The Base Rate Fallacy in Belief Reasoning∗

Audun Jøsang Stephen O’Hara University of Oslo 21CSI, USA [email protected] [email protected]

Abstract – The base rate fallacy is an error that occurs probability of the antecedent x together with the condition- when the conditional probability of some hypothesis y given als. The expression p(yx) thus represents a derived value, some evidence x is assessed without taking account of the whereas the expressions p(y|x) and p(y|x) represent input ”base rate” of y, often as a result of wrongly assuming values together with p(x). Below, this notational convention equality between the inverse conditionals, expressed as: will also be used for belief reasoning. p(y|x)=p(x|y). Because base rates and conditionals are The problem with conditional reasoning and the base rate not normally considered in traditional Dempster-Shafer be- fallacy can be illustrated by a simple medical example. As- lief reasoning there is a risk of falling victim to the base sume a pharmaceutical company that develops a test for a rate fallacy in practical applications. This paper describes particular infectious disease will typically determine the re- the concept of the base rate fallacy, how it can emerge in liability of the test by letting a group of infected and a group belief reasoning and possible approaches for how it can be of non-infected people undergo the test. The result of these avoided. trials will then determine the reliability of the test in terms of its sensitivity p(x|y) and false positive rate p(x|y), where x: Keywords: Belief reasoning, base rates, base rate fallacy, “Positive Test”, y: “Infected” and y: “Not infected”. The subjective logic conditionals are interpreted as: 1 Introduction • p(x|y): “Probability of positive test given infection” Conditional reasoning is used extensively in areas where • p(x|y): “Probability of positive test in the absence of conclusions need to be derived from evidence, such as in infection”. medical reasoning, legal reasoning and intelligence analysis. The problem with applying these reliability measures in a Both binary logic and probability calculus have mechanisms practical setting is that the conditionals are expressed in the for conditional reasoning. In binary logic, Modus Ponens opposite direction to what the practitioner needs in order to (MP) and Modus Tollens (MT) are the classical operators apply the expression of Eq.(1). The conditionals needed for which are used in any field of logic that requires conditional making the diagnosis are: deduction. In probability calculus, binomial conditional de- duction is expressed as: • p(y|x): “Probability of infection given positive test” • p(y|x) p(yx)=p(x)p(y|x)+p(x)p(y|x) (1) : “Probability of infection given negative test” but these are usually not directly available to the medical where the terms are interpreted as follows: practitioner. However, they can be obtained if the base rate p(y|x): conditional probability of y given x is TRUE of the infection is known. p(y|x): conditional probability of y given x is FALSE The base rate fallacy [2] consists of making the erroneous p(x): probability of the antecedent x assumption that p(y|x)=p(x|y). Practitioners often fall p(x): probability of the antecedent’s complement victim to this the reasoning error, e.g. in medicine, legal (= 1 - p(x)) reasoning (where it is called the prosecutor’s fallacy) and in p(yx): deduced probability of the consequent y intelligence analysis. While this reasoning error often can produce a relatively good approximation of the correct diag- The notation yx, introduced in [1], denotes that the truth nostic probability value, it can lead to a completely wrong or probability of proposition y is deduced as a function of the result and wrong diagnosis in case the base rate of the dis- ∗In the proceedings of the 13th International Conference on Information ease in the population is very low and the reliability of the Fusion (FUSION 2010), Edinburgh, July, 2010 test is not perfect. This paper describes how the base rate fallacy can occur p(x|y)=0.001 (i.e. a non-infected person will test pos- in belief reasoning, and puts forward possible approaches itive in 0.1% of the cases). Let the base rate of infection for how this reasoning error can be avoided. in population A be 1% (expressed as a(yA)=0.01) and let the base rate of infection in population B be 0.01% (ex- 2 Reasoning with Conditionals pressed as a(yB)=0.0001). Assume that a person from pop- A Assume a medical practitioner who, from the description ulation tests positive, then Eq.(4) and Eq.(1) lead to the p(y x)=p(y |x)=0.9099 of a medical test, knows the sensitivity p(x|y) and the false conclusion that A A which in- positive rate p(x|y) of the test, where x: “Positive Test”, y: dicates a 91% likelihood that the person is infected. As- B “Infected” and y: “Not infected”. sume that a person from population tests positive, then p(y x)=p(y |x)=0.0909 The required conditionals can be correctly derived by in- B B which indicates only a 9% verting the available conditionals using Bayes rule. The in- likelihood that the person is infected. By applying the cor- verted conditionals are obtained as follows: rect method the base rate fallacy is avoided in this example. ⎧ ⎪ p(x|y)= p(x∧y) 2.2 Example 2: Intelligence Analysis ⎨ p(y) p(y)p(x|y) ⇒ p(y|x)= . (2) A typical task in intelligence analysis is to assess and ⎩⎪ p(x∧y) p(x) p(y|x)= p(x) compare a set of hypotheses based on collected evidence. Stech and El¬asser of the MITRE corporation describe how On the right hand side of Eq.(2) the base rate of the dis- analysts can err, and even be susceptible to deception, when ease in the population is expressed by p(y). By applying determining conditionals based on a limited view of evi- Eq.(1) with x and y swapped in every term, the expected dence [3]. rate of positive tests p(x) in Eq.(2) can be computed as a An example of this is how the reasoning about the detec- function of the base rate p(y). This is expressed in Eq.(3) tion of Krypton gas in a middle-eastern country can lead to below. the erroneous conclusion that the country in question likely has a nuclear enrichment program. A typical ad-hoc approx- p(x)=p(y)p(x|y)+p(y)p(x|y) (3) imation is: a(x) a(y) In the following, and will denote the base rates p(Krypton | enrichment) = high x y of and respectively. The required positive conditional is: ⇓ (6) a(y)p(x|y) p(enrichment | Krypton) = high p(y|x)= . a(y)p(x|y)+a(y)p(x|y) (4) The main problem with this reasoning is that it does In the general case where the truth of the antecedent is not consider that Krypton gas is also used to test pipelines expressed as a probability, and not just binary TRUE and for leaks, and that being a middle-eastern country with oil FALSE, the opposite conditional is also needed as specified pipelines, the probability of the gas being used outside of a in Eq.(1). In case the negative conditional is not directly nuclear program is also fairly high, i.e. available, it can be derived according to Eq.(4) by swapping | x and x in every term. This produces: p(Krypton not enrichment) = medium to high (7) a(y)p(x|y) so that a more correct approximation is: p(y|x)=a(y)p(x|y)+a(y)p(x|y) (5) p(Krypton | enrichment) = high = a(y)(1−p(x|y)) . a(y)(1−p(x|y))+a(y)(1−p(x|y)) p(Krypton | not enrichment) = medium to high ⇓ (8) The combination of Eq.(4) and Eq.(5) enables conditional p(enrichment | Krypton) = uncertain reasoning even when the required conditionals are expressed in the reverse direction to what is needed. This additional information should lead the analyst to the A medical test result is typically considered positive or conclusion that there is a fair amount of uncertainty of a negative, so when applying Eq.(1), it can be assumed that nuclear program given the detection of Krypton. The as- p(x) p(x) either = 1 (positive) or = 1 (negative). In case the signment of the ‘high’ value to p(enrichment | Krypton) p(yx)= patient tests positive, Eq.(1) can be simplified to neglects the fact that an oil-rich middle-eastern country is p(y|x) so that Eq.(4) will give the correct likelihood that the likely to use Krypton gas Ð regardless of whether they have patient actually has contracted the disease. a nuclear program. 2.1 Example 1: Probabilistic Medical Rea- 2.3 Deductive and Abductive Reasoning soning The term frame1 will be used with the meaning of a tradi- Let the sensitivity of a medical test be expressed as tional state space of mutually disjoint states. We will use the p(x|y)=0.9999 (i.e. an infected person will test posi- tive in 99.99% of the cases) and the false positive rate be 1Usually called frame of discernment in traditional belief theory term “parent frame” and “child frame” to denote the rea- 3 Review of Belief-Based Conditional soning direction, meaning that the parent frame is what the Reasoning analyst has evidence about, and probabilities over the child frame is what the analyst needs. Defining parent and child In this section, previous approaches to conditional reason- frames is thus equivalent with defining the direction of the ing with beliefs and related frameworks are briefly reviewed. reasoning. 3.1 Smets’ Disjunctive Rule and Generalized Forward conditional inference, called deduction, is when Bayes Theorem the parent frame is the antecedent and the child frame is An early attempt at articulating belief-based conditional the consequent of the available conditionals. Reverse condi- reasoning was provided by Smets (1993) [4] and by Xu tional inference, called abduction, is when the parent frame & Smets [5, 6]. This approach is based on using the so- is the consequent, and the child frame is the antecedent. called Generalized Bayes Theorem as well as the Disjunc- Causal and derivative reasoning situations are illustrated tive Rule of Combination, both of which are defined within in Fig.1 where x denotes the cause state and y denotes the Dempster-Shafer belief theory. the result state. Causal conditionals are expressed as In the binary case, Smets’ approach assumes a conditional Θ p(effect | cause). connection between a binary parent frame and a binary child frame X defined in terms of belief masses and con- ditional plausibilities. In Smets’ approach, binomial deduc- tion is defined as:

p( x ) Cause reasoning Causal

Derivative reasoning Derivative pl(x)= m(θ)pl(x|θ)+m(θ)pl(x|θ) +m(Θ)(1 − (1 − pl(x|θ))(1 − pl(x|θ))) pl(x)= m(θ)pl(x|θ)+m(θ)pl(x|θ) Causal p( y | x ) p( y | x ) Conditionals +m(Θ)(1 − (1 − pl(x|θ))(1 − pl(x|θ))) pl(X)= m(θ)pl(X|θ)+m(θ)pl(X|θ) +m(Θ)(1 − (1 − pl(X|θ))(1 − pl(X|θ))) p( y ) Effect (9) The next example illustrates a case where Smets’ deduc- Figure 1: Visualizing causal and derivative reasoning tion operator produces inconsistent results. Let the condi- tional plausibilities be expressed as: pl(x|θ)= 1 pl(x|θ)= 3 pl(X|θ)=1 4 4 The concepts of “causal” and “derivative” reasoning can Θ → X : be meaningful for obviously causal conditional relationships 1 3 pl(x|θ)= 4 pl(x|θ)= 4 pl(X|θ)=1 between states. By assuming that the antecedent causes the (10) consequent in a conditional, then causal reasoning is equiv- Eq.(10) expresses that the plausibilities of x are totally alent to deductive reasoning, and derivative reasoning is independent of θ because pl(x|θ)=pl(x|θ) and pl(x|θ)= A equivalent to abductive reasoning. Derivative reasoning re- pl(x|θ). Let now two basic belief assignments (bbas), mΘ p( | ) B quires derivative conditionals expressed as cause effect . and mΘ on Θ be expressed as: Causal reasoning is much simpler than derivative reasoning ⎧ ⎧ A B because it is much easier to estimate causal conditionals than ⎨ mΘ(θ)=1/2 ⎨ mΘ(θ)=0 A A B B derivative conditionals. However, a large part of human in- mΘ : mΘ(θ)=1/2 mΘ : mΘ(θ)=0 ⎩ A ⎩ B tuitive reasoning is derivative. mΘ(Θ) = 0 mΘ(Θ) = 1 (11) In medical reasoning for example, the infection causes the This results in the following plausibilities pl, belief test to be positive, not the other way, so derivative reasoning masses mX and pignistic probabilities E,onX in Table 1: must be applied to estimate the likelihood of being affected A B by the disease. The reliability of medical tests is expressed State Result of mΘ on Θ Result of mΘ on Θ as causal conditionals, whereas the practitioner needs to ap- pl mX E pl mX E ply the derivative inverted conditionals. Starting from a pos- x 1/4 1/4 1/4 7/16 1/16 1/4 itive test to conclude that the patient is infected therefore x 3/4 3/4 3/4 1/16 9/16 3/4 represents derivative reasoning. Most people have a ten- X 1 0 n.a. 1 6/16 n.a. dency to reason in a causal manner even in situations where derivative reasoning is required. In other words, derivative Table 1: Inconsistent results of deductive reasoning with situations are often confused with causal situations, which Smets’ method provides an explanation for the tendency of the base rate fal- lacy in medical diagnostics, legal reasoning and intelligence Because X is totally independent of Θ according to analysis. Eq.(10), the bbas on X (mX ) should not be influenced by the bbas on Θ. It can be seen from Table 1 that the prob- mining whether conditional deduction properties are satis- ability expectation values E are equal for both bbas, which fied for a set of conditional constraints. seems to indicate consistency. However, the belief masses The surveyed literature on creedal networks and non- are different, which shows that Smets’ method [4] can pro- monotonic probabilistic reasoning only describe methods duce inconsistent results. It can be mentioned that the frame- for deductive reasoning, although abductive reasoning un- work of subjective logic does not have this problem. der these formalisms would theoretically be possible. In Smets’ approach, binomial abduction is defined as: A philosophical concern with imprecise probabilities in general, and with conditional reasoning with imprecise pl(θ)= m(x)pl(x|θ)+m(x)pl(x|θ)+m(X)(pl(X|θ))) , probabilities in particular, is that there can be no real up- pl(θ)= m(x)pl(x|θ)+m(x)pl(x|θ)+m(X)pl(X|θ))) , per and lower bound to probabilities unless these bounds are pl(Θ)= m(x)(1 − (1 − pl(x|θ))(1 − pl(x|θ))) set to the trivial interval [0, 1]. This is because probabilities +m(x)(1 − (1 − pl(x|θ))(1 − pl(x|θ))) about real world propositions can never be absolutely cer- +m(X)(1− (1− pl(X|θ))(1− pl(X|θ))) . tain, thereby leaving the possibility that the actual observed (12) probability is outside the specified interval. For example, Eq.(12) fails to take the base rates on Θ into account and Walley’s Imprecise Dirichlet Model (IDM) [11] is based on would therefore unavoidably be subject to the base rate fal- varying the base rate over all possible outcomes in the frame lacy, which would also be inconsistent with probabilistic of a Dirichlet distribution. The probability expectation value reasoning as e.g. described in Example 1 (Sec.2.1). It can of an outcome resulting from assigning the total base rate be mentioned that abduction with subjective logic is always (i.e. equal to one) to that outcome produces the upper prob- consistent with probabilistic abduction. ability, and the probability expectation value of an outcome resulting from assigning a zero base rate to that outcome 3.2 Halpern’s Approach to Conditional Plau- produces the lower probability. The upper and lower prob- sibilities abilities are then interpreted as the upper and lower bounds for the relative frequency of the outcome. While this is an Halpern (2001) [7] analyses conditional plausibilities interesting interpretation of the Dirichlet distribution, it can from an algebraic point of view, and concludes that con- not be taken literally. According to this model, the upper ditional probabilities, conditional plausibilities and con- and lower probability values for an outcome x i are defined ditional possibilities share the same algebraic properties. as: Halpern’s analysis does not provide any mathematical meth- r(x )+W ods for practical conditional deduction or abduction. P (x )= i IDM Upper Probability: i k (13) W + i=1 r(xi) 3.3 Conditional Reasoning with Imprecise r(x ) P (x )= i Probabilities IDM Lower Probability: i k (14) W + r(xi) Imprecise probabilities are generally interpreted as proba- i=1 bility intervals that contain the assumed real probability val- where r(xi) is the number of observations of outcome x i, ues. Imprecision is then an increasing function of the in- and W is the weight of the non-informative prior probability terval size [8]. Various conditional reasoning frameworks distribution. based on notions of imprecise probabilities have been pro- It can easily be shown that these values can be mislead- posed. ing. For example, assume an urn containing nine red balls Creedal networks introduced by Cozman [9] are based on and one black ball, meaning that the relative frequencies of creedal sets, also called convex probability sets, with which red and black balls are p(red)=0.9 and p(black)=0.1. upper and lower probabilities can be expressed. In this the- The a priori weight is set to W =2. Assume further that an ory, a creedal set is a set of probabilities with a defined upper observer picks one ball which turns out to be black. Accord- 1 and lower bound. There are various methods for deriving ing to Eq.(14) the lower probability is then P (black)= 3 . creedal sets, e.g. [8]. Credal networks allow creedal sets It would be incorrect to literally interpret this value as the to be used as input in Bayesian networks. The analysis of lower bound for the relative frequency because it obviously creedal networks is, in general, more complex than the anal- is greater than the actual relative frequency of black balls. ysis of traditional probabilistic Bayesian networks because This example shows that there is no guarantee that the actual it requires multiple analyses according to the possible prob- probability of an event is inside the interval defined by the abilities in each creedal set. Various algorithms can be used upper and lower probabilities as described by the IDM. This to make the analysis more efficient. result can be generalized to all models based on upper and Weak non-monotonic probabilistic reasoning with con- lower probabilities, and the terms “upper” and “lower” must ditional constraints proposed by Lukasiewicz [10] is also therefore be interpreted as rough terms for imprecision, and based on probabilistic conditionals expressed with upper not as absolute bounds. and lower probability values. Various properties for condi- Opinions used in subjective logic do not define upper and tional deduction are defined for weak non-monotonic prob- lower probability bounds. As opinions are equivalent to abilistic reasoning, and algorithms are described for deter- general Dirichlet probability density functions, they always cover any probability value except in the case of dogmatic The algebraic expressions for conditional deduction and opinions which specify discrete probability values. abduction in subjective logic are relatively long and are therefore omitted here. However, they are mathematically 3.4 Conditional Reasoning in Subjective simple and can be computed extremely efficiently. A full presentation of the expressions for binomial conditional de- Logic duction in is given in [1] and a full description of multi- Subjective logic [12] is a probabilistic logic that takes nomial deduction is provided in [13]. Only the notation is A opinions as input. An opinion denoted by ω x =(b, d, u, a) provided here. expresses the relying party A’s belief in the truth of state- Let ωx, ωy|x and ωy|x be an agent’s respective opinions ment x. Here b, d, and u represent belief, disbelief and un- about x being true, about y being true given that x is true, certainty respectively, where b, d, u ∈ [0, 1] and b + d + u = and about y being true given that x is false. Then the opin- 1. The parameter a ∈ [0, 1] is called the base rate, and ion ωyx is the conditionally derived opinion, expressing the is used for computing an opinion’s probability expectation belief in y being true as a function of the beliefs in x and the A value that can be determined as E(ωx )=b + au. In the ab- two sub-conditionals y|x and y|x. The conditional deduc- sence of any specific evidence about a given party, the base tion operator is a ternary operator, and by using the function rate determines the a priori trust that would be put in any symbol ‘’ to designate this operator, we write: member of the community. ω = ω  (ω ,ω ) . The opinion space can be mapped into the interior of yx x y|x y|x (15) ω = an equal-sided triangle, where, for an opinion x Subjective logic abduction is described in [1] and requires (b ,d ,u ,a ) b d u x x x x , the three parameters x, x and x deter- the conditionals to be inverted. Let x be the parent node, and mine the position of the point in the triangle representing let y be the child node. In this situation, the input conditional the opinion. Fig.2 illustrates an example where the opinion opinions are ωx|y and ωx|y. That means that the original x about a proposition from a binary state space has the value conditionals are expressed in the opposite direction to what ω =(0.7, 0.1, 0.2, 0.5) x . is needed. The inverted conditional opinions can be derived from Uncertainty ω ω 1 knowledge of the supplied conditionals, x|y and x|y, and knowledge of the base rate of the child, a y. Example opinion: Given knowledge of the base rate ay of the child state ω = (0.7, 0.1, 0.2, 0.5) vac x where ωy is a vacuous subjective opinion about the base rate of the hypothesis, defined as ⎧ 0 0 ⎪ b =0 0.5 ⎨⎪ y vac dy =0 ωy =(by,dy,uy,ay) (16) ⎪ uy =1 ω Projector ⎩ 0.5 0.5 x ay = base rate of y ω ,ω 1 0 1 and given the logical conditionals x|y x|y, then the Disbelief Belief ω ,ω 01ax E( x ) inverted conditionals y|x y|x can be derived using the Probability axis following formula vac ω ·ωx|y ω = y Figure 2: Opinion triangle with example opinion y|x vac ωy (ωx|y,ωx|y) (17) The top vertex of the triangle represents uncertainty, the vac ω ·¬ ωx|y bottom left vertex represents disbelief, and the bottom right ω = y y|x ωvac(¬ ω ,¬ ω ) vertex represents belief. The parameter bx takes value 0 on y x|y x|y the left side edge and takes value 1 at the right side belief vertex. The parameter dx takes value 0 on the right side edge  and takes value 1 at the left side disbelief vertex. The param- The abduction operator, , is written as follows: eter ux takes value 0 on the base edge and takes value 1 at ωyx = ωx  ωx|y,ωx|y,ay (18) the top uncertainty vertex. The base of the triangle is called the probability axis. The base rate is indicated by a point The advantage of subjective logic over probability calcu- on the probability axis, and the projector starting from the lus and binary logic is its ability to explicitly express and opinion point is parallel to the line that joins the uncertainty take advantage of ignorance and belief ownership. Subjec- vertex and the base rate point on the probability axis. The tive logic can be applied to all situations where probability point at which the projector meets the probability axis deter- calculus can be applied, and to many situations where prob- mines the expectation value of the opinion, i.e. it coincides ability calculus fails precisely because it can not capture de- A with the point corresponding to expectation value E(ω x ). grees of ignorance. Subjective opinions can be interpreted as probability density functions, making subjective logic a ditionals are determined, so it is not necessarily a weakness simple and efficient calculus for probability density func- in the presented models. However, in practice it is crucial tions. An online demonstration of subjective logic can be that thresholds and conditionals are adequately determined. accessed at: http://folk.uio.no/josang/sl/. Simply setting the thresholds in an ad-hoc way is not ade- quate. ω 4 Base Rates in Target Recognition It would be possible to set conditionals (yj |xi) in a prac- tical situation by trial and error by assuming that the true Applications hypothesis yj were known during the learning period. How- Evidence for recognizing a target can come from different ever, should the base rate of the occurrence of the hypothesis sources. Typically, some physical characteristics of an ob- change, the settings would no longer be correct. Similarly, ject are observed by sensors, and the observations are trans- it would not be possible to reuse the settings of conditionals lated into evidence about a set of hypotheses regarding the in another situation whether the base rates of the hypothesis nature of the object. The reasoning typically is based on are different. conditionals, so that belief about the nature of objects can be By using the sound method of determining the condition- derived from the observations. Let yj denote the hypothesis als, their values can more easily be adjusted to new situa- that the object is of type yj , and let xi denote the observed tions by simply determining the base rates of the hypothe- characteristic xi. In subjective logic notation the required sis in each situation and deriving the required conditionals ω ω conditional can be denoted as (yj |xi). The opinion about (yj |xi) accordingly. the hypothesis yj can then be derived as a function of the ω ω opinion xj and the conditional (yj |xi) according to Eq.15. 5 Base Rates in Global Warming It is possible to model the situation where several sensors Models observe the same characteristics, and to fuse opinions about an hypothesis derived from different characteristics. In or- As an example we will test two hypotheses about the con- der to correctly model such situations the appropriate fusion ditional relevance between CO2 emission and global warm- operators are required. ing. Let x:“Global warming” and y:“Man made CO2 emis- A common problem when assessing a hypotheses y j is sion”. We will see which of these hypotheses lead to rea- CO that it is challenging to determine the required conditionals sonable conclusions about the likelihood of man made 2 ω emission based on observing global warming. There have (yj |xi) on a sound basis. Basically there are two ways to determine the required conditionals, one complex but sound, been approximately equally many periods of global warm- and the other ad-hoc but unsafe. The sound method is based ing as global cooling over the history of the earth, so the on first determining the conditionals about observed char- base rate of global warming is set to 0.5. Similarly, over CO acteristics given specific object types, denoted by ω(x |y ). the history of the earth, man made 2 emission has oc- i j a =0.1 These conditionals are determined by letting the sensors ob- curred very rarely, meaning that y for example. Let serve the characteristics of known objects, similarly to the us further assume the evidence of global warming, i.e. that way the reliability of medical tests are determined by ob- an increase in temperature can be observed, expressed as: serving their outcome when applied to known medical con- ωx =(0.9, 0.0, 0.1, 0.5) (19) ditions. In order to correctly apply the sensors for target recognition in an operative setting these conditionals must • IPCC’s View be inverted by using the base rates of the hypotheses accord- According to the IPCC (International Panel on Climate ing to Eq.(4) and Eq.(5) in case of probabilistic reasoning, Change) [16] the relevance between CO 2 emission and and according to Eq.(17) in case of subjective logic reason- global warming is expressed as: ing. In the latter case the derived required conditionals are ω ωIPCC =(1.0, 0.0, 0.0, 0.5) (20) denoted (yj |xi). This method is complex because the reli- x|y ability of the sensors must be empirically determined, and IPCC ωx|y =(0.8, 0.0, 0.2, 0.5) (21) because the base rate of the hypothesis must be known. (22) The ad-hoc but unsafe method consists of setting the re- quired conditionals ω(y |x ) by guessing or by trial-and- j i According to the IPCC’s view, and by applying the for- error, in principle without any sound basis. Numerous exam- malism of abductive belief reasoning of Eq.(18), it can ples in the literature apply belief reasoning for target recog- be concluded that whenever there is global warming on nition with ad hoc conditionals or thresholds for the obser- Earth there is man made CO2 emission with the likeli- vation measurements to determine whether a specific object IPCC hood ω =(0.62, 0.00, 0.38, 0.10), as illustrated has been detected, e.g. [14] (p.4) and [15] (p.450). Such yx thresholds are in principle conditionals, i.e. in the form of in Fig.3. ”hypothesis yj is assumed to be TRUE given that the sensor This is obviously a questionable conclusion since all provides a measure xi greater than a specific threshold”. but one period of global warming during the history Examples in the literature typically focus on fusing beliefs of the earth has taken place without man made CO 2 about some hypotheses, not on how the thresholds or con- emission. Figure 3: Likelihood of man made CO2 emission with IPCC’s conditionals

Figure 4: Likelihood of man made CO2 emission with the skeptic’s conditionals

• The Skeptic’s View Based on this analysis the IPCC’s hypothesis seems un- Martin Duke is a journalist who produced the BBC reasonable. As global warming is a perfectly normal phe- documentary “The Great Global Warming Swindle” nomenon, it is statistically very unlikely that any observed and who is highly skeptical about the IPCC. Let us ap- global warming is caused by man-man CO 2 emission. ply skeptic Martin Dukin’s view that we don’t know anything about whether a reduction in man-made CO 2 6 Discussion and Conclusion emission would have any effect on global warming. Applications of belief reasoning typically assesses the ef- This is expressed as: fect of evidence on some hypothesis. We have shown that Skeptic common frameworks of belief reasoning commonly fail to ωx|y =(1.0, 0.0, 0.0, 0.5) (23) properly consider base rates and thereby are unable to han- ωSkeptic =(0.0, 0.0, 1.0, 0.5) x|y (24) dle derivative reasoning. The traditional probability projec- tion of belief functions in form of the Pignistic transforma- From this we conclude that global warming occurs with tion assumes a default subset base rate equal to the sub- man-made CO2 emission together with a likelihood of set’s relative atomicity. In other words, the default base rate ωSkeptic =(0.08, 0.01, 0.91, 0.10) yx , as illustrated in of a subset is equal to the relative number of singletons in Fig.4. This conclusion seems more reasonable in light the subset over the total number of singletons in the whole of the history of the earth. frame. Subsets also have default relative base rates with re- spect to every other fully or partly overlapping subset of the [4] Ph. Smets. Belief functions: The disjunctive rule of frame. Thus, when projecting a bba to scalar probability val- combination and the generalized Bayesian theorem. ues, the Pignistic probability projection dictates that belief International Journal of Approximate Reasoning, 9:1Ð masses on subsets contribute to the projected probabilities 35, 1993. as a function of the default base rates on those subsets. However, in practical situations it would be possible and [5] H. Xu and Ph Smets. Reasoning in Evidential Net- useful to apply base rates different from the default base works with Conditional Belief Functions. Inter- rates. For example, consider the base rate of a particular national Journal of Approximate Reasoning, 14(2Ð infectious disease in a specific population, and the frame 3):155Ð185, 1996. {“infected”, “not infected”}. Assuming an unknown per- [6] H. Xu and Ph Smets. Evidential Reasoning with Con- son entering a medical clinic, the physician would a pri- ditional Belief Functions. In D. Heckerman et al., ed- ori be ignorant about whether that person is infected or itors, Proceedings of Uncertainty in Artificial Intelli- not, which intuitively should be expressed as a vacuous be- gence (UAI94), pages 598Ð606. Morgan Kaufmann, lief function, i.e. with the total belief mass assigned to San Mateo, California, 1994. (“infected” ∪ “not infected”). The probability projection of a vacuous belief function using the Pignistic probability [7] J.Y. Halpern. Conditional Plausibility Measures and projection would dictate the a priori probability of infec- Bayesian Networks. Journal of Artificial Intelligence tion to be 0.5. Of course, the base rate is normally much Research, 14:359Ð389, 2001. lower, and can be determined by relevant statistics from a [8] P. Walley. Statistical Reasoning with Imprecise Prob- given population. Traditional belief functions are not well- abilities. Chapman and Hall, London, 1991. suited for representing this situation. Using only traditional belief functions, the base rate of a disease would have to be [9] F.G. Cozman. Credal networks. Artif. Intell., expressed through a bba that assigns some belief mass to 120(2):199Ð233, 2000. either “infected” or “not infected” or both. Then after as- sessing the results of e.g. a medical test, the bba would have [10] T. Lukasiewicz. Weak nonmonotonic probabilistic log- to be conditionally updated to reflect the test evidence in or- ics. Artif. Intell., 168(1):119Ð161, 2005. der to derive the a posteriori bba. Unfortunately, no compu- [11] P. Walley. Inferences from Multinomial Data: Learn- tational method for conditional updating of traditional bbas ing about a Bag of Marbles. Journal of the Royal Sta- according to this principle exists. The methods that have tistical Society, 58(1):3Ð57, 1996. been proposed, e.g. [5], have been shown to be flawed [13] because they are subject to the base rate fallacy [2]. [12] A. J¿sang. A Logic for Uncertain Probabilities. Subjective logic is the only belief reasoning framework International Journal of Uncertainty, Fuzziness and which is not susceptible to the base rate fallacy in condi- Knowledge-Based Systems, 9(3):279Ð311, June 2001. tional reasoning. Incorporating base rates for belief func- tions represents a necessary prerequisite for conditional be- [13] A. J¿sang. Conditional Reasoning with Subjective lief reasoning, in particular for abductive belief reasoning. Logic. Journal of Multiple-Valued Logic and Soft In order to be able to handle conditional reasoning within Computing, 15(1):5Ð38, 2008. general Dempster-Shafer belief theory, it has been proposed [14] Najla Megherbi, Sebatsien Ambellouis, Olivier Colˆot, to augment traditional bbas with a base rate function [17]. and Franc¸ois Cabestaing. Multimodal data association We consider this to be a good idea and propose as a research based on the use of belief functions for multiple target agenda the definition of conditional belief reasoning in gen- tracking. In Proceedings of the 8th International Con- eral Dempster-Shafer belief theory. ference on Information Fusion (Fusion 2005), 2005. References [15] Praveen Kumar, Ankush Mittal, and Padam Kumar. Study of Robust and Intelligent Surveillance in Visi- [1] A. J¿sang, S. Pope, and M. Daniel. Conditional deduc- ble and Multimodal Framework. Informatica, 31:447Ð tion under uncertainty. In Proceedings of the 8th Eu- 461, 2007. ropean Conference on Symbolic and Quantitative Ap- proaches to Reasoning with Uncertainty (ECSQARU [16] Intergovernmental Panel on Climate Change. Climate 2005), 2005. Change 2007: The Physical Science Basis. IPCC Sec- retariat, Geneva, Switzerland, 2 February 2007. url: [2] Jonathan Koehler. The Base Rate Fallacy Recon- http://www.ipcc.ch/. sidered: Descriptive, Normative and Methodological Challenges. Behavioral and Brain Sciences, 19, 1996. [17] Audun J¿sang, Stephen O’Hara, and Kristie O’Grady. Base Rates for Belief Functions. In Proceedings of the [3] Frank J. Stech and Christopher El¬asser. Midway revis- Workshop on the Theory on Belief Functions(WTBF ited: Deception by analysis of competing hypothesis. 2010), Brest, April 2010. Technical report, MITRE Corporation, 2004.