Likelihood of Global Warming Given X.

Introduction:

I’m going to profile a rather wacky article here. A Norwegian physics student applied Bayes’ Theorem to a number of things in a peripheral way in order to provide a basis for applying the theory to global warming. In “Testing Hypotheses about Climate Change: the Bayesian Approach”, Kristoff Rypdal applies Bayes to Russian roulette, the learning process of individuals, hurricanes and global warming and the melting of arctic ice caps. Essentially the author applied probability theory and hypothesis testing, where the concept of probability is defined subjectively as “a degree of knowledge" about a hypothesis. He defines knowledge as something generated by four processes: 1) the inspired formulation of new hypotheses 2) prediction (here deductio

n enters a central element) 3) collection of new data through experiment or observation 4) verification/falsification by comparing predictions and observations.

Summary:

I’ll go ahead and skip Kristoff’s lengthy explanation of what Bayes’ theory does via a parable about Mafiosos and their desire to watch him play Russian roulette. After by passing his .38, Kristoff takes us to Section VI. Hurricanes and Global Warming. He essential states that for the sake of the formula, the existence of human caused global warming is bivalent, 50/50 (H,H^N). The scientific community (again for the sake of the formula) thinks that the odds of a massive hurricane occurring more than once per century in the absence of global warming is 10%, i.e. p(B|H^N)=0.1. He also states that the scientific community believed that in the presence of global warming, massive hurricanes will occur more than once per century is 50%, i.e. p(B|H)=0.5.

Kristoff then basically applies the exact same theory to the arctic ice caps, in Section VIII: Bayesian Learning and Arctic Ice Cap Melting. He states that there has been a well-documented scientific effort regarding the monitoring of the arctic ice caps (which is true) and that these scientists have observed a large reduction in summer sea ice (which is also true). He then gives the “scientific estimates of probability” for the sake of applying the situation to Bayes’ Theorem. He states that the probability of the arctic ice caps melting w/o human caused global warming is 10% or, p(C|H^N)=.1. He also states that the odds of this kind of melting occurring in conjunction with the presence of human caused global warming is 50% or, p(C|H)=0.5. These two measurements are identical as the one above and result in p(H|C)=.83 or 83%. When combined however, the author states that p(H^N)=1-p(H)=.17. When combining this to p(H|B) we get .96, thus turning the results of the previous equation from ‘highly likely’ to ‘virtually certain’.

Conclusion:

Overall I thought Kristoff’s application was interesting though only theoretical in nature. It would be interesting to conduct polling studies on the topic of global warming in both the scientific community and the general public and apply Bayes theorem to the results. This could actually be applied to any polling of public perception really, so long as there was actual information to prove something correct or incorrect. As a study of ‘the causes of global warming’ Kristoff’s article is speculative and not particularly edifying, but as a study of human perception and likely, it is rather interesting.

Source:

Rypdal, K. (2008). Testing hypotheses about climate change: the Bayesian approach. Department of Physics and Technology, University of Troms, 9037 Troms, Norway

http://web.me.com/kristofferrypdal/Themes_Site/Courses_files/Bayesian%20approach%20to%20Climate%20Change.pdf

Summary Of Findings: Bayesian Analysis (4 out of 5 Stars)

Note: This post represents the synthesis of the thoughts, procedures and experiences of others as represented in the 12 articles read in advance of (see previous posts) and the discussion among the students and instructor during the Advanced Analytic Techniques class at Mercyhurst College on 6 MAY 2009 regarding Bayesian Analysis specifically. This technique was evaluated based on its overall validity, simplicity, flexibility and its ability to effectively use unstructured data.

Description:
Bayesian analysis is a method that uses Bayesian statistics to assess the likelihood of an event happening in light of new evidence. It generates an estimate and the use of Bayesian statistics in Intelligence analysis allows for the uncertainty of the traditional intelligence data set to be understood in a scientifically valid manner.

Strengths:

*can limit analyst biases by reducing the weight of evidence simply because it is new or vivid

*forces the analyst to resassess evidence and consider alternative possibilites

*adheres to rigid mathematical formulas

*provides a numerical likelihood

*provides audit trail and ability to reproduce results

Weaknesses:

*Probabilities are based largely on subjectivity

*Susceptible to biases

*Highly complex problems require heavy computations

*Can be mathematically complex

*Not always useful as a stand alone method (works well in tandem with methods like Delphi); may require SMEs for determining probability distributions

*Some reliance on ambiguous validities

*"Negative evidence"--absence of positive evidence

How-To:

This method loosely follows the guidance suggested by his line of research into the use of natural frequencies in teaching and explaining Bayes to beginners.

1.) Create a 2x2 matrix. Label the quadrants with the respective information that creates true positive, false negative, false positive, and true negative quadrants.

2.) Take the given information, the base line (for example, 100 out of 1,000) with the new information (for example, a new document that is 90% credible saying that war is immiment) which means that your true positive and your false negative must equal 100 and the false positive and true negative must equal 900.

3.) To calculate the true positive quandrant, take 90% of the 100 from the base line (which equals 90).

4.) To calculate the false negative quadrant, take the numerator of the base line (100) and subtract the true positive quadrant (90), creating an answer of 10.

5.) To calculate the true negative quadrant, take 90% of your non-war cases (900), equalling 810.

6.) To calculate the false positives, subtract the sum of the three quadrants known from the total number of cases (1,000), which equals 90.

7.) To calculate the new probablitiy, divide what the numerator of the base line (100) from the new total of positive caes (90+90=180), which equals 55.5%

The 55.5% means that there is a 55% probability that countries X and Y are likely to go to war.

Experience:

To understand the basic mathematical principles behind Bayes, the class worked through some sample problems. One of the problems was based on a medical test with an 80% accuracy rate for a cancer with 2% affliction rate in the general population. The class applied this to a sample population of 1000 cases. We established a matrix and assessed the true positive, false positive, false negative, and true negative quantities (16, 116, 4, and 784 respectively). We plugged these numbers into the appropriate matrix fields. We then divided the number of actual cases of cancer (20 or 2% of 1000) into the number of positive tests (132--the 16 true positives and 116 false postives). The result was 15% rate of those who have the cancer from the positive tests, a rather stark difference from the 2% base rate! This problem actually reflects the number of breast cancer rates from a medical treatment from around a two decades ago!

Note: see the matrix for a synopsis of another of the problems we worked through (a peice of evidence emerging suggesting a cause for war).

The class also used a Bayesian application to assess the likelihood we would contract swine flu. We started with the initial hypothesis that we would contract swine flu or we would not contract swine flu, and assigned an initial probability to each hypothesis (the latter >5%) We then added weighted evidence which influenced the base rate of the hypothesis. After all the evidence was entered, the class assessed the likelihood of contracting swine flu.

Bayes' Formula

Author's Note: This is a great video for teaching Bayes' Theorem in its simplest form.

Summary
In order to illustrate the utility of Bayes’ Theorem, the author draws upon two simple scenarios. First, suppose someone faces the decision of needing to choose between three doors. If the person making the decision does not have any prior knowledge about the situation, the scenario creates an unconditional probability. But, once the person receives new information about the scenario, the rational person should reconsider his/her decision and subsequent probabilities.

Bayes’ Theorem is about the introduction of new information used to adjust probabilities and create conditional probabilities. In the formula, P(G/U), P is the probability that G will occur, if U happens.

To illustrate the application of Bayes’ Theorem and conditional probabilities, the author illustrates a second scenario. Pretend that there is a 70% probability that the economy will grow and a 30% probability that the economy will slow (an unconditional probability). The author owns a stock that has an 80% chance of increasing if the economy grows. That same stock, however, only has a 30% chance of increasing if the economy slows. The 80% and the second 30% are conditional probabilities; they are based on the condition that the economy will grow or slow.

The author can then determine the scenario’s four conditional probabilities:
1) What is the probability that the economy will grow and the stock will increase?
2) What is the probability that the economy will grow and the stock will decrease?
3) What is the probability that the economy will slow and the stock will increase?
4) What is the probability that the economy will slow and the stock will decrease?

To answer these questions and determine their probabilities, the author uses the equation: P(UG) = P(U/G)P(G). Notice that this equation is longer than the first because this one incorporates two conditions: the economy will grow/slow and the stock will increase/decrease.

Bayes' Theorem for Intelligence Analysis

Jack Zlotnick
CIA Historical Review Program

Author’s Note: Released by the CIA’s Historical Program in the early 1990’s, Jack Zlotnick wrote this piece in the 1970’s. At the time, the CIA was still in the process of testing Bayes’ Theorem. Due to the ongoing testing period (at that time), Zlotnick does not offer a position on the utility and validity of the Bayesian method with regards to intelligence. In fact, Zlotnick spends a considerable amount of time in the article discussing the ways the theorem should continue to be tested.

Summary
Due to the very nature of intelligence, analysts should be naturally interested in the Bayesian Theorem. Intelligence is probabilistic in nature. Intelligence analysts usually conduct their analysis based on incomplete evidence in which they must address probabilities (thus WOEP’s).

For intelligence applications, Bayes’ Theorem is represented by the equation R=PL. “R” is the revised estimate of the odds favoring one hypothesis over another competing hypothesis (the odds of a particular hypothesis occurring after new evidence is entered into the equation). “P” is the prior estimate on the hypotheses probabilities (the odds before considering the new evidence entered into the equation). The analyst must offer judgments about “L” or the likelihood ratio. This variable is the analyst’s evaluation of the “diagnosticity” of an item of evidence. For instance, if a foreign power mobilizes its troops, what are the chances that “X” will happen over “Y”.

The principle features of the Bayes Theorem distinguish it from conventional intelligence analysis in three ways. First, it forces analysts to quantify judgments that are not ordinarily expressed in numeric terms. Second, the analyst does not take the available evidence as given and draw conclusions. And third, the analyst makes his/her own judgments about the bits and pieces of evidence. He/she does not sum up the evidence as he/she would if he/she had to judge its meaning for a final conclusion. The mathematics does the summing up.

The author is skeptical that the complex tasks analysts are forced to consider can be reduced to numeric values. Bayes’ Theorem, however, may be useful for examining strategic warning by uncovering patterns of activity by foreign powers.