6CCS3AIN, Tutorial 03 (Version 1.0)
Several of these questions make use of the Bayesian network in Figure 1.
1. Write down a Bayesian network that captures the knowledge that (a) Smoking causes cancer; and (b) Smoking causes bad breath. Given the information that:
P (cancer|smoking) = 0.6 P (badBreath|smoking) = 0.95
and P (smoking) = 0.2 use a Naive Bayes model to determine P (smoking, cancer, badBreath).
Assume that all three variables are binary: Smoking has values smoking and ¬smoking, Cancer has
values cancer and ¬cancer, and BadBreath has values badBreath and ¬badBreath.
2. Write down a Bayesian network that captures the knowledge that (a) a Late Start can cause a student to Fail their Project, and (b) Ignoring Advice can cause a student to Fail their Project. Use the binary variables LateStart, with values l and ¬l, IgnoreAdvice, with values i and ¬i, and FailProject, with values f and ¬f.
3. How many numbers do we need to specify all the necessary probability values for the network in Figure 1? How many would we need if there were no conditional independencies (if we didn’t have the network)?
4. Compute the joint probability P (¬m, ¬t, ¬h, ¬s, ¬c)
5. Use the enumeration algorithm to compute the probability of P (m|h, s).
6. Use prior sampling to compute the joint probability over m, t, h, s and c.
Use the sequence of random numbers in Table 1. Give the results of the first 5 samples only.
7. Use rejection sampling to compute P (m|h, s)
Use the sequence of random numbers in Table 1, starting at the beginning of the sequence. This time you
should show the results of the first 5 samples that aren’t rejected.
If you need more random numbers than exist in Table 1, start that sequence again from the beginning.
8. Use likelihood sampling to compute P(T |h, s).
Use the sequence of random numbers in Table 1, starting at the beginning of the sequence, and again report
the results of the first five samples.
9. For the optional computational part of the tutorial, download the file wetGrass.py from KEATS.
This provides code, using pomegranate1, that implements the “Wet Grass” example from Lecture/Week 3. You can run the example using:
python wetGrass.py
That should give you this as output:
Evidence is: Grass is wet.
Here’s the prediction:
Cloudy, Sprinkler Off, Rain, Wet grass.
And by the numbers:
Cloudy: ((’c’, 0.6010250059300832), (’l’, 0.3989749940699168))
Sprinkler: ((’f’, 0.5976922222293125), (’n’, 0.40230777777068755))
Rain: ((’r’, 0.7772459441092383), (’n’, 0.2227540558907617))
1For details on pomegranate see Tutorial 2.
1
run 1 run 2 run 3 run 4 run 5
0.14 0.57 0.50 0.12 0.49 0.67 0.21 0.34 0.99 0.99
0.01 0.43 0.59 0.54 0.97 0.51 0.96 0.55 0.89 0.54 0.97 0.51 0.99 0.99 0.99
Table 1: Some random numbers between 0 and 1.
The first part of wetGrass.py sets up the Bayesian network. This is a straightforward extension of what we had for underwear.py.
The second part then runs a query on the model. The variable scenario: scenario = [[None, None, None, ’w’]]
specifies the observed values (the evidence), if any, of the variables in the model — the order of the variable is the order specified in the model.add_states() command. Here we are setting the variable WetGrass to value w, denoting that the grass is wet, and not saying anything about the values of Cloudy, Sprinkler and Rain. This:
Evidence is: Grass is wet.
reflects the input back — code does this in a somewhat general way2.
The code uses two methods to do inference given the evidence. First model.predict() identifies the most
likely (highest probability) value of each variable in the model:
Here’s the prediction:
Cloudy, Sprinkler Off, Rain, Wet grass.
and then model.predict_proba() is used to generate the probabilities of all the values of the non-evidence variables:
And by the numbers:
Cloudy: ((’c’, 0.6010250059300832), (’l’, 0.3989749940699168))
Sprinkler: ((’f’, 0.5976922222293125), (’n’, 0.40230777777068755))
Rain: ((’r’, 0.7772459441092383), (’n’, 0.2227540558907617))
It turns out that model.predict_proba() handles evidence variables differently to non-evidence variables, and so the code doesn’t output any information about evidence variables (we already know the value taken by evidence variables after all).
The final part of the code does some minimal formatting on what is generated by model.predict() and model.predict_proba() to produce the output above.
(a) Use wetGrass.py to calculate the probability of the variables in the model if:
i. The grass is wet and the sprinkler is on. ii. The sprinkler is on and the sky is cloudy.
iii. It is raining.
(b) Use pomegranate to build the model in Figure 1. Use this model to calculate P(m|h,s).
2The documentation for pomegranate is not great, but from what I can tell, there is no function that gives access to the states in a model once the model has been built, so we can’t retrieve the variables, and their names and values in a general way. (We could build our own structures to keep track of this, of course). Hence the way that they are hard-coded.
2
3
altered serum calcium
S T P(C|S,T) T T 0.95
T F 0.85
T P (H | T )
(S)
headaches
(H) FT0.85 F0.7
coma (C)
metastatic cancer (M)
brain tumor (T)
M P(T|M)
Figure 1: The example Bayesian network.
P(M) 0.1
M P(S|M)
T 0.8 T 0.7 F 0.2 F 0.1
FF 0.01
T 0.9