# CS代考 Bayes’ Rule – cscodehelp代写

Bayes’ Rule

Probability that some hypothesis h is true, given that some event e has occured. P ph|eq “ P pe|hqP phq

P peq

P peq “ P pe|hqP phq ` P pe|␣hqP p␣hq

Now, since:

we can rewrite this in a form in which it is commonly applied:

P ph|eq “ P pe|hqP phq

P pe|hqP phq ` P pe|␣hqp1 ́ P phqq

where variables are binary valued.

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Some examples

From:

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The wrong underwear

You live with your partner.

You come home from a business trip to discover a strange pair of un- derwear in your drawer.

What should you think?

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In popular culture

(Metro-Goldwyn-Mayer)

Major plot device in the 2003 romantic comedy.

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The wrong underwear

What do you think?

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What do you think?

What would you guess is the probability? Just a rough estimate…

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The wrong underwear

Probability that some hypothesis is true, given that some event has occured. P ph|eq “ P pe|hqP phq

P pe|hqP phq ` P pe|␣hqp1 ́ P phqq

event = underwear hypothesis = partner cheating To apply Bayes rule we need:

‚ Ppe|hq

‚ Ppe|␣hq ‚ Pphq

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The wrong underwear

P pe|hq

If they cheated, how likely is the underwear?

Well, if they cheated, then maybe it is likely the underwear would appear. But also, wouldn’t they be more careful?

Say 50% chance.

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The wrong underwear

P pe|␣hq

Is there an innocent explanation?

A friend stayed over? Something was left in the dryer? Say 5% chance

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The wrong underwear

P phq?

Prior probability they cheated — before there was any evidence. Hard to quantify.

Studies suggest about 4% of married partners cheat in a given year. Say 4%.

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The wrong underwear

The verdict?

P ph|eq “ P pe|hqP phq

P pe|hqP phq ` P pe|␣hqp1 ́ P phqq

0.5 ̈ 0.04

“ 0.5 ̈0.04`0.05 ̈p1 ́0.04q

“ 0.294 Low because of the low prior.

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The wrong underwear

Now it happens again.

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The wrong underwear

What do you think?

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The wrong underwear

Well now the prior is 0.294

P ph|eq “ P pe|hqP phq

P pe|hqP phq ` P pe|␣hqp1 ́ P phqq

0.5 ̈ 0.294

“ 0.5 ̈0.294`0.05 ̈p1 ́0.294q

“ 0.806 and the verdict is pretty clear.

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A medical example

Let M be meningitis, S be stiff neck:

P pm|sq “

P ps|mqP pmq P psq

0.8 ̈ 0.0001 0.1

“

“ 0.0008

Posterior probability of meningitis still very small, again because of low prior. Common pattern in medical test results.

Note that here we used a slightly different formulation of Bayes Rule.

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Bayes rule & knowledge representation

Useful for assessing diagnostic probability from causal probability:

PpCause|Effectq “ PpEffect|CauseqPpCauseq PpEffectq

We saw this in both examples.

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Bayes rule & knowledge representation

Often easier to assess causal probabilities.

PpCause|Effectq “ PpEffect|CauseqPpCauseq PpEffectq

Can visualise this as:

Cause Effect

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Bayes rule & knowledge representation

Often easier to assess causal probabilities.

PpMeasles|Spotsq “ PpSpots|MeaslesqPpMeaslesq P pSpotsq

Can visualise this as:

Measles Spots

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Mathematical!

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What we learned so far

Probability is a rigorous formalism for uncertain knowledge

Joint probability distribution specifies probability of every atomic event

Queries can be answered by summing over atomic events

For nontrivial domains, we must find a way to reduce the joint size

Independence and conditional independence provide the tools for efficient computation along with Bayes’ rule.

Next week we’ll look at how they are used.

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