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Warm-up Question

Your friend texts you and says they woke �up with a headache after staying up late �last night. You’re worried about whether �they have Covid.
They don’t have a fever or a cough, and �have a negative antigen test.
How likely is your friend to be sick, relative �to a random person at Berkeley?

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Bayes’ Rule

p(A|B) = p(B|A) * p(A) / p(B)
A = “has Covid”
B = “headache, no cough, no fever, negative test”

Sometimes more useful form (especially for binary outcomes):

p(A=1|B) / p(A=0|B) = (p(B|A=1)/p(B|A=0)) * p(A=1)/p(A=0)

“Posterior probability ratio = Likelihood ratio * Prior ratio”

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Likelihood Ratios

Likelihood ratio: (0.4/0.2) x (0.65/0.95) x (0.7/0.98) x (0.3/0.95) = 0.31
LR is < 1: less likely to be sick than a random student.

Mainly because of negative test

Observation	Prob \| Covid	Prob \| No Covid
Headache	40%	20%
No Cough	65%	95%
No Fever	70%	98%
Negative Test	30%	95%

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Example: Russia-Ukraine Evidence

For HW5 predictions, here were 4 pieces of evidence:

Russia sends “peacekeeping” forces into eastern Ukraine
Biden condemns Putin’s actions
Deputy Secretary of State calls Russian actions an invasion
Biden urges all Americans in Ukraine to leave immediately

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Other Examples

Will Ethereum go up or down next week, given that it went up last week?
Did this senator embezzle funds, given that they said they didn’t embezzle funds?
Will my friend be on time, given that they said they’ll be on time?
Am I right about this, given that I think I’m right about this?

In several cases above, Bayes’ rule provides a way to integrate base rates with other sources of evidence.

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Aristotle vs. Galileo

Images from �https://arbital.com/p/bayes_science_virtues/

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Bacteria: Genetic vs. Acquired Immunity

https://academic.oup.com/genetics/article/28/6/491/6033179

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Exercise: Bayesian Romance

You go on a date. The conversation goes well and they laugh at your jokes. The date was only scheduled for an hour but you stick around for two and a half hours.��The next day you text them about a second date. It’s now been five days and you haven’t heard back. What’s the probability that they’re ghosting you?

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Coda: Incorporating Weak Evidence

Sometimes evidence is pretty weak–how do we tell if likelihood ratio is 1.05, 1.1, or 1.2?
Idea: think of many “similar” types of evidence
Example: for Campos vs. Haney prediction, I came up with 6 minor considerations that all felt ~equally important.
If all 6 had gone the same way, I’d have given 2:1 odds (e.g. 2/3 probability on the favored candidate)
So each individual consideration should have a likelihood ratio of 2^(⅙) = 1.12.

If they split 5-1 instead of 6-0, then LR = 2^(4/6) = 1.59, so P = 1.59/2.59 = 61%
Similarly, 4-2 split gives LR = 2^(2/6) = 1.26, so P = 1.26/2.26 = 56%.

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