Here’s an example given in section 1.1 of Mathematical Theory of Hints by Jurg Kohlas and Paul-Andre Monney. Suppose we have implication \(a_1 \vee a_2 \implies b\) while \(a_1\), \(a_2\) are not known to be true for certain. Let \(p_1 = 0.3\) be the probability that \(a_1\) is true and \(p_2 = 0.4\) the probability that \(a_2\) is true. This is an example of combining “pure arguments” by Jacob Bernoulli in Ars Conjectandi.
First, we use function bcaRel to define the implication relation in its disjunctive form \(b \vee (\neg a_1 \land \neg a_2)\). The required binary table can also be obtained from https://web.stanford.edu/class/cs103/tools/truth-table-tool/.
tt <- matrix(c(0,1,0,1,0,1,
1,0,1,0,1,0,
1,0,1,0,0,1,
0,1,1,0,0,1,
1,0,0,1,0,1,
1,1,1,1,1,1), nrow = 2 + 3 + 1, ncol = 6, byrow = TRUE, dimnames = list(NULL,c("a1 no", "a1 yes", "a2 no", "a2 yes", "b no", "b yes")))
spec <- matrix(c(1,1,1,1,1,2,1,1,1,1,1,0), nrow = 5 + 1, ncol = 2)
infovar <- matrix(c(1,2,3,2,2,2), nrow = 3, ncol = 2)
varnames <- c("a1","a2","b")
bcaRel1<-bcaRel(tt,spec,infovar,varnames)
cat("The implication relation","\n")## The implication relation## bcaRel1
## 1 a1 no a2 no b no + a1 no a2 no b yes + a1 no a2 yes b yes + a1 yes a2 no b yes + a1 yes a2 yes b yes
## specnb mass
## 1 1 1Reliability_Proof_Machinery.R Second, we use function bca to define the probabilities such that each of the assumptions are true. For \(a_1\) is true, probability 0.3 is given to “a1 is true” and 0.7 given to the whole first frame.
tt <- matrix(c(0,1,1,1), nrow = 2, ncol = 2, dimnames = list(NULL, c("a1 no", "a1 yes")))
m <- c(0.3,0.7)
varnames <- "a1"
idvar <- 1
bca1 <- bca(tt, m, idvar=idvar, varnames=varnames)
bcaPrint(bca1)## bca1 specnb mass
## 1 a1 yes 1 0.3
## 2 frame 2 0.7Reliability_Proof_Machinery.R For \(a_2\) is true, probability 0.4 is given to “a2 is true” and 0.6 given to the whole second frame.
tt <- matrix(c(0,1,1,1), nrow = 2, ncol = 2, dimnames = list(NULL, c("a2 no", "a2 yes")))
m <- c(0.4,0.6)
varnames <- "a2"
idvar <- 2
bca2 <- bca(tt, m, idvar=idvar, varnames=varnames)
bcaPrint(bca2)## bca2 specnb mass
## 1 a2 yes 1 0.4
## 2 frame 2 0.6Reliability_Proof_Machinery.R Now we combine the two bca’s. To do that we need to first extend the two bca’s are they’re defined on the marginal frames. Using function extmin, bca1 can be extended to the whole frame of the product space of the three variables (a1, a2, b) as:
## bca1_extmin
## 1 a1 yes a2 no b no + a1 yes a2 no b yes + a1 yes a2 yes b no + a1 yes a2 yes b yes
## 2 frame
## specnb mass
## 1 1 0.3
## 2 2 0.7Reliability_Proof_Machinery.R Likewise, bca2 can be extended to the whole frame of the product space as:
## bca2_extmin
## 1 a1 no a2 yes b no + a1 no a2 yes b yes + a1 yes a2 yes b no + a1 yes a2 yes b yes
## 2 frame
## specnb mass
## 1 1 0.4
## 2 2 0.6Reliability_Proof_Machinery.R Having extended the marginal bca to the whole frame, we can use function dsrwon to perform Dempster’s rule of combination them.
## bca12_extmin
## 1 a1 yes a2 yes b no + a1 yes a2 yes b yes
## 2 a1 yes a2 no b no + a1 yes a2 no b yes + a1 yes a2 yes b no + a1 yes a2 yes b yes
## 3 a1 no a2 yes b no + a1 no a2 yes b yes + a1 yes a2 yes b no + a1 yes a2 yes b yes
## 4 frame
## specnb mass
## 1 1 0.12
## 2 2 0.18
## 3 3 0.28
## 4 4 0.42Reliability_Proof_Machinery.R
Remember that at the beginning, we defined a relation \(a_1 \vee a_2 \implies b\). This relation must now be combined with the combined bca’s to yield the final bca in the product space of (a1, a2, b).
## bca12_extmin_dsrwon_bcaRel1
## 1 a1 yes a2 yes b yes
## 2 a1 yes a2 no b yes + a1 yes a2 yes b yes
## 3 a1 no a2 yes b yes + a1 yes a2 yes b yes
## 4 a1 no a2 no b no + a1 no a2 no b yes + a1 no a2 yes b yes + a1 yes a2 no b yes + a1 yes a2 yes b yes
## specnb mass
## 1 1 0.12
## 2 2 0.18
## 3 3 0.28
## 4 7 0.42Reliability_Proof_Machinery.R
Now we can get the marginal bca of variable b. To do so, we need to summarize the other variables on this dimension. We do so by eliminating (deleting) the other dimensions than b, that is a1 and a2. We choose to eliminate dimension 1 (a1) first, using function elim.
## bca12_extmin_elim1 specnb mass
## 1 a2 yes b yes 1 0.4
## 2 a2 no b yes + a2 yes b yes 2 0.18
## 3 a2 no b no + a2 no b yes + a2 yes b yes 5 0.42Reliability_Proof_Machinery.R Likewise, we eliminate dimension 2.
## bca12_extmin_elim12 specnb mass
## 1 b yes 1 0.58
## 2 frame 2 0.42Reliability_Proof_Machinery.R Having obtained the marginal bca of variable b, we can now evaluate belief and plausibility, using function belplau.
## bel disbel unc plau rplau
## b yes 0.58 0 0.42 1 2.380952
## frame 1.00 0 0.00 1 InfReliability_Proof_Machinery.R Note the result: bel(yes) = 0.58;
which is the result one will obtain by applying the combination rule developed by Bernoulli:
\[bel(b) = 1 - (1-p1) \cdot (1-p2)\] \[= 1 - (1-0.3 \cdot (1-0.4) = 1 - 0.42 = 0.58.\] Alternatively, instead of using the OR gate, one can build up the graph by defining the two implications separately. First, we define the first implication.
tt <- matrix(c(0,1,0,1,
1,0,0,1,
1,0,1,0,
1,1,1,1), nrow = 4, ncol = 4, byrow = TRUE, dimnames = list(NULL,c("a1 no", "a1 yes", "b no", "b yes")))
spec <- matrix(c(1,1,1,2,1,1,1,0), nrow = 4, ncol = 2)
infovar <- matrix(c(1,3,2,2), nrow = 2, ncol = 2)
varnames <- c("a1","b")
bcaRel1<-bcaRel(tt,spec,infovar,varnames)
bcaPrint(bcaRel1)## bcaRel1 specnb mass
## 1 a1 no b no + a1 no b yes + a1 yes b yes 1 1Reliability_Proof_Machinery.R
Similarly, we can define the second implication as follows.
tt <- matrix(c(0,1,0,1,
1,0,0,1,
1,0,1,0,
1,1,1,1), nrow = 4, ncol = 4, byrow = TRUE, dimnames = list(NULL,c("a2 no", "a2 yes", "b no", "b yes")))
spec <- matrix(c(1,1,1,2,1,1,1,0), nrow = 4, ncol = 2)
infovar <- matrix(c(2,3,2,2), nrow = 2, ncol = 2)
varnames <- c("a2","b")
bcaRel2<-bcaRel(tt,spec,infovar,varnames)
bcaPrint(bcaRel2)## bcaRel2 specnb mass
## 1 a2 no b no + a2 no b yes + a2 yes b yes 1 1Reliability_Proof_Machinery.R
Then we extend, combine, and eliminate variables. For the first variable and the first implication, we obtain:
bca1_extmin <- extmin(bca1,bcaRel1)
bca1_extmin_bcaRel1_dsrwon <- dsrwon(bca1_extmin, bcaRel1)
bca1_extmin_bcaRel1_dsrwon_elim <- elim(bca1_extmin_bcaRel1_dsrwon, 1)
bcaPrint(bca1_extmin_bcaRel1_dsrwon_elim)## bca1_extmin_bcaRel1_dsrwon_elim specnb mass
## 1 b yes 1 0.3
## 2 frame 2 0.7Reliability_Proof_Machinery.R Similarly for the second variable and the second implication, we obtain:
bca2_extmin <- extmin(bca2,bcaRel2)
bca2_extmin_bcaRel2_dsrwon <- dsrwon(bca2_extmin, bcaRel2)
bca2_extmin_bcaRel2_dsrwon_elim <- elim(bca2_extmin_bcaRel2_dsrwon, 2)
bcaPrint(bca2_extmin_bcaRel2_dsrwon_elim)## bca2_extmin_bcaRel2_dsrwon_elim specnb mass
## 1 b yes 1 0.4
## 2 frame 2 0.6Reliability_Proof_Machinery.R
Now, we combine the two results
## bca12 specnb mass
## 1 b yes 1 0.58
## 2 frame 2 0.42Reliability_Proof_Machinery.R
Next, evaluate belief and plausibility.
## bel disbel unc plau rplau
## b yes 0.58 0 0.42 1 2.380952
## frame 1.00 0 0.00 1 InfReliability_Proof_Machinery.R