1. Decision Analysis Lecture 12
Tony Cox My
Course web site:

2. Agenda Correction to notes for Lecture 9, page 36
Problem set 10 solution
Recommended readings
Causal analytics

3. Result is a posterior beta distribution with updated n = 21 and x = 6
Bayesian estimation of proportion: Example with n = 20 trials, x = 5 successes
Choose uniform prior
Initial n = 1, x = 1, pbeta(q, x, n-x)
w = c(0:100)/100
y = dbeta(w, 1, 1)
Result is a posterior beta distribution with updated n = 21 and x = 6
z = beta(x + 1, n - x + 1) = dbeta(w, 6, 16)
Posterior mean = (x + 1)/(n + 2) = 6/22 = 0.27

4. Homework #10 (optional) (Due by 4:00 PM, April 18 if you want it graded)
A deep sea oil drilling platform has a normally distributed lifetime (until failure) with a mean of 30 years and a standard deviation of 4 years.
While it is operating, the platform produces oil worth $10M per year.
Voluntarily stopping operations and closing down the platform costs $0.
Having the platform fail while still in use leads to involuntary closure and a cost of $50M.
At what age should the platform be voluntarily shut down (if it has not yet failed)?
Hint: Continue until marginal benefit < expected marginal cost of continuing
Hint: Use a hazard function calculator for normally distributed lifetimes, e.g.,

5. Oil platform solution
At time t, expected marginal cost from continuing for an additional interval of length dt years is 50*h(t)dt million.
Expected marginal benefit of operating for another dt years is 10*dt million dollars.
Marginal benefit = marginal cost when 50*h(t) = 10, or h(t) = 0.2.
For T ~ normal with mean 30 years, s.d. 4 years, h(30) = 0.2 (using calculator at
In R: h(t) = f(t)/(1 – F(t)), dnorm(30, 30, 4)/(1-pnorm(30, 30, 4)) = 0.2
So, voluntarily shut down platform at age 30 years

6. Causal analytics library
BayesianNetworksWithoutTearsCharniak1991.pdf
OverviewPearl2009.pdf
CausalDiscoverySystemsBiologyLagani2016.pdf
CausalGraphicalModelsElwert2013
CausalOrderingDashDruzdzel2008.pdf
CausalParametersHeckman1999
DoesXcauseY_Mooij2016.pdf
GrangerCausalityEconomicPolicyWhite2010.pdf
HealthPolicyAnalysisTerza2011.pdf
LiNGAMShimuzi2014.pdf

7. Recommended readings
Statistics and causality, Pearl (2009)
Graphical causal models (Elwert, 2013)
Probabilistic computational causal discovery for systems biology (Lagani et al., 2016)

8. Causal Analytics: Agenda
Technical background
Probabilistic causation
Netica software and Bayesian network (BN) technology
CAT software and information-based algorithms: CART trees, random forests, partial dependence plots, BN learning
Causal analytics for risk management: Vision
Learning causal DAG models from data: Information-based algorithms for causal discovery and inference
Making the algorithms useful: R packages and the Causal Analytics Toolkit (CAT), dagitty software
Applications to causal inference, prediction, attribution, optimization, explanation
Summary, conclusions, and perspectives on causal analytics for risk analysis

9. Causal analytics in context
Analytics Goal: Use data to learn to act more effectively
Descriptive analytics: What’s happening?
What’s new? What’s changed, why? What to worry about?
Predictive analytics: What’s likely to happen next?
What will (probably) happen if we don’t change what we’re doing?
Causal analytics: What can we do about it?
What will (probably) happen next if we do things differently?
Prescriptive analytics: What should we do?
Evaluation analytics: How well is that working?
Learning analytics: How might we do better?
Collaborative analytics: How to do better together?

10. Technical background

11. Interpreting X  Y: What does “cause” mean?
“X is a cause of Y” denoted X  Y, has been used to mean many different things, not all of them consistent
Probabilistic causation
Associational causation
Attributive causation
Counterfactual/potential outcomes causation
Computational (simulation, calculation order) causation
Predictive causation
Manipulative causation
Mechanistic/explanatory causation

12. Probabilistic causality
Key idea: Causes make their effects more likely, or changes their pdfs.
“X changes the probability of Y” is necessary but not sufficient for causation
Usual meaning: Observing X changes the probability distribution for observed Y values
Example: A high X value for an individual makes a high Y value for that individual more probable
Desired meaning: Changing X changes the probability distribution of Y values
Example: Reducing X makes high Y less probable
This is manipulative causation, not probabilistic inference

13. All agree on what X  Y means for how to calculate probabilities of Y values
In the model X  Y, suppose that input X is a random variable and that the model gives Pr(outputs | inputs) = Pr(y | x) = Pr(Y = y | X = x). Then: Pr(Y = y) = xPr(Y = y | X = x)*Pr(X = x).
Prediction formula for value of Y given uncertain input X
Pr(y | x) and Pr(x) are stored in tables for Y and X in a Bayesian Network (BN) model
Basis for “forward Monte Carlo” risk assessment
Extension allows composition of probabilistic relations: X  Y = (X  Z)(Z  Y) in X  Z  Y
Conditional: Pr(Y = y | X = x) = zPr(Z = z | X = x)*Pr(Y = y | Z = z)
Unconditional: Pr(y) = xPr(y | x)*Pr(x)

14. Inference and prediction via Bayesian Network (BN) Solver
Netica DAG model: “True state  Observation”
Store “marginal probabilities” at input nodes (having output arrows only)
Store “conditional probability tables” (CPTs) at all other nodes.
Make observations
View results (or make query, then view)
Solver calculates conditional probabilities

15. Charniak’s BN (1991)

16. Example: Family Out BN
P(family out | lights-on = T, hear-bark = F) = ?

17. P(y | x), x = evidence/observations (or what-if assumptions)
P(family out | lights-on = T, hear-bark = F) = 0.501

18. Netica gives Bayesian update of all other probabilities, given evidence
P(family out | hear-bark = T, bowel problem = T) = 0.153

19. Lessons from example All probabilities of events are conditional!
No such thing as “the” probability that family is out. Depends on what one observes.
All uncertain quantities have conditional probabilities for their values
Easy to break problem into parts, specify each part, and propagate evidence to draw inferences or make probabilistic predictions using BN solver

20. Dissatisfactions with probabilistic causality
By Bayes’ Rule, if observing X makes Y more likely, then observing Y makes X more likely
Example: Disease X  Symptom Y
Proof: By conditional probability,
Pr(X | Y) = Pr(Y | X)*Pr(X)/Pr(Y)
This implies Pr(X | Y)/Pr(X) = Pr(Y | X)/Pr(Y)
Probabilistic causality does not establish the direction of a causal link
No guarantee of manipulative causality

21. Wrap-up on probabilistic causation
Useful as a screening rule: Effects should depend on their causes  Probability distributions of effects should depend on values of their causes
Conversely, if Y is conditionally independent of X given the values of other variables, then, then X is not a direct cause of Y.
This is the key information shown in DAG models such as X  Y  Z.
Z is conditionally independent of X given Y.
In a DAG model, effects are never conditionally independent (no arrow) of the their direct causes.
Challenge: How to develop better screens for manipulative causality?

22. Essential machine-learning (ML) algorithms
Classification and Regression Trees (CART trees)
Random Forest (RF) ensembles
Partial dependence plots
Bayesian Networks (BNs)
Some key themes and principles
Conditional independence, information network structure
Conditional probabilities for quantifying dependencies
Non-parametric methods
Model ensembles, model averaging

23. CART Trees
A powerful, popular method for data mining, machine learning, and CPT estimation
In R, party, ctree, rpart, and other algorithms provide CART (Classification and Regression Tree) algorithms
Can download applet from:
Basic idea: “Always ask the most informative question next” for reducing uncertainty (conditional entropy) about the dependent variable.
Partitions a set of cases into groups or clusters (the leaf nodes) with similar conditional probabilities for dependent variable.

24. How to read a CART tree
Tips of the tree (“leaf nodes” give the conditional expected value (for continuous variables) or conditional distributions (for discrete variables) of the dependent variable, given the value ranges of the variables along the path from the top of the tree (the “root node”) to the leaf node.
The dependent variable is called “y” by default in the tree
Branches (“splits”) show the value ranges being conditioned on
The tips of the tree also show how many cases in the data set are described by the combination of values leading to that leaf node.
These are the “n” values at the leaf nodes
Example:
n = 92 days had tmin < 43 degrees and PM2.5 < 14 g/m3.
An average of y = elderly people per day died on days with this description.

25. How classification and regression (CART or C&RT) trees works
Basic idea: Always ask the most informative question next, given answers so far.
Questions are represented by splits in tree
Leaf nodes show conditional means (or conditional distributions) of dependent variable
Internal nodes show significance level for split: how significant are differences between conditional distributions
Trees can handle continuous, categorical, ordered categorical, and binary variables and missing values

26. How tree-growing works: Recursive partitioning, stopping rules
Add new questions/splits if doing so improves predictions
Goal: Reduce prediction errors by conditioning on relevant information
Stop this “recursive partitioning” when further questions (splits in tree) do not significantly improve prediction.
Classification & Regression Tree (CART) algorithm
Recursive partitioning = tree-growing by successive splitting
Which variables are informative for predicting the dependent variable can depend on what other predictors are included.
Omitting month makes PM2.5 informative.

27. Improving a tree: Some refinements in advanced algorithms
Grow a large tree and prune back to minimize cross-validation error
Fit multiple trees to random subsets of data and let them vote for best splits (“bagging”)
Over-train on mis-predicted cases (“boosting”)
Select random subsets of columns to use as predictors to diversify and de-correlate predictions
Average predictions from many trees (“RandomForest” ensemble prediction)
Join prediction “patches” together smoothly (MARS)
Search for best subsets of predictors instead of myopically choosing the next splits

28. From concepts to software: Creating a CART tree in CAT
Select dependent variable first, by clicking on a column heading in the Data sheet
Select predictor columns using Ctrl + click
Can select in any order
Click on desired analysis (Tree)
Output appears on new tab
(If unsure what analyses to do after columns selected, click on Analyze.)

29. Classification/Decision Tree Algorithms in Practice
Different decision tree (= classification tree) algorithms use different splitting, stopping, and pruning criteria.
Examples of tree algorithms:
CHAID: chi-square automatic interaction detection
CART: Classification and regression trees, allows continuous as well as discrete variables
MARS: Multiple adaptive regression splines, smooths the relations at tree-tips to fit together smoothly
KnowledgeSeeker allows multiple splits, variable types
ID3, C5.0, etc.

30. Interpreting a tree causally
The measured direct causes of a variable in a DAG model (its “parents” in the DAG) should appear in most/all CART trees for that dependent variable
Indirect causes (its more remote ancestors) should not.
Direct children should
More remote descendants and siblings should not
Manipulative causation: How would changing the answers to some of the questions change the frequency distributions of cases at leaf nodes?
Li et al.,
Athey and Imbens, 2015,

31. Classification-tree tests of conditional independence
In a classification tree, the dependent variable is conditionally independent of all variables not in the tree, given the variables that are in the tree (at least as far as the tree-growing heuristic can discover).
Starting with a childless node (output node), we can recursively seek direct parents of all nodes.

32. Using trees to identify potential parents for each node
Smoking  Heart disease
sex, age  smoking

33. Generating CPTs with Trees
Classification trees grown using parents of a node provide exactly the information needed to fill in its Conditional Probability Table (CPT)
More efficient than exhaustive tables because of “don’t care” conditions (fewer leaves than logical combinations of values of variables)
Each leaf node gives conditional probabilities or a conditional expected value of the dependent variable, given the combination of values of its parents.

34. Decision psychology and practice: “Fast, Frugal Heuristics”
Classification trees provide a basis for quick, effective decision-making that often compares favorably to more computationally intensive statistical procedures.
In the real world, simpler decision rules are often better/more robust than more complex prediction and optimization models and algorithms
Todd and Gigerenzer,

35. Generalizations: Ensembles of trees (Random Forest)
Averaging over hundreds of trees gives more robust results and reduces prediction errors compared to any single tree
Random Forest ensemble is “go to” black-box method for most predictive analytics

36. Partial dependence plots
A partial dependence plot summarizes how the dependent variable is predicted to change as one variable is changed, keeping all other variables with their real values

37. Bayesian Networks (BNs) show dependence relations among variables
A BN provides a high-level roadmap for descriptive, predictive, and causal analytics
“A tree at every node”
Each node has a conditional probability table (CPT) or regression model, CART tree, etc. describing how the conditional probabilities of its values depend on other variables.

38. Bayesian Networks (BNs) also show independence relations among variables
If no arrow connects two variables, they are conditionally independent of each other, given the other variables in the BN.
Omitted variables can create statistical dependencies
Conditioning on variables can also sometimes create dependencies
Information principle for causality: Direct causes are not conditionally independent of their effects.

39. How to create BNs, CART trees, random forests, partial dependence plots in CAT
Select dependent variable
Click on column heading
Select main predictor
Used in partial dependence plot, dagitty software
Select other predictors
In advanced BN modeling, can insert constraints on allowed, forbidden, an required arrows
Click on “Analyze” or on names of analyses
Tree, Bayesian Network, Importance Plot, Sensitivity Plot
Outputs appear on new tabs.

40. Advanced features in CAT
Simulation-based power calculations for Bayesian Networks (BNs) and CART trees: How large would an effect of X on dependent variable have to be to be detected with high probability?
Choice of algorithms for BN learning
Predictive analytics toolkit with library of prediction algorithms, out-of-sample performance assessment

41. Information-based algorithms
Information-based algorithms automatically find dependencies and conditional independence relationships among variables in data.
Example of a Bayesian network DAG. An arrow between two variables shows that they are mutually informative about each other, i.e., statistically dependent.
Directions of arrows indicate a way to compute joint probabilities of variables from marginal distributions at input nodes and conditional probability tables at other nodes

42. Wrap-up: Information-based ML algorithms discover relationships in data
CART trees predict one variable from others
Classification & Regression Trees
RandomForest ensembles average predicted effects over many CART trees
Partial dependence plots
BN learning algorithms relate multiple variables at once
BN interpretation (dagitty)

43. Causal analytics for improving decisions: Vision

44. Causal model: Pr(y | do(x), z)
Basic causal model
Decision-maker (DM) chooses action or policy from a choice set of possible alternatives
Outcome probabilities are determined both by the DM’s choice and by covariates and by events not controlled by the decision-maker
DM’s choice affects probabilities of outcomes
Causal models reveal how, with uncertainty bands
action X  outcomes Y  other variables Z
Causal model: Pr(y | do(x), z)
dependence plot of E(Y | do(x))
E(Y | do(x)) = total effect of do(x) on E(Y), allowing other variables to adjust

45. Vision for causal analytics
Represent understanding of how the world works by a causal model.
Use causal model to quantify how probabilities of consequences and expected utilities change as decision variables or policies are changed
Given preferences for consequences and risk attitude (social utility function), solve for best policy

46. Benefits of causal analytics: Use data to answer…
What works? (policy evaluation)
How well? (effect size estimation)
For whom, under what conditions? (effects of covariates )
How do changes in inputs affect output probabilities?
How have they in the past?
Causal explanation, mediation analysis, path analysis
How to cause desired changes? (decision analysis)
What might work better? (trials, learning)
What is the best achievable result? (optimization)
What will happen if we make changes? (causal prediction)
How sure can we be? (uncertainty analysis)
What information would improve answers? (value of information (VOI) analysis)

47. Representing understanding via a causal graph (DAG)
Modular structure
Each variable is conditionally independent of its more remote ancestors, given the values of its parents in the “DAG” (directed acyclic graph)
Dependencies are quantified by conditional probability tables (CPTs) or trees

48. Components for causal DAG modeling: Representing knowledge via networks
A causal graph model is represented as a set of variables in a directed acyclic graph (DAG).
Choice nodes (green rectangle) = decision (policy, controlled) variables
Chance nodes (yellow) = random (state, uncontrolled) variables
Value nodes (pink hexagon = objective function, value or utility function
Deterministic functions, constants, conditional probability tables

49. Interpreting DAG structure
A variable’s probability distribution of values depends on values of variables pointing into it (“direct parents”).
Structure: Each variable is conditionally independent of its more remote ancestors, given the values of its parents
Example: Health Damage depends on Emissions Reduction only through Concentration and is CI of Emissions Reduction given Concentration
DAG provides modular organization of knowledge and expertise

50. Causal Graph Modeling with DAGs
Quantification: Dependencies are quantified by CPTs or regression models
A conditional probability table (CPT), regression model, CART tree, randomForest ensemble, etc. at each node calculates conditional probabilities for its values, given values of parents
Nonlinear CPTs and interactions are allowed.

51. Using a causal DAG: Solve for best policy by probabilistic simulation of outcomes
Use causal model to quantify how probabilities of consequences change as decision variables or policies are changed
Given preferences for consequences, solve for best policy
Simulation-based (total) dependence plot: Decision variable is varied over a range of alternative (counterfactual) values. Other variables are drawn from their conditional distributions (CPTs) for each value of the decision variable.

52. Selecting “Probability Bands” in Analytica yields:
If correct causal model is known, then Analytica’s probabilistic simulation-based uncertainty analysis can be used to inform risk management decisions.

53. Wrap-up on example Analytica model
Run time = ~1 sec.
Method = Monte-Carlo propagation of input probability distributions through influence diagram. 
Conclusion: An emissions reduction factor of 0.7 is a robust optimum for this decision problem, given the probabilistic model (DAG influence diagram model) mapping inputs to output probabilities.

54. Analytica Wrap-Up
If the causal model is right and the values are agreed to, then this policy is approximately optimal with high confidence.
Q1: But how sure can we be that the causal model is (approximately) right? What happens if it isn’t?
Q2: How to learn useful causal model from data?

55. Summary: Vision for causal analytics
Represent understanding of how the world works by a causal model.
Learn, validate, and document models with data
Use causal model to quantify how probabilities of consequences and expected utilities change as decision variables or policies are changed
Policies can be actions/interventions that change inputs, or decision rules that map data to actions
Given preferences for consequences and risk attitude (social utility function), solve for best policy
Set policy variables to maximize expected utility
Decision analysis
Perform sensitivity analyses, value-of-information (VOI) analyses; optimize timing of interventions
Evaluate results, adaptively learn and improve policies