Use Case Titanic
In the following example we will learn the structure on the Titanic dataset.
import bnlearn as bn
# Load example mixed dataset
df = bn.import_example(data='titanic')
# Convert to onehot
dfhot, dfnum = bn.df2onehot(df)
# Structure learning
# model = bn.structure_learning.fit(dfnum, methodtype='cl', black_list=['Embarked','Parch','Name'], root_node='Survived', bw_list_method='nodes')
model = bn.structure_learning.fit(dfnum)
# Plot
G = bn.plot(model, interactive=False)
# Compute edge strength with the chi_square test statistic
model = bn.independence_test(model, dfnum, test='chi_square', prune=True)
# Plot
bn.plot(model, interactive=False, pos=G['pos'])
# Parameter learning
model = bn.parameter_learning.fit(model, dfnum)
|
|
At this point we can also start making inferences:
# Make inference
query = bn.inference.fit(model, variables=['Survived'], evidence={'Sex':True, 'Pclass':True})
print(query)
# +----+------------+----------+
# | | Survived | p |
# +====+============+==========+
# | 0 | 0 | 0.555427 |
# +----+------------+----------+
# | 1 | 1 | 0.444573 |
# +----+------------+----------+
#
# +-------------+-----------------+
# | Survived | phi(Survived) |
# +=============+=================+
# | Survived(0) | 0.5554 |
# +-------------+-----------------+
# | Survived(1) | 0.4446 |
# +-------------+-----------------+
print(query.df)
# Survived p
# 0 0.555427
# 1 0.444573
# Another inference using only sex for evidence
query = bn.inference.fit(model, variables=['Survived'], evidence={'Sex':0})
print(query)
# +----+------------+----------+
# | | Survived | p |
# +====+============+==========+
# | 0 | 0 | 0.406634 |
# +----+------------+----------+
# | 1 | 1 | 0.593366 |
# +----+------------+----------+
print(query.df)
# Print model
CPDs = bn.print_CPD(model)
# All CPDs are now stored in the dict CPD which contain the CPD for each node.
print(CPDs.keys())
# dict_keys(['Pclass', 'Survived', 'Embarked', 'Sex', 'SibSp', 'Parch'])
CPDs['Survived']
# Survived Pclass Sex p
# 0 0 0 0 0.331202
# 1 0 0 1 0.555427
# 2 0 1 0 0.368132
# 3 0 1 1 0.634709
# 4 0 2 0 0.500000
# 5 0 2 1 0.746269
# 6 1 0 0 0.668798
# 7 1 0 1 0.444573
# 8 1 1 0 0.631868
# 9 1 1 1 0.365291
# 10 1 2 0 0.500000
# 11 1 2 1 0.253731
Use Case Medical domain
In this section I will describe the use-case to analyse patients treatment regarding shortness-of-breath (dyspnoea). In this context you may readily know some associatons from literature and/or experience, like smoking is related to dyspnoea. In this use-case I will demonstrate how to use your expert-knowledge in a bayesian model. Furthermore, the data set is small (few variables) and synthetic from Lauritzen and Spiegelhalter (1988), and is about lung diseases (tuberculosis, lung cancer or bronchitis) and visits to Asia.
Description
- Motivation
“Shortness-of-breath (dyspnoea) may be due to tuberculosis, lung cancer or bronchitis, or none of them, or more than one of them. A recent visit to Asia increases the chances of tuberculosis, while smoking is known to be a risk factor for both lung cancer and bronchitis. The results of a single chest X-ray do not discriminate between lung cancer and tuberculosis, as neither does the presence or absence of dyspnoea.”
- Source
Lauritzen S, Spiegelhalter D (1988). Local Computation with Probabilities on Graphical Structures and their Application to Expert Systems (with discussion). Journal of the Royal Statistical Society
Import data
The first step is to import the data set. If you have unstructured data, use the df2onehot functionality bnlearn.bnlearn.df2onehot(). The Examples section contains examples how to import a raw data set followed by (basic) structering approaches (section: Start with RAW data). In my case I will load the data from bnlearn, which is readily a structured dataset.
import bnlearn as bn
# Load dataset with 10.000 samples
df = bn.import_example('asia', n=10000)
# Print to screen
print(df)
smoke |
bronc |
lung |
asia |
tub |
either |
dysp |
xray |
|
|---|---|---|---|---|---|---|---|---|
0 |
0 |
1 |
1 |
1 |
1 |
1 |
0 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
0 |
2 |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
1 |
… |
… |
… |
… |
… |
… |
… |
… |
… |
9999 |
0 |
1 |
1 |
1 |
1 |
1 |
0 |
1 |
This data set contains 8 variables with discrete values, meaning that the variables have the state yes/no, true/false or 1/0 values. bnlearn can handle multiple catagories (also non-numerical, Start with RAW data). In this example we generate 10.000 samples (representing the patients). Note that the number of variables depends on the complexity of the data set (number of variables and the catagories per variable). If you want to get feeling of the performance of bayesian models, I would advice to play arround with various example data sets in bnlearn and determine when you can re-construct the entire DAG given the complexity of the data set. As an example, 1000 samples is sufficient for the sprinkler data set because there are only 4 variables, each with state yes/no. Some other data sets (such as alarm) are way more complicated and 1000 samples would not be sufficient.
Make inferences when you have data and know-how
Expert knowledge can be included in bayasian models by using graphs in the form of a Directed Acyclic Graphs (DAG, Directed Acyclic Graphs). The DAG describes the relationships between variables. Lets create a custom DAG, and make inferences Inference.
- Aim: Make inferences about shortness-of-breath (dyspnoea) when:
You have measured data and imported: Import data.
You have know-how/expert knowledge.
Create a custom Directed Acyclic Graph
My knowledge about dyspnoea is limited to: smoking is related to lung cancer, smoking is related to bronchitis, and if you have lung or bronchitus you may need an xray examination. Basically, I will create a simple DAG. Note that bayesian modeling is especially fun because you can make very complex DAGs. Note that the direction is very important. The first column is “from” or “source” and the second column “to” or “destination”. Note, this is a very simple model that is designed for demonstration purposes only.
edges = [('smoke', 'lung'),
('smoke', 'bronc'),
('lung', 'xray'),
('bronc', 'xray')]
Plot the Bayesian DAG.
# Create the DAG from the edges
DAG = bn.make_DAG(edges)
# Plot and make sure the arrows are correct.
bn.plot(DAG)
Compute Conditional Probability Distributions (CPDs)
At this point we have the data set in our dataframe (df), and we have the DAG based on your expert knowledge. The next step is to connect your brains (DAG) to the data set. We can do this with the function bnlearn.bnlearn.parameter_learning.fit() which will compute the CPDs. See section Parameter learning to learn more about conditional probability distributions (CPDs) and how parameters can be learned. In general; it is the task to estimate the values of the CPDs in the DAG based on the input data set. How cool is that!
Parameter learning on the expert-DAG using the input data set.
# Check the current CPDs in the DAG.
CPDs = bn.print_CPD(DAG)
# [bnlearn] >No CPDs to print. Tip: use bn.plot(DAG) to make a plot.
# This is correct, we dit not yet specify any CPD.
# Learn the parameters from data set.
# As input we have the DAG without CPDs.
DAG = bn.parameter_learning.fit(DAG, df, methodtype='bayes')
# Print the CPDs
CPDs = bn.print_CPD(DAG)
# At this point we have a DAG with the learned CPDs
The learned Conditional Probability Distributions are depicted in the tables below. As an example, the probability that a patient does not smoke is P(smoke=0)=0.49 whereas the probability of a patient smoking is P(smoke=1)=0.5.
CPD of smoke:
smoke(0) |
0.495273 |
smoke(1) |
0.504727 |
Slightly more complicated are the patients that smoke and have lung-cancer which is basically the intersection. The more edges towards a node the more complicated the CPD becomes. Luckily we have bnlearn to do the heavy lifting!
CPD of lung:
smoke |
smoke(0) |
smoke(1) |
lung(0) |
0.13913362701908957 |
0.05457492795389049 |
lung(1) |
0.8608663729809104 |
0.9454250720461095 |
CPD of bronc:
smoke |
smoke(0) |
smoke(1) |
bronc(0) |
0.5936123348017621 |
0.3114193083573487 |
bronc(1) |
0.4063876651982379 |
0.6885806916426513 |
CPD of xray:
bronc |
bronc(0) |
bronc(0) |
bronc(1) |
bronc(1) |
lung |
lung(0) |
lung(1) |
lung(0) |
lung(1) |
xray(0) |
0.7651245551601423 |
0.08089070665757782 |
0.7334669338677354 |
0.08396533044420368 |
xray(1) |
0.23487544483985764 |
0.9191092933424222 |
0.2665330661322645 |
0.9160346695557963 |
Make inferences
When you are at this part, you combined your expert knowledge with a data set! Now we can make inferences which allows to ask questions to the model. Let me demonstrate a few questions.
Question 1
What is the probability of lung-cancer, given that we know that patient does smoke? The model returns that the probability of lung-cancer or lung(1) is 0.94 when the patient does smoke; P(lung=1 | smoke=1)=0.94.
q1 = bn.inference.fit(DAG, variables=['lung'], evidence={'smoke':1})
print(q1.df)
# Finding Elimination Order: : 100% 2/2 [00:00<00:00, 401.14it/s]
# Eliminating: bronc: 100%| 2/2 [00:00<00:00, 200.50it/s]
# [bnlearn] >Variable Elimination..
lung |
phi(lung) |
|---|---|
lung(0) |
0.0546 |
lung(1) |
0.9454 |
Question 2
What is the probability of bronchitis, given that we know that patient does smoke? The model returns that the probability of bronchitis or bronc(1) is 0.68 when the patient does smoke; P(bronc=1 | smoke=1)=0.68.
q2 = bn.inference.fit(DAG, variables=['bronc'], evidence={'smoke':1})
# Finding Elimination Order: : 100% 2/2 [00:00<00:00, 286.31it/s]
# Eliminating: lung: 100% 2/2 [00:00<00:00, 143.26it/s]
# [bnlearn] >Variable Elimination..
bronc |
phi(bronc) |
|---|---|
bronc(0) |
0.3114 |
bronc(1) |
0.6886 |
Question 3
Lets add more information to our inference. What is the probability of lung-cancer, given that we know that patient does smoke and also has bronchitis?
q3 = bn.inference.fit(DAG, variables=['lung'], evidence={'smoke':1, 'bronc':1})
# Finding Elimination Order: : 100% 1/1 [00:00<00:00, 334.31it/s]
# Eliminating: xray: 100% 1/1 [00:00<00:00, 338.47it/s]
# [bnlearn] >Variable Elimination..
lung |
phi(lung) |
|---|---|
lung(0) |
0.0546 |
lung(1) |
0.9454 |
Question 4
Lets specify the question even more. What is the probability of lung-cancer or bronchitis, given that we know that patient does smoke but did not had xray?
q4 = bn.inference.fit(DAG, variables=['bronc','lung'], evidence={'smoke':1, 'xray':0})
lung |
bronc |
phi(lung,bronc) |
|---|---|---|
lung(0) |
bronc(0) |
0.1092 |
lung(0) |
bronc(1) |
0.2315 |
lung(1) |
bronc(0) |
0.2001 |
lung(1) |
bronc(1) |
0.4592 |
The highest probability for the patient under these condition is that lung-cancer is true and bronchitus is true too (P=0.45). Note that, if you put xray=1, then the probability becomes even higher (P=0.67).
Determine causalities when you have data
Suppose that we have the medical records of hundreds or even thousands patients treatment regarding shortness-of-breath (dyspnoea). Our goal is to determine the causality across variables given the data set.
- Steps to take
Import the data set.
Compute Directed Acyclic Graph by means of structure learning.
Compare to DAG to that of the expert-DAG.
Compute Directed Acyclic Graph from data
Import and process teh data set (Import data). For this use-case we will compute the best performing DAG given the data set. You only need to provide the data set into bnlearn bnlearn.bnlearn.structure_learning.fit(). More about Directed Acyclic Graphs can be found in the section Directed Acyclic Graphs.
# Structure learning on the data set
model = bn.structure_learning.fit(df)
# [bnlearn] >Computing best DAG using [hc]
# [bnlearn] >Set scoring type at [bic]
# Compute significance
model = bn.independence_test(model, df, prune=True)
# [bnlearn] >Edge [lung <-> tub] [P=0.540506] is excluded because it was not significant (P<0.05) with [chi_square]
The computations can take seconds to days or even never-ending, depending on the complexity of your data set and the method in bnlearn you choose. This use-case contains only 8 variables, each with two states and will be computed within seconds. If your data set is huge, and readily have suspicion you can use the black_list or white_list parameters (Black and white lists).
Lets plot the learned DAG and examine the structure!
# Plot the DAG
bn.plot(model, interactive=False)
bn.plot(model, interactive=True)
# Plot differences between expert-DAG and the computed-DAG
bn.compare_networks(model, DAG)
A comparison with our initial expert-DAG shows few differences in red. As an example, we did not include the either variable, which describes either being lung-cancer or bronchitus.
Make inference when you have data
In this scenario we the goal is to make inferences across variables given the data set.
- Steps to take
Import the data set
Compute Directed Acyclic Graph (DAG)
Compute Conditional Probability Distributions (CPDs)
The first step is to import and pre-process the data set as depicted in Import data. Then we compute the DAG by means of structure learning as depicted in Compute Directed Acyclic Graph from data. To make inferences, we first need to compute the CPDs which we can do with bnlearn.bnlearn.parameter_learning.fit().
# Learning the CPDs using parameter learning
model = bn.parameter_learning.fit(model, df, methodtype='bayes')
# Print the CPDs
CPDs = bn.print_CPD(model)
CPD of smoke:
smoke(0) |
0.495455 |
smoke(1) |
0.504545 |
CPD of bronc:
smoke |
smoke(0) |
smoke(1) |
bronc(0) |
0.6009174311926605 |
0.31675675675675674 |
bronc(1) |
0.39908256880733944 |
0.6832432432432433 |
CPD of lung:
smoke |
smoke(0) |
smoke(1) |
lung(0) |
0.138348623853211 |
0.05333333333333334 |
lung(1) |
0.861651376146789 |
0.9466666666666667 |
CPD of dysp:
bronc |
bronc(0) |
bronc(0) |
bronc(1) |
bronc(1) |
either |
either(0) |
either(1) |
either(0) |
either(1) |
dysp(0) |
0.7508090614886731 |
0.7821064552661382 |
0.6189591078066915 |
0.12156934978817462 |
dysp(1) |
0.24919093851132687 |
0.21789354473386183 |
0.38104089219330856 |
0.8784306502118254 |
CPD of either:
lung |
lung(0) |
lung(0) |
lung(1) |
lung(1) |
tub |
tub(0) |
tub(1) |
tub(0) |
tub(1) |
either(0) |
0.5098039215686274 |
0.8427672955974843 |
0.648876404494382 |
0.01302897644361059 |
either(1) |
0.49019607843137253 |
0.15723270440251572 |
0.351123595505618 |
0.9869710235563894 |
CPD of tub:
tub(0) |
0.0555455 |
tub(1) |
0.944455 |
CPD of xray:
either |
either(0) |
either(1) |
xray(0) |
0.7716262975778547 |
0.0750711093051605 |
xray(1) |
0.22837370242214533 |
0.9249288906948395 |
From this point on we can start making inferences given the DAG and the CPDs. For demonstration purposes I will repeat question 4.
Question
What is the probability of lung-cancer or bronchitis, given that we know that patient does smoke but did not had xray?
q = bn.inference.fit(DAG, variables=['bronc','lung'], evidence={'smoke':1, 'xray':0})
lung |
bronc |
phi(lung,bronc) |
|---|---|---|
lung(0) |
bronc(0) |
0.0797 |
lung(0) |
bronc(1) |
0.1720 |
lung(1) |
bronc(0) |
0.2370 |
lung(1) |
bronc(1) |
0.5113 |
The highest probability for the patient under these condition is that lung-cancer is true and bronchitus is true too (P=0.51).