Epistemic probability

The family of distributions Fi(y) represents the uncertainty in the unknown y due to uncertainty in the models structure it can be probabihstically combined in a summary measure by means of a standard Bayesian approach (Apostolakis, 1993). Indicating with p Mi) the epistemic probability which expresses the analyst s confidence in the set of assumptions underpinning the /-th model (here the term epistemic... 

The mean conditional probability that an I C cable in the fire compartment is damaged during the fire scenario was estimated to be 1.76 E-02 based on failure criterion 1 and 4.10 E-03 based on criterion 2. Figure 5 shows the epistemic uncertainty of the cable damage probability for each criterion. It can be seen that the epistemic probability distributions due to both criteria differ significantly. Since just a sample of 60 values is available from the simulations of MCDET and FDS, the one-sided upper (95%, 95%) tolerance limit was calculated to quantify the epistemic uncertainty of the cable damage probability (see section 3.2). With failure criterion 1, the (95%, 95%) tolerance limit is 0.5. The (95%, 95%) tolerance limit based on criterion 2 is 1.53 E-02. 

Figure 8. Epistemic probability distribution function on the mean farm availability-informed capability averaged over first 5 years of operation.

Hora SC. 1996. Aleatory and epistemic uncertainty in probability elicitation with an example from hazardous waste management. Reliability Eng Syst Saf 54 217-223. 

The probabilities of different outcomes can thus be seen as resulting from the causal powers and capacities of the system and their arrangement. This makes probability a function of the nature of the system, not merely a statement of degrees of belief or the frequency with which an outcome occurs. We can account for the observed probability (in a frequency sense) by the interplay of capacities or causal powers, and we can estimate a probability (in the epistemic sense) if we know something about the capacities of the things that may influence the outcome. 

Instead of assessing risk, I suggest that we should try to assess riskiness in the everyday sense of this term, where it refers to the epistemic possibility of harm, not merely probabilities of identified types of harm. Whereas risk relates to outcomes, riskiness is a property of a thing, situation or activity and is relative to our knowledge about it. I suggest that what are normally termed precautionary approaches are concerned with riskiness, rather than just risk they are concerned with whether, for all we know, there is a possibility of harm, not just with the probabilities of known, specifiable types of harm. 

All parameters mentioned are either stochastic or uncertain because of lack of knowledge. Hence, they either lead to aleatory or epistemic uncertainties in the calculations to be performed. That is why they are treated with probability distributions whose selection is indicated and justified below. 

The literature indicates numerous, often widely different values for assigning probabilities to scenarios such as those of Figs. 10.2, 10.3 and 10.4. This reflects uncertainties due to lack of knowledge (epistemic) and stochastic (aleatory) effects. If properly treated these uncertainties should be represented by probability distributions. Table 10.15 gives an overview of conditional probabilities for the consequences of a puff release of a pressurized flammable gas from various sources. Not only are the differences in probabilities evident but also the differences as to the endpoints. 

This is so on accounts of laws on which a condition on law-hood is that laws are instance confirmable since confirmation is an epistemic notion. If confirmation is understood in subjective Bayesian terms, then this point is obvious. A subjective probability distribution on which M R... 

The two reasons for pluralism, multiple aims and epistemic limits, thus act in concert. Chemistry (and probably any science faced with complexity issues) can partially overcome its epistemic limits by diverging according to its different aims. 

In this Section, the Dempster-Shafer Theory (DST) of Evidence (Shafer, 1976) is considered for the representation of the epistemic uncertainty affecting the expert knowledge of the probability P Mi) that the alternative model Mi, I = 1,..., be correct. In the DS framework, a lower and an upper bound are introduced for representing the uncertainty associated to P (Ml). The lower bound, called behef, Bel (Mi), represents the amount of belief that directly supports M at least in part, whereas the upper bound, called plausibility, Pl Mi), measures the fact that M could be the correct model up to that value because there is only so much evidence that contradicts it. 

In the so-called probability of frequency approach (Kaplan Garrick 1981, Aven 2003), relative frequency-based probabilities are used to describe aleatory uncertainty and subjective probabilities to describe epistemic uncertainty. The probability of frequency approach differs fundamentally in philosophy but not much in practice fiwm a standard Bayesian approach (Aven 2003). In the Bayesian approach all uncertainty is epistemic, and probability is always considered an expression of belief it is not a property of the world in the way that a relative frequency-based probability is. The notion of aleatory uncertainty, sometimes just referred to as variation in the Bayesian approach (Aven 2003), is captured by the concept of chmce, defined as the limit of a relative frequency in an exchangeable, infinite Bernoulli series (Lind-ley 2006). A chance distribution is then the limit of... 

In the following, probability refers to the relative frequency-based concept whenever followed by the word chance in parenthesis, and to the epistemic (subjective, lack of knowledge-based) concept when used alone. 

Irrespective of the taxonomy used, epistemic and aleatory uncertainty, or uncertainty and variation, alternatives to probability have been suggested for the representation of the epistemic concept. These include interval or imprecise probabihty (Coolen 2004, Coolen Utkin 2007, Utkin Coolen 2007, Weichselberger 2000), fuzzy set theory and the associated theory of possibility (Zadeh 1965, Zadeh 1978, Unwin 1986), and the theory of behef functions (Shafer 1976), also known as evidence theory or the Dempster-Shafer theory of evidence. 

Specifically, it has been suggested that a possibilistic representation of epistemic uncertainty may be more adequate when sufficiently informative data are not available for statistical analysis and thus one has to resort mostly to information of qualitative namre. In particular, in cases where an expert does not have sufficiently refined knowledge or opinion to characterize the epistemic uncertainty in terms of probability distributions, possibility theory uses two measures of... 

There often seems to be a tacit assumption made that knowledge-based or subjective probability is an appropriate representation of epistemic uncertainty only when sufficient data exist, on which to base the probability or probability distribution in question. However, this is a misconception. Rather, the question is whether a comparison with the standard described in the previous section can be made. Faced with an expert/assessor who is not willing to specify a single number p or a single distribution G as his/her probability or probability distribution, respectively, but who will provide an interval or a set of distributions, an analyst may choose to define a possibUify distribution to reflect the imprecise input from the expert. Using a possibility distribution n implies that no particular probability distribution G is selected and assigned from among the family ( r) of probabihty distributions compatible with 7r. Compatible here means that for any interval... 

Parameters uncertainty is generally epistemic in nature as one simply does not know the correct values for the model s input parameters. Probability is always a measure of the degree of belief of the analyst. 

Fortunately, the Bayesian approach allows incorporation of all available relevant information into the assessment of probabilities, ft allows quantification of both epistemic as well as aleatory imcertainties, and the combination of their effects into a single probability value of an undesirable event, or into a single probability distribution for the consequences of assumed risk. Once the uncertainties are determined, they can be propagated through the risk model using techniques such as Monte Carlo simulation. 

Epistemic uncertainty is uncertainty that comes from lack of knowledge. This lack of knowledge comes from many sources, for example, inadequate understanding of the processes, incomplete knowledge of the phenomena, imprecise evaluation of the related charactetistics, etc. Epistemic uncertainties affect the values of the probabilities and frequencies of the events included in the accident scenarios, such as mechanical failure and repair rate, probability of failure on demand for a control system, or human error. There are three different cases in this regard ... 

Main subject of the analysis presented here was the fire dynamics as simulated by the FDS code in interaction with stochastic factors affecting the fire over time. The mixture of MCDET and FDS was used to simulate many different time series of quantities of the fire dynamics associated with corresponding conditional occurrence probabilities. From these results, distributions referring to the temporal evolution of target temperatures and quantifications of the influence of epistemic uncertainties could be derived. 

Figure 5. Epistemic uncertainty of the conditional probability of damaged I C cables.

Expert judgment is a key source of data especially in relation to the failure process and the quantification of epistemic uncertainties. For example, we need to specify and to determine the values of target variables, such as the baseline probability and risk reduction proportions, which we assume are fixed. The shock rate, wear-out parameters and onset of premature wear-out also require to be assessed. For the epistemically uncertain parameters, such as the onset of aging, we use the whole uncertainty distribution as assessed by experts. 

We can focus on the effect of epistemic uncertainty only by calculating the mean farm availability averaged over the whole early life from the inner-loop iterations of the model. Figure 8 shows the empirical probability distribution of this mean. 

MUSTADEPT provides in output the probability with which the SCC occupies the different degradation states over time, and the uncertainty associated to this evaluation. In particular, such uncertainty encodes both aleatory (i.e., inherent to the SSC stochastic behaviors) and epistemic (i.e., due to the lack of precise knowledge of the model parameters and the Monte Carlo estimation) contributions. 

For each time instant, the specific solutions of the DSTE approach give the bounds of the probabilities of finding the SSC in the 4 degradation states, and the associated uncertainties, described in terms of Belief and Plausibility functions. These encompass both epistemic and aleatory uncertainties and, roughly speaking, can be regarded as the lower and upper bounds, respectively, of all the possible CDFs encoded by the imprecise estimations provided by the experts on the model parameters. 

Kaplan (1997) proposes the so-called probability of frequency approach to risk assessment, based on a risk concept in line with risk definition C6 (R = P C), where subjective probabilities are used to express uncertainty about true frequen-tist probabilities. The assessment thus focuses on quantifying uncertainty about an underlying true risk, which is estimated. Kaplan s view is strongly tied to realism, as the risk description focuses on a true risk as determined by experts. Closely related perspectives are those where uncertainty is quantified around a true risk, such as in the traditional Bayesian perspective where uncertainty is quantified in relation to model parameters (Aven Heide 2009). Such uncertainty quantification can also be done using non-probabilistic representations of epistemic uncertainty (Helton Johnson 2011). These methods typicdly consider a risk problem in a highly mathematized form. 

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