# Solutions 4

### Exercise 2.5:

See the Hugin network.

### Exercise 2.22:

#### Part i:

When inserting and propagating the three pieces of evidence, Hugin responds with the error message: "Propagation of inconsistent evidence has been attempted, or underflow has occurred".

#### Part ii:

After modifying the potentials (see the resulting Hugin network) we get the following probability distributions:
P(Cold?|e) = (0.03(y),0.97(n))
P(Angina?|Cold?=n,e) = (0,0.97,0.03)
P(Angina?|Cold?=y,e) = (0,1,0)

By multiplying the latter two probability distributions with P(Cold?|e) we get P(Angina?,Cold?|e):
no mild severe 0 0.03 0 0 0.94 0.03

#### Part iii:

By using max-propagation in Hugin (or by max-marginalizing out the variables from e.g. the table above) we get: P(Cold?|e)=(0.03,0.94) and P(Angina?|e)=(0,0.94,0.03). This shows that (Cold?=no,Angina?=mild) is the most probable posterior configuration.

#### Part iv:

Conf(e)=log2((0.978*0.002*0.965)/7.513*10-7)=11.3 which indicates a conflict.

#### Part v:

If P(Angina?=mild,e)=x(s)=a*s+b and P(e)=y(s)=c*s+d, then P(Angina=mild|e)=x(s)/y(s).

With s=0.01 we have P(e)=7.513*10-7 we get c*0.01+d=7.513*10-7. By also entering Angina?=mild we get a*0.01+b=7.298*10-7.

Next we change s to 0.02 which gives us: c*0.02+d=1.48*10-6 and a*0.02+b=1.46*10-6. We can now solve the two pairs of linear equations, and we get: a=7.3*10-5, b=2.11*10-8, c= 7.287*10-5 and d=2.3*10-8. Hence, P(Angina?=mild|e)=(7.3*s+0.00211)/(7.28*s+0.0023).