2019) for an argument in support of this view.↵

^{14}For an introduction to many-valued logics, see Rescher (1969). Well-known many-valued logical systems include those developed by Post (1921), Kleene (1952), Łukasiewicz and Tarski (1956), and Belnap (1976, 1977), among others.**DISCLAIMER**This article is for informational purposes only and should not be construed as legal, tax, investment, financial, or other advice. Nothing contained in the article constitutes a solicitation, recommendation, endorsement, or offer by the author to buy or sell any securities or other financial instruments.

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