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Before we decide how we will capture this change, let us first specify the domain for the standard form of Bayesianism. This set-up will also prove fruitful to formalize the notions of hypothesis, evidence, and the catch-all, which prepares us for the subsequent treatment of domain changes and associated changes in probability.

Dynamics: time stamps In Bayesian confirmation theory, we model the rational degrees of belief of an agent by a probability function. The sample space can be a Cartesian product of sets, which allows us to represent very different types of empirical data. In the Bayesian framework, probability functions range over evidence and hypotheses. The hypotheses are only specified up to their empirical content.

The scientific theories that motivate them are not brought into view. Because the empirical content of hypotheses is spelled out in terms of probability functions over the data, the hypotheses are called statistical. Under a hypothesis we may also subsume an entire family i. Hypotheses that correspond with singleton sets will be called elementary hypotheses, others will be called composite.

We do not discuss in detail the scientific theories themselves, or even how they lead to statistical hypotheses. Standard Bayesianism lacks an analogous procedure for revising the probability function in light of a new hypothesis. We will now discuss how the presence of the catch-all allows us to represent the dynamics of the set of hypotheses.

This prepares us for the proposal of open-minded Bayesianism in the next section. We briefly recapitulate our approach so far and our use of the following terms: scientific theory, statistical hypothesis, sample space, evidence, and catch-all. A scientific theory together with background assumptions produces an empirical, or statistical, hypothesis. How this happens requires engaging with the details of a scientific theory, which falls outside the scope of our current framework. Such an empirical or statistical hypothesis is a set, possibly a singleton, of probability functions.

In order to compare hypotheses produced by different theories in the light of a common body of empirical data and thus to compare their measures of confirmation or evidential support , their probability functions need to have a common domain. This domain is called the evidence space: it is an algebra on a sample space which may be a Cartesian product set to allow for the representation of mutually independent measurable quantities.

The catch-all hypothesis is included to express that many other hypotheses are conceivable, each associated with a probability assignment or a set of such assignments over the evidence. With the idea of a catch-all hypothesis in place, we can now turn to a full specification of open-minded Bayesianism. The inclusion of a catch-all hypothesis makes room for modeling the introduction of new hypotheses, namely by shaving them off from the catch-all. But this in itself is not sufficient: we still need to specify how shaving off influences probability assignments over the hypotheses.

This is the task undertaken in the next section. In the previous section, we have introduced the formal framework of open-minded Bayesianism. It is a form of Bayesianism that requires the set of hypotheses to include a catch-all hypothesis. In the current section, we develop the probability kinematics for open-minded Bayesianism.

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Two versions will be considered: vocal and silent. The two approaches suggest slightly different rules for revising probability functions upon theory change. In open-minded Bayesianism, hypotheses are represented as sets of probability functions. If prior probabilities are assigned to the functions within a set, then a single marginal probability function can be associated with the set.

But without such a prior probability assignment within the set, the set specifies so-called imprecise probabilities see, for instance, Walley We first clarify probability assignments over explicitly formulated hypotheses. In standard Bayesianism, prior probabilities are assigned to the hypotheses, which are all explicitly formulated. We can furthermore assign priors over the individual probability functions contained within composite hypotheses, if there are any. We call such priors within a composite hypothesis sub-priors. The use of sub-priors leads to a marginal likelihood function for the composite hypothesis.

Now recall that in open-minded Bayesianism, the space of hypotheses also contains a catch-all, which is a composite hypothesis encompassing all statistical hypotheses that are not explicitly specified. In standard Bayesianism, this catch-all hypothesis is usually not mentioned, and all probability mass is concentrated on the hypotheses that are formulated explicitly. Within the framework of open-minded Bayesianism, we will represent this standard form of Bayesianism by setting the prior of the catch-all hypothesis to zero. Silent open-minded Bayesianism assigns no prior or likelihood to the catch-all hypothesis, not even symbolically.

As detailed in the foregoing, we aim to represent probability assignments of an agent that change over time. Informally, this is often presented as if the probability function changes over time, but it is more accurate to say that the entire probability function gets replaced by a different probability function at certain points in time. Accordingly, subsequent functions need not even have the same domain. It amounts to restricting the algebra to those sets that intersect with the evidence just obtained.

Equivalently, it amounts to setting all the probability assignments outside this domain to zero. On both views, the new algebra is larger i. What is still missing from our framework is a principle for determining the probability over the larger algebra.

Viewed in this way, our proposal does not introduce any radical departure from standard Bayesianism.

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As detailed below, silent and vocal open-minded Bayesianism will give a slightly different rendering of this rule. In this section, we consider how the probability function ought to change upon the introduction of a new hypothesis after some evidence has been gathered. We first consider an abstract formulation of a reduction and extension of the domain, as well as an example of such an episode in the life of an epistemic agent.

After that, we consider both versions of open-minded Bayesianism as developments of the standard Bayesian account. We will consider these questions in the context of standard Bayesianism and both forms of open-minded Bayesianism. And since the prior of this hypothesis is zero, the confirmation of this hypothesis is zero as well.

In other words, standard Bayesianism simply does not allow us to represent new hypotheses other than by the empty set. In this sense, the ensuing problem of old evidence does not even occur: new theories cannot be taken into account in the first place. In this view, none of the previous probability assignments change upon theory change, but additional probabilities can be expressed and earlier expressions can be rewritten accordingly.

We are still left with two terms that have different unknown factors, which do not simply cancel out. We return to this point below. On this view, when the theoretical context changes, new conditional probabilities become relevant to the agent. Let us briefly motivate the silent version as an alternative to vocal open-mindedness. We have seen that the vocal version comes with a heavy notational load. Given that, in the end, we can only compute comparative probabilities, it seems desirable to dispense with the symbolic assignment of a prior and a likelihood to the catch-all hypothesis.

In particular, we can engage in the kind of reconstructive work as is done in vocal open-mindedness, but this is not mandatory here. Importantly, we can compute all known confirmation measures using the priors and posteriors that are conditional on a particular theoretical context. Once the context changes, this clearly impacts on the confirmation allotted to the respective hypotheses. The natural use of a degree of confirmation thus becomes comparative: it tells us which hypothesis among the currently available ones is best supported by the evidence, but there is no attempt to offer an absolute indication of this support.

In this section we critically evaluate open-minded Bayesianism. We clarify our views on it, and conclude that it provides a handle on the problem of old evidence: it explains how old evidence can be used afresh without violating Bayesian coherence norms. Towards the end, we sketch a number of ideas and problems that deserve further exploration.

It may be argued that open-minded Bayesianism fails to provide us with the required normative guidance. In the silent version, it only concerns suppositional reasoning and hence cannot inform our unconditional beliefs. In metaphorical terms, the worry is that the agent cannot keep hiding behind the conditionalization stroke. Either way, it may seem that the agent must come clean on her absolute commitments at some point. We respond to this worry by biting the bullet. If we want to allow new theories to enter the conceptual scene, then we will have to provide room for this in our formal framework.

There are attempts to accommodate other forms of theory change in a Bayesian framework that employ fully specified probability assignments e. In this paper, by contrast, we have offered a framework that creates room for new theories by leaving part of the probability assignment unspecified. We accept that this leads to a model that only concerns conditional belief.

We should emphasize that we are not alone in preaching an open-minded form of Bayesianism. We already mentioned the proposal for tempered Bayesianism by Shimony , who suggested the use of a catch-all hypothesis in this context. This suggestion was also discussed by Earman , who introduced the evocative terminology of shaving off new hypotheses from the catch-all.

Furthermore, our proposal of humble Bayesianism is related to earlier work by Salmon and Lindley The latter paper lends further support to open-minded Bayesianism. After all, a standard Bayesian will have the probabilities of the hypotheses under consideration add up to one, and so judges herself to be perfectly calibrated cf.

Dawid The standard Bayesian is overly confident, hence a more open-minded form of Bayesianism seems called for. The price to pay is that the epistemic attitudes for which the framework of the open-minded Bayesian provides the norms are different: they have a conditional nature.

Whether we spell out the details using a vocal or a silent open-mindedness, the normative framework tells the agent what to believe only if she temporarily supposes, without committing to it, that the true theory is among those currently under consideration. Now that we have bitten the bullet, we better make sure that we do so for good reasons. In this section, we argue that open-minded Bayesianism provides a new handle on the problem of old evidence, by explaining how old evidence can be re-used.

The strong point of open-minded Bayesianism is that this reconsideration of the posteriors does not render the agent probabilistically incoherent. When writing about the operation of shaving off new hypotheses, Earman , p. Notice that we do not assign an explicit value to the prior of the current theoretical context. We may think of the prior associated with the catch-all hypothesis as a number extremely close to unity—and the humbler we are, the closer to unity we can imagine it to be.

But lacking such a definite value, 24 the problem that the catch-all gets crowded out by explicit hypotheses does not arise. There are, however, differences in how the vocal and silent approaches to open-minded Bayesianism deal with reassessing the posteriors, and in what role they give to old evidence.

The other posteriors are obtained via a renormalization. One cannot unlearn the evidence that has been gathered, but it is still possible to use base rates or other sources of objective information to determine the priors retroactively. The point here is rather subtle. Since the silent approach remains silent precisely on this prior, it is hard to see how we can retroactively decompose it. So in this approach, it is not clear whether old evidence ever confirms new theories.

In silent Bayesianism, the old evidence is therefore not given a new role. Now that we have discussed the role of evidence in two forms of open-minded Bayesianism, it is time to take stock. Both approaches suffer some drawbacks. The vocal proposal comes with the complication of a heavy notational load that hampers the evaluation of the degree of confirmation.

The silent proposal allows too much freedom in the assignment of a posterior to the new hypothesis—so much freedom, that it is not clear that the old evidence has any impact. For these reasons, we propose a hybrid approach to open-minded Bayesianism, that combines the best elements of both. On our hybrid proposal, the open-minded Bayesian remains in the silent phase, 28 except for the times at which her theoretical context changes. Unlike a standard Bayesian, the open-minded Bayesian is allowed to change the algebra to which probabilities are assigned and thus to assign non-zero probabilities to the new hypothesis, which is impossible without a catch-all.

Then she enters the vocal phase: she engages in assigning a prior to the new hypothesis retroactively and computing its posterior given the evidence also retroactively and renormalizing the other priors. Open-minded Bayesianism thus offers a particular perspective on the use of old evidence for confirming a new theory.

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On the conceptual level, it shows how our perception of evidence and confirmation changes if we move from one theoretical context to another. Relative to one set of hypotheses, the data were telling towards one particular candidate hypothesis, and so counted as evidence that confirms this candidate. But with the inclusion of a new hypothesis, the data may tell against the formerly best candidate, and so count as evidence that disconfirms it. We take it to be a virtue of our model that it brings out this context-sensitivity of evidence and confirmation. To make our proposal for a hybrid approach more vivid, we apply it to the food inspection example.

Initially, when the food inspector implicitly assumes her equipment to be working properly, she can be described by the silent approach to open-minded Bayesianism. Within the initial context, she only needs to take into account two explicit hypotheses: the kitchen is clean or it is not. She assigns prior probabilities to these hypotheses and she computes posteriors, but these assignments are conditional on her implicit assumption that the testing strips are uncontaminated as well as the many other background assumptions collected in the theoretical context.

So far, she acts much like any Bayesian would; her open-mindedness will surface only when provoked. The result, that five dishes out of five appear to be infected, was initially unlikely on both of her explicit hypotheses. If the priors were equal or at least of the same order of magnitude , then on any measure of confirmation, the evidence provides very strong confirmation for the hypothesis that the kitchen was unclean.

Our framework for open-minded Bayesianism is able to represent this formally. In the vocal phase, the agent shaves off her third hypothesis from the catch-all and revises her probability assignments: she retroactively assigns a prior to the new hypothesis, adjusts the priors of the two old hypotheses by a suitable factor, and computes the likelihood of the old evidence on the new hypothesis as described in Sect.

All this leads her to reassess the posteriors of the old hypotheses and to assign a posterior to the new hypothesis. Assuming equal priors, the final result is this: within the new theoretical context, the posterior of the new hypothesis given the old evidence is more than three thousand times higher than that of the hypothesis that was best confirmed within the old theoretical context.

Irrespective of the details of the confirmation measure and assuming priors of at least equal orders of magnitude, this implies that the old evidence strongly confirms the new hypothesis and disconfirms the others. This illustrates that it is the shift in theoretical context itself that may cause old evidence to confirm a new hypothesis.

Once the agent is satisfied that, for the evidence currently at hand, the new theoretical context includes all the relevant hypotheses, she may start to conditionalize all her findings on this context and thereby enter a new silent phase. The remaining catch-all hypothesis need not be mentioned again until new doubts arise. In Kuhnian terminology, the silent version of open-minded Bayesianism is sufficient for describing episodes of normal science and if the conditionalization on the theoretical context remains implicit, it is indistinguishable from the usual Bayesian picture , but the vocal version of open-minded Bayesianism is required to model revolutionary changes in the theoretical context.

With the foregoing, we believe we have only scratched the surface of the matter at hand. Many avenues for further research lay open for exploration. In what follows, we briefly mention a number of these avenues. With this we showcase our ongoing research on this, we invite the reader to join in, but mostly we indicate where we ourselves feel that our account is lacking. One important consideration that has received relatively little attention in the foregoing concerns degrees of confirmation. Our goal with this paper was to show that we can accommodate the introduction of a new theory and hence a new empirical hypothesis in the Bayesian framework, and that old evidence can play a role in the determination of the posterior probability of this new hypothesis without violating probabilistic coherence.

We have been mostly silent on how the posteriors may be used to compute a degree of confirmation, so that the impact of old evidence can be expressed more precisely: any such story will supervene on the probability assignments. However, a complete account of open-minded Bayesianism might involve more detail on degrees of confirmation. Another aspect of the process of theory change targeted in this article certainly deserves a more detailed normative treatment: the decision to introduce a new theory.

In the foregoing, we have treated this decision as completely external to the model. However, we also indicated that the search for new theories may be motivated by a so-called statistical model selection criterion, e. We think that our account, which may provide rationality constraints on the transition from one theoretical context to another, can be combined fruitfully with an account of how theoretical contexts are evaluated and selected. Furthermore, we should stress that we have only considered one type of theory change—a change that may be captured by shaving off new hypotheses from a catch-all hypothesis.

One captivating question concerns the exact reach of our account of new theory and old evidence. An answer to this question requires us to survey a rich landscape of theory changes as moves in an encompassing space of possible theories. We would like to mention one other aspect to theory change that is related to two issues discussed above, namely the decision to introduce a new theory and the type of theory change effected by that.

It concerns the notion of awareness. Hill and Dietrich and List have argued that a decision problem obtains new dimensions when the agent is made aware of considerations that were previously not live to her. We think that roughly the same can be said about the epistemic problems an agent faces, and that the foregoing offers a natural model for an agent that becomes aware of a theory while performing a predictive, or more generally an epistemic task.

It seems natural to combine the frameworks for modeling awareness. Finally, we briefly mention two possibilities that open-minded Bayesianism offers, when it is combined with ideas on relative infinitesimals in the sense of Wenmackers On one side of the spectrum, the framework allows us to model radically skeptical yet empiricist epistemic attitudes: all the priors and posteriors of explicit hypotheses, old and new ones, may be very small, indeed infinitesimally small, compared to the probabilities associated with the catch-all.

Despite that, a particular theory may have a large prior or posterior relative to the other theories in the theoretical context. The framework thus allows us to model a radical sceptic who is nevertheless sensitive to differences in empirical support. On the other side of the spectrum, the framework of open-minded Bayesianism allows us to model practical certainty without spilling over into dogmatism.

We may be aware of the existence of certain hypotheses, but we might choose not to include them in our considerations: they may seem irrelevant to the kinds of evidence under study assuming statistical independence , they are deemed highly unlikely, 32 including them requires too high a number of computations, or other pragmatic reasons. However, upon receiving falsifying or strongly disconfirming evidence, we might want to reconsider some of these omissions. Falsifying or strongly disconfirming evidence may lead to a situation in which the probability of the catch-all is no longer regarded as a relative infinitesimal: the marginal likelihood becomes so small that it becomes comparable to the probability of the catch-all.

The above list of research topics indicates that our resolution to the problem of old evidence and new theories leaves much to be done. However, the list also suggests that the framework of open-minded Bayesianism provides access to several interesting aspects of belief dynamics that fall outside the scope of standard Bayesianism. See for instance Easwaran for a recent overview of approaches to the problem of old evidence.

Yet, this has been noted in the literature. See for example Gillies We also agree with Sprenger that, if we intend to capture objective confirmation in a scientific context, the relevant credence function belongs to an abstract agent representing any unbiased scientist in the relevant context, rather than a particular historical person. But we follow a different approach. Considering the algebra spanned by the cylindrical subsets of this sample space allows us to represent measurements as initial segments of infinitely long streams of data.

Using this terminology, this article deals with the problem of new hypotheses, rather than the problem of new theories. It is clear that an indeterministic theory can generate statistical predictions about measurable quantities. In the case of deterministic theories, such as Newtonian mechanics, it may be less clear how they lead to hypotheses that are expressed in terms of a probability assignment.

However, when we combine such a theory with measured values for masses, velocities, etc. Typically, this will happen because the evidence was surprising according to the hypotheses currently under consideration, as witnessed by a very low likelihood i. A principled decision to introduce a new theory may be based on the computation of a model score, or on the application of a model selection tool.

But such scores and tools fall outside the scope of the present paper. The procedure for deciding to introduce a new theory is not intended to be a part of our model. See for instance Duhem , p. In statistics this is known as hierarchical modeling cf. However, if we represent Bayesianism without a catch-all within an open-minded framework, a probability has to be assigned to the catch-all and its value has to be zero: see Sect. Although we do not advocate this here, the vocal formalism is compatible with assigning a definite prior to the catch-all.

See Sect. Readers only interested in the gist of our account may skip this subsection and continue reading at Sect. Since the inspector assumes that the test is perfect, instead of representing the test results, she may just as well represent these data in terms of dishes being infected or not such that 0 means that a dish is not infected and 1 that a dish is infected.

This illustrates how data and evidence may come apart: we regard evidence as interpreted data, where the interpretation depends on the sample space that is used in a hypothesis. The assumption of equal priors is not essential for the framework. The agent may assign different priors, based on considerations that are external to the Bayesian framework, such as relevant base rates where the usual reference class problem emerges; cf.

The binomial distribution only applies to situations that can be thought of as having a fixed bias and producing independent outcomes. The catch-all should be large enough to allow the agent to reconsider even these assumptions at a later point in time. Christensen ; Joyce , for which the factors do cancel out. Recall from Sect. Or assuming it to be unity minus an infinitesimal: see Sect. Again, the review shows that majority of papers analyzed examined CSR in Ghana in multiple sectors this accounted for Additionally, telecommunication sector, forestry sector, and the oil industry constituted approximately Our analysis of the methodological approaches used in CSR studies in Ghana shows that various methods have been adopted to explore the issues of CSR in Ghana.

Our analysis shows that majority of the papers adopted quantitative methodology such as correlation analysis, factor and exploratory analysis, analysis of variance, and regression analysis. Case study and content analysis techniques were some of the few qualitative methodologies used in CSR research in Ghana. In conclusion, single and two authorship dominated CSR's research in Ghana. Although the study is based on CSR papers on Ghana, yet authors collaborate from different institutions in different countries. Of the papers, authors that are based on the University of Ghana dominated the sample constituting Here, we examine the level of analysis for CSR research in Ghana.

We classified the level of analysis into institutional and organizational levels. Our analysis shows that greater portion of the reviewed articles focused on organizational level i. Our analysis further indicated that very few papers i. An implication of this is that, unlike countries in East Africa, such as Kenya and Uganda, tourism in Ghana is at its infant stage, explaining why tourism CSR related activities are the least.

Also, individual level themes such as CSR and leadership, among others are currently emerging but are not being addressed in Ghana Strand, Consistent with the state of the Ghanaian economy, Ghana is driven by the service sector. Consequently, these sectors have received more research attention because they have organized structures and so most researchers have acquaintances in these firms e.

Lack of adequate research funding support to academics in Ghana compared with their counterparts in the developing economies might account for uneven attention to sectors with regard to the state of CSR research in Ghana. In terms of methods, exploratory and descriptive analysis dominates. There is a low level of advanced statistical techniques such as structural equation modelling, logit, and probit regression on CSR's studies on Ghana. With regard to the level of analysis, unlike Aguinis and Glavas and Lunenberg et al. Both predictors, outcome studies, and moderation and mediation of CSR's outcome relationship studies are lacking in CSR's studies on Ghana.

Of the institution of author collaborations, scholars from the University of Ghana lead the studies. This confirms many characteristics about the University of Ghana. First, the University of Ghana is the premier university in Ghana. Second, the University of Ghana is the largest University in Ghana. By ranking, the University of Ghana is the leading among universities in Ghana. The results show that organizational level CSR performance related topics, including CSR and management as well as CSR practices, have received much research attention.

Overall how CSR is organized in the advanced countries e. The results from the review show that public policy, academic, and practical interests in CSR on Ghana are being enhanced. This collaborates with similar heightened interest in the topic in Africa in general. According to Cheruiyot and Maru , the sustainable development agenda by the United Nations across the world, including Africa, is one of the key factors contributing to the enhanced interest in CSR in Africa. Following Idemudia research contributions from different disciplines e.

In terms of issues explored from the Ghanaian studies, the findings reflect both CSR debates in Africa and to a lesser extent globally. In addition, CSR's practices and activities, which emerged from the issues explored on Ghana, also collaborate with Hinson and Ndhlovu in Africa. Though studies such as CSR disclosure in the developed and the developing countries are connected to both Africa and the global community, Ali et al. Some important gaps identified through this review, which set the tune for further studies, include first the lacuna in the institutional level analysis of CSR.

At the institutional level i. Second, there is a gap in theory and conceptual framework concerning CSR activities in Ghana.

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Third, CSR's studies that integrate different disciplines e. Consequently, a fuller understanding of CSR is lacking regarding issues in Ghana. Although sectoral and individual level analysis seems so crucial in the field, current studies on Ghana deal with more of organizational level analysis to the neglect of the other two. Although hospitality is booming in Ghana because of the globalization process, few papers have focused on tourism CSR and CSR and the telecommunication companies. From a scholarly viewpoint, current studies on CSR look too descriptive, and therefore, future studies must seek to employ advanced statistics to validate existing findings.

The current review complements and builds on extant literature with regard to the CSR's review on countries in Africa. The paper makes three contributions. Consequently, a full understanding of various CSR's studies that offer a comprehensive state of knowledge in the field is currently a challenge. The present review, therefore, in part fills a significant gap in the extant literature on the topic.

Second, by developing a comprehensive review of the specific body of literature on CSR in the context of Ghana and for that matter, a developing country and situating the study within the broader CSR and management literature create a significant opportunity for researchers interested in African issues to work with this review in the light of future studies.

With most other sectors of Ghana presently growing e. This can be made possible by making funds available for research and tying specific funds to CSR's specific sector research. The full text of this article hosted at iucr. If you do not receive an email within 10 minutes, your email address may not be registered, and you may need to create a new Wiley Online Library account. If the address matches an existing account you will receive an email with instructions to retrieve your username. Tools Request permission Export citation Add to favorites Track citation.

Share Give access Share full text access. Share full text access. Please review our Terms and Conditions of Use and check box below to share full-text version of article. Abstract There are recent calls to pay attention to the institutional requirement or the configurations of the national business system because it eventually results in the different manifestation of corporate social responsibility CSR in different contexts.

Figure 1 Open in figure viewer PowerPoint. Figure 2 Open in figure viewer PowerPoint. Figure 3 Open in figure viewer PowerPoint. Type of analysis Frequency Percentage Exploratory analysis 43 Nature of collaboration Frequency Percentage Single authorship 38 Name of school Frequency Percentage University of Ghana 52 Employee perceptions of corporate social responsibility and organisational citizenship behaviours: A comparative Ghanaian Doctoral dissertation, University of Ghana.