a bayesian dive

A Bayesian DiveSomik Raha, Vedika Research

Question

If someone is a haemophiliac, what is your probability that this person is a male?

If someone is a male, what is your probability that this person is a haemophiliac?

Question

If someone is a haemophiliac, what is your probability that this person is a male?

If someone is a male, what is your probability that this person is a haemophiliac?

Haemophilia A (clotting factor VIII deficiency) is the most common form of the disorder, present in about 1 in 5,000–10,000 male births.

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Male%given%Haemophilia%

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Haemophilia*given*Male*

Male given Hemophiliac (n=9) Haemophiliac given Male (n=9)

44%44%

12%

88%

12%

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44%44%

12%

88%

12%

Willing to be shot if you are wrong!

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44%44%

12%

88%

12%

Willing to be shot if you are wrong! And, you are wrong!

Placing a 100% probability on anything implies you are willing to be shot if you are wrong.

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Male given Hemophiliac (n=9) Haemophiliac given Male (n=9)44%44%

12%

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88%

12%

What if you thought that P(Haemophiliac|Male) = P(Male|Haemophiliac)?

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44%44%

12%

instead of

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Male given Hemophiliac (n=9) Haemophiliac given Male (n=9)44%44%

12%

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88%

12%

What if you thought that P(Haemophiliac|Male) = P(Male|Haemophiliac)?

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44%44%

12%

instead of

Associative Logic Error

Examples of Associative Logic Error

Naseeruddin Shah in Court Scene of “Khuda Key Liye”

Deen mey daari hai, daari mey deen nahiThe faithful have beards, but the beard does not have any faith


What is the essence of Jainism?

Cultural Jains: Non-violence and vegetarianism


Mahavira




Mahavira



Essence: Aliveness of the Universe


Mahavira



Essence: Aliveness of the Universe

You cannot die vs

Suicide

Question

If someone has lung cancer, what is your probability that this person was a smoker?

If someone is a smoker, what is your probability that this person will get lung cancer?

Smoker given Lung Cancer (n=9) Lung Cancer given Smoker (n=9)

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Smoker,(given(lung(cancer(

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Lung%cancer,%given%smoker%

What do you notice?

33%22%

44%

33%22%

33%

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44%44%

12%

88%

12%

Smoker given Lung Cancer (n=9) Lung Cancer given Smoker (n=9)

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What do you notice?

33%22%

44%

33%22%

33%

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44%44%

12%

88%

12%

1 in 5000 males is haemophiliac

95% of all hemophilia cases are male

Condition Gender Joint

0.01% * 95% = 0.01% * 5% =

99.99% * 50% = 99.99% * 50% =

The Math of Probability

LikelihoodPrior

1 in 5000 males is haemophiliac

95% of all hemophilia cases are male

Joint

0.01% * 95% = 0.01% * 5% =

99.99% * 50% = 99.99% * 50% =


0.0095%0.0005%

49.995%49.995%

Condition Gender

LikelihoodPrior

Joint

0.01% * 95% = 0.01% * 5% =


0.0095%0.0005%

?

???

Condition Gender

99.99% * 50% = 99.99% * 50% =

49.995%49.995%

LikelihoodPrior

PosteriorPre-Posterior

Joint

0.01% * 95% = 0.01% * 5% =


0.0095%0.0005%

Condition Gender

99.99% * 50% = 99.99% * 50% =

49.995%49.995%

LikelihoodPrior


0.0095%

49.995%

??

Joint

0.01% * 95% = 0.01% * 5% =


0.0095%0.0005%

Condition Gender

99.99% * 50% = 99.99% * 50% =

49.995%49.995%

LikelihoodPrior


0.0095%

49.995%

49.995%

0.0005%

Joint

0.01% * 95% = 0.01% * 5% =


0.0095%0.0005%

Condition Gender

99.99% * 50% = 99.99% * 50% =

49.995%49.995%

LikelihoodPrior


0.0095%

49.995%

49.995%

0.0005%50%

50% = 50% * ?= 50% * ?

= 50% * ?= 50% * ?

Joint

0.01% * 95% = 0.01% * 5% =


0.0095%0.0005%

Condition Gender

99.99% * 50% = 99.99% * 50% =

49.995%49.995%

LikelihoodPrior


0.0095%

49.995%

49.995%

0.0005%50%

50%99.998%

0.019%

0.001%

99.999%

Joint


0.0095%0.0005%

Condition Gender

49.995%49.995%

LikelihoodPrior


0.0095%

49.995%

49.995%

0.0005%50%

50%99.998%

0.019%

0.001%

99.999%



0.0095%

49.995%

49.995%

0.0005%50%

50%99.998%

0.019%

0.001%

99.999%

In this case, your intuition matched the math!



0.0095%

49.995%

49.995%

0.0005%50%

50%99.998%

0.019%

0.001%

99.999%

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44%44%

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88%

12%

In this case, your intuition matched the math!

Smoker given Lung Cancer (n=9)

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Now let’s work this example Instructions:1. Fill in the prior and likelihood 2. Calculate joint probability 3. Flipped tree 4. Place joints correctly 5. Calculate pre-posterior

probability (add up joints) 6. Calculate posterior probability

(divide joint by pre-posterior) 7. Report probability of lung

cancer given smoker

Put the probability you thought of over here

Smoker given Lung Cancer (n=9)

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Now let’s work this example

CDC: 19.3% of all Americans are smokers (2010) Na<onal Cancer Ins<tute: 226,000 Americans in 2012 will be diagnosed with lung cancer US Census Bureau: 313 million people in the US as of Apr 21,2012 % with lung cancer: 0.07% Lung Cancer Prognosis: 8.9% of around 25,000 lung cancer pa<ents were never smokers; therefore 91.1% of lung cancer pa<ents were smokers

Lung Cancer given Smoker (n=9)

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33%22%

33%

Bayesian mathematics is how our brain is actually wired.

It is the math of common sense.

So what’s the big deal about all this?

Core of machine learning

Spam filters

Alison Gopnik, TED Talk, “What Do Babies Think?”

Turns out this is how we normally learn

https://www.youtube.com/watch?v=cplaWsiu7Yg

Reflections?

Suggested further reading: “The Theory That Would Not Die”

Ayurvedic probabilistic fun

Arthritis Diagnostic EngineOsteo

Rheumatoid Gout

From Vedika’s Research Labs

Swelling( Symptoms(

Pain(Arthri4s(

Diges4ve(Problems(

Star4ng(Loca4on(

Effect(of(Oiling(

Tongue(Coa4ng(

Relevance Diagrams (this is an exact computational tree)

Swelling Symptoms Pain Digestive ProblemsTight, Inflamed

Inflammation & Redness

General Swelling

Cracking of joints

Loss of appetite

Skin issues

Fixed

Sharp Shooting

Present

Absent

Tongue Coating

No Tongue Coating

…

Trace one pathway of logic

Osteo Arthritis

Rheumatoic Arth

Gout

Cracking of joints

Fixed Pain

Digestive Problems Present

pOsteo

p1p2

p3Joint

= pOsteo * p1 * p2 * p3 = pJoint

Cracking of joints

Fixed Pain

Digestive Problems Present

p1p2

p3= pJointOsteo Arthritis

Rheumatoic Arth

Gout

Flip it! pOsteo*

pOsteo* = pJointp1 * p2 * p3

Let’s try it

Need a volunteer Vaidya

Rest of you please follow along and answer the questions as well

Getting into continuous land

PDF and CDF

No new idea, really!

The thing about binomialsToss n independent coins

p : probability of 1 heads

p(k,n) : probability of getting k heads in n tosses

p(k,n) = C(n,k) * p^k * (1-p)^(n-k)

Appendix

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Historical (n=30) Vaidya Scientists (n = 9)

Male given Hemophilia Haemophilia given Male

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Historical (n=30) Vaidya Scientists (n = 9)63% 100%


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Historical (n=30) Vaidya Scientists (n = 9)63% 100%

66% 89%


a bayesian dive

Education

cultural jains

essence of jainism

lung cancer n

smoker n

male births

vedika researchquestionif

common form

vs suicidequestionif