To solve cases, sherlock holmes would Induce or form many different hypotheses of what could have happened and then use deduction to narrow down to the correct hypothesis. He is quoted in saying "When you have eliminated the impossible, whatever remains, however impossible, must be the truth" which exemplifies his problem-solving strategy. His use of induction was one of his biggest skills and kept his clients coming back.



Sherlock Holmes Example

Both Induction and Deduction are necessary within casual inference because they are two ways of approaching a problem and often go hand in hand. Induction goes from evidence -> hypothesis while deduction works in the forward direction; cause (hypothesis) -> effect (theory, conclusion). Artificial Intelligence has started automating Induction and Deduction, an example is Bayesian networks. 

University of Michigan - Ann Arbor

Causal inference refers to making conclusions about finding the reasonings behind the relationship of two or more variables. After coming to some conclusions about statistical data, we ask why such relationships exist between variables.

Causal Inference

Eliminating different hypothesis to draw a conclusion.  The conclusion logically follows from "premises".

Deductive reasoning

Reaching conclusions from evidence. Makes broad generalizations from specific observations. Think of inductive reasoning as bottom-up logic, as it involves widening specific premises out into broader generalizations.

Inductive Reasoning

Pearl, J. & Mackenzie D. (2018). The Book of Why: The New Science of Cause and Effect. Basic Books.

The Book of Why

A machine passes this test if a human could perceive their interactions with the machine as human, indicating that the machine has successfully emulated human reasoning. 

Turing Test

- Causal Inference Books
- Causal Inference Papers

Causal Inference References

The calculus of causation is formed of:
1. Causal diagrams (the language of knowledge)
2. Causal symbolic language (the language of queries)

The calculus of causation

Organisms, as well as ML, can be divided into 3 levels of causation corresponding to three cognitive abilities: seeing, doing, and imagining.
1. Association
2. Intervention
3. Counterfactuals


Ladder of Causation 

Bayes Theorem is about predicting the probability of an event A occurring given an event B has already occurred. We can use the inverse probability of this statment and the probability of each event to find this value. 
P(A) is called prior probability, P(B|A) is Likelihood, P(B) is evidence, and P(A|B) is posterior probability.


Bayes Theorem Overview

Most of the statistical methods are entirely objective, which means reasoning exclusively from data and experiments. But causal analysis and Bayesian analysis are different in that they involves some subjectivity.

Causal analysis requires some subjective commitments based on one's knowledge in the field before the causal digram is drawn and data is explored.

Similarly, Bayesian analysis combines objective evidence with the prior subjective belief.

From objectivity to subjectivity

Stages of Casual Inference: Induction and Deduction

Logical reasoning is the process of using a rational, systematic series of steps based on sound certain mathematical procedures and statements to arrive at a conclusion. 

There are three types of the reasoning:
- Abductive Reasoning 
- Deductive Reasoning 
- Inductive Reasoning

Reasoning

Hill's Criteria is a collection of nine informal guidelines ("viewpoints" as supposed to requirements) used to determine whether a causal relationship exists within an observed association.  Initially, five out of the nine viewpoints were proposed by the surgeon general's advisory committee, to which Bradford Hill later added four more.  Hill's Criteria is not meant to be the necessary conditions to support or refute a cause-and-effect hypothesis.  The debate over the causality between cigarette and lung cancer prompted the criteria, which were used for the proponent of the causal relationship to justify their claim that cigarette smoking is indeed causally related to lung cancer.  

Hill's Criteria

1. necessary causation (but-for causation)
2. sufficient causation
3. necessary-and-sufficient causation

Three different kinds of causation

1. The Law of Counterfactuals
2. The Law of Conditional Independence

The Two Fundamental Laws of Causal Inference

A Randomized Controlled Trial (RCT) or a Controlled Experiment is a method to eliminate unmeasurable confounders effect on causal inference by implementing an intervention on a randomly selected group of individuals (treatment/intervened group) while keeping other individuals unchanged (controlled group, placebo, not-intervened group) to observe the changes in outcomes due to the intervention effect. The RCTs are intended to "disarm" confounders (confounding variables).

Randomized Controlled Trial (RCT) = Controlled Experiment

List of techniques that help solve intractable inference problems in machine learning that usually arise from interactions between latent variables in a structured graphical model.

-Expectation Maximization

-MAP Inference and Sparse Coding

-Variational Inference and Learning

-Learned Approximate Inference



Approximate Inference 

"The estimand is the target quantity—the purpose of the statistical analysis ... the thing we are estimating" " that connects statistical evidence to theory."

It refers "to both the theoretical and the empirical estimands."

Estimand

1- "Set a **theoretical estimand**, clearly connecting this quantity to theory"

2- "Link to an **empirical estimand**, which is informative about the theoretical estimand under some identification assumptions"

3- Use "an **estimation strategy** to learn the empirical estimand (e.g., a regression model) ... from data."

Three Critical Choices in Causal Inference

- Correlation does not imply causation.
- Causation does not imply correlation either.

Correlation vs. Causation

In the case of Sherlock Holmes, deductive reasoning was his ability to investigate and then eliminate different hypotheses to eventually find the correct one and solve the case.

Sherlock Holmes' use of deductive reasoning

Consists of premises followed by a conclusions.

Syllogism

People apply mental models using their imagination to solve deductive reasoning problems. The mental model is the representation of a situation in someones mind and can help determine validity of syllogisms.

Mental model approach

In order to test a rule it is necessary to look to falsify the rule. The canonical problem experiment that shows this is the Wason four-card problem, where students are asked to test the validity of a rule given four cards and must try to turn over cards to confirm and falsify the rule to determine if its valid.

Falsification principle

In the case of Sherlock Holmes, inductive reasoning was his ability to look at the evidence at hand and formulate new leads and hypothesis that would eventually help solve the case.

Example of Inductive reasoning

Children below the age of 9 show capability in reasiong about diverse evidence. By age 5, children were able to reason that more options will be more respresentitive of a person's favorite toy preference. In addition, the way that children categorize groups as diverse is different from how adults do. Children have less exposure, so small differences in belief or grouping might seem like large differences for children. 

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