Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan...
Transcript of Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan...
![Page 1: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/1.jpg)
Modals, conditionals, and probabilistic generative models
Topic 1: intro to probability & generative
models; a bit on modality
Dan Lassiter, Stanford Linguistics Université de Paris VII, 25/11/19
![Page 2: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/2.jpg)
4 lectures: The plan
1. probability, generative models, a bit on epistemic modals
2. indicative conditionals
3. causal models & counterfactuals
4. reasoning about impossibilia
Mondaysexcept#3–it’llbeWednesday11/11,nomeetingMonday11/9!
![Page 3: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/3.jpg)
Today: Probabilistic generative models
• widespread formalism for cognitive models • allow us to
– integrate model-theoretic semantics with probabilistic reasoning
– make empirical, theoretical advances in conditional semantics & reasoning
– make MTS procedural, with important consequences for counterfactuals & representing impossibilia
today
3,4
2
![Page 4: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/4.jpg)
How we’ll get there …
• probability – aside on epistemic modals
• exact and approximate inference • kinds of generative models
– (causal) Bayes nets – structural equation models – probabilistic programs
![Page 5: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/5.jpg)
Probability theory
![Page 6: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/6.jpg)
What is probability?
La théorie de probabilités n’est au fond, que le bon sens réduit au calcul: elle fait apprécier avec exactitude ce que les esprits justes sentent par une sorte d’instinct, sans qu’ils puissent souvent s’en rendre compte. -Laplace (1814)
Probability is not really about numbers; it is about the structure of reasoning. -Shafer (1988)
![Page 7: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/7.jpg)
What is probability?
• probability is a logic
• usually built on top of classical logic – an enrichment, not a competitor!
• familiar style of semantics, combining possible worlds with degrees
![Page 8: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/8.jpg)
Interpretations of probability
• Frequentist: empirical/long-run proportion • Propensity/intrinsic chance • Bayesian: degree of belief
All are legitimate for certain purposes. For cognitive modeling, Bayesian interpretation is most relevant
![Page 9: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/9.jpg)
intensional propositional logic
Syntax
Semantics
For i 2 N, pi 2 L�, 2 L ) ¬� 2 L
) � ^ 2 L) � _ 2 L) �! 2 L
J�K ✓ W
J¬�K = W � J�KJ� ^ K = J�K \ J KJ� _ K = J�K [ J KJ�! K = J¬�K [ J K
Truth: � is true at w i↵ w 2 J�K� is true (simpliciter) i↵ w@ 2 J�K
![Page 10: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/10.jpg)
Classical (‘Stalnakerian’) dynamics
C is a context set (⇡ information state).
If someone says “�”, choose to update or reject.
Update: C[�] = C \ J�K
C[�] entails i↵ C[�] ✓ J K
![Page 11: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/11.jpg)
from PL to probability
(Kolmogorov,1933)
For sets of worlds substitute probability distributions:
P: Prop ! [0, 1], where1. Prop ✓ }(W )2. Prop is closed under union and complement3. P(W) = 13. P (A [B) = P (A) + P (B) if A \B = ;
Read P (J�K) as “the degree of belief that � is true”i.e., that w@ 2 J�K
![Page 12: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/12.jpg)
conditional probability
One could also treat conditional probability as basic and use it to define conjunctive probability:
P (A|B) =P (A \B)
P (B)
P (A \B) = P (A|B)⇥ P (B)
![Page 13: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/13.jpg)
probabilistic dynamics
A core Bayesian assumption:
For any propositions A and B, your degree of belief P(B), after observing that A is true, should be equal to your conditional degree of belief P(A|B) before you made this observation.
Dynamics of belief are determined by the initial model (‘prior’) and the data received.
![Page 14: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/14.jpg)
probabilistic dynamics
This assumption holds for Stalnakerian update too. Bayesian update is a generalization: 1) Eliminate worlds where observation is false. 2) If using probabilities, renormalize.
C1 =)observe �
C2 = C1 \ J�K
P1(J K) =)observe �
P2(J K) = P1(J K|J�K)
![Page 15: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/15.jpg)
random variables
a random variable is a partition on W – equiv., a Groenendijk & Stockhof ‘84 question meaning.
rain? = [|is it raining?|]= {{w|rain(w)}, {w|¬rain(w)}}
Dan-hunger =[|How hungry is Dan?|]={{w|¬hungry(w)(d)},{w|sorta-hungry(w)(d)},{w|very-hungry(w)(d)}}
![Page 16: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/16.jpg)
joint probability
We often use capital letters for RVs, lower-case for specific answers.
P(X=x): prob. that the answer to X is x
Joint probability: a distribution over all possible combinations of a set of variables.
P (X = x ^ Y = y) — usu. written — P (X = x, Y = y)
![Page 17: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/17.jpg)
2-RV structured model
rain norain
nothungry
sortahungry
veryhungry
Ajointdistributiondeterminesanumberforeachcell.ChoiceofRVsdeterminesthemodel’s‘grain’:whatdistinctionscanitsee?
![Page 18: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/18.jpg)
marginal probability
• obvious given that RVs are just partitions • P(it’s raining) is the sum of:
– P(it’s raining and Dan’s not hungry) – P(it’s raining and Dan’s kinda hungry) – P(it’s raining and Dan’s very hungry)
P (X = x) =X
y
P (X = x ^ Y = y)
![Page 19: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/19.jpg)
independence
• X and Y are independent RVs iff: – changing P(X) does not affect P(Y)
• Pearl: independence judgments cognitively more basic than probability estimates – used to simplify inference in Bayes nets – ex.: traffic in LA vs. price of beans in China
X |= Y , 8x8y : P (X = x) = P (X = x|Y = y)
![Page 20: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/20.jpg)
2-RV structured model
rain norain
nothungry
sortahungry
veryhungry
Here,letprobabilitybeproportionaltoarea.rain,Dan-hungerindependent• probably,it’sraining
• probably,Danissortahungry
![Page 21: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/21.jpg)
2-RV structured model
rain norain
nothungry
sortahungry
veryhungry
rain,Dan-hungernotindep.:rainreducesappetite• Ifrain,Dan’sprobablynothungry
• Ifnorain,Dan’sprobablysortahungry
![Page 22: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/22.jpg)
inference
Bayes’ rule:
Exercise: prove from the definition of conditional probability.
P (A|B) =P (B|A)⇥ P (A)
P (B)
![Page 23: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/23.jpg)
Why does this formula excite Bayesians so?
Inference as model inversion: – Hypotheses H: {h1, h2, …} – Possible evidence E: {e1, e2, …}
Intuition: use hypotheses to generate predictions about data. Compare to observed data. Re-weight hypotheses to reward success and punish failure.
P (H = hi|E = e) =P (E = e|H = hi)⇥ P (H = hi)
P (E = e)
![Page 24: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/24.jpg)
some terminology
P (H = hi|E = e) =P (E = e|H = hi)⇥ P (H = hi)
P (E = e)
posterior
normalizingconstant
likelihood prior
![Page 25: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/25.jpg)
more useful versions
P(e) typically hard to estimate on its own – how likely were you, a priori, to observe what you did?!?
P (H = hi|e) =P (e|H = hi)⇥ P (H = hi)Pj P (e|H = hj)⇥ P (H = hj)
P (e) =X
j
P (e, hj)
=X
j
P (e|hj)P (hj)
worksiffHisapartition!
![Page 26: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/26.jpg)
more useful versions
Frequently you don’t need P(e) at all: To compare hypotheses,
P (hi|e) / P (e|hi)⇥ P (hi)
P (hi|e)P (hj |e)
=P (e|hi)
P (e|hj)⇥ P (hi)
P (hj)
![Page 27: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/27.jpg)
example
You see someone coughing. Here are some possible explanations:
– h1: cold – h2: stomachache – h3: lung cancer
Which of these seems like the best explanation of their coughing? Why?
![Page 28: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/28.jpg)
example
cold beats stomachache in the likelihood cold beats lung cancer in the prior
=> P(cold|cough) is greatest => both priors and likelihoods important!
P (cold|cough) / P (cough|cold)⇥ P (cold)
P (stomachache|cough) / P (cough|stomachache)⇥ P (stomachache)
P (lung cancer|cough) / P (cough|lung cancer)⇥ P (lung cancer)
![Page 29: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/29.jpg)
A linguistic application: epistemic modals
![Page 30: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/30.jpg)
Modality & probability
![Page 31: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/31.jpg)
Lewis-Kratzer semantics
![Page 32: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/32.jpg)
The disjunction problem
What if likelihood = comparative possibility? Then we validate: Exercise: generate a counter-example.
![Page 33: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/33.jpg)
Probabilistic semantics for epistemic adjectives
An alternative: likelihood is probability. – fits neatly w/a scalar semantics for GAs
Exercise: show that probabilistic semantics correctly handles your counter-model from previous exercise: Key formal difference from comparative possibility?
![Page 34: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/34.jpg)
Other epistemics
Ramifications throughout the epistemic system
– logical relations with must, might, certain, etc – make sense of weak must
Shameless self-promotion:
![Page 35: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/35.jpg)
Inference & generative models
![Page 36: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/36.jpg)
holistic inference: the good part
probabilistic models faithfully encode many common-sense reasoning patterns. e.g., explaining away: evidential support is non-monotonic non-monotonic inference:
– If x is a bird, x probably flies. – If x is an injured bird, x probably doesn’t fly.
(seePearl,1988)
![Page 37: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/37.jpg)
holistic inference: the bad part
• with N worlds we need 2n-1 numbers – unmanageable for even small models
• huge computational cost of inference: update all probabilities after each observation
• is there any hope for a model of knowledge that is both semantically correct and cognitively plausible?
![Page 38: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/38.jpg)
Generative models
We find very similar puzzles in: – possible-worlds semantics – formal language theory
Languages: cognitive plausibility depends on representing grammars, not stringsets
– ‘infinite use of finite means’ Generative models ~ grammars for distributions
– and for possible-worlds semantics!
![Page 39: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/39.jpg)
Kinds of generative models
• Causal Bayes nets • Structural equation models • Probabilistic programs
![Page 40: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/40.jpg)
Causal Bayes nets
(Pearl,1988)
rain
sprinkler
wetgrass
wetgrassdependentonrainandsprinklerrainandsprinklerindependent(butdependentgivenwetgrass!!)uponobservingwetgrass=1,updateP(V):=P(V|wetgrass=1)highprobabilitythatatleastoneenableristrue
![Page 41: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/41.jpg)
Demo!
![Page 42: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/42.jpg)
sketch: approx. inference in CBNs
rain
sprinkler
wetgrass
1.Repeatmanytimes: a.sampleavaluefor nodeswithnoparents P(rain) P(sprinkler)
b.workdownward,samplingvaluesforeachnode conditionalonitsparents
P(wetgrass|rain,sprinkler)
2.analyzeacceptedsamples
![Page 43: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/43.jpg)
Demo!
![Page 44: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/44.jpg)
explaining away
Multiple possible causes leads to the inference pattern explaining away.
1. observe that wet grass is true: => P(rain) increases => P(sprinkler) increases 2. observe that sprinkler is true => P(rain) goes back to prior
![Page 45: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/45.jpg)
Demo!
![Page 46: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/46.jpg)
intransitivity of inference
• if rain, infer wet grass • if wet grass, infer sprinkler • NOT: if rain, infer sprinkler
We can’t avoid holistic beliefs; best we can do is exploit independence relationships
![Page 47: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/47.jpg)
exact & approximate inference
A vending machine has one button, producing bagels with probability p and cookies otherwise. H: the probability p is either .2, .4, .6, or .8, with equal prior probability. You hit the button 7 times and get
B B B B C B B What is p?
![Page 48: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/48.jpg)
exact inference
exactcalculation
Prior: 8h : P (h) / 1
L’hood: P (seq|p) = pNB(seq)(1� p)NC(seq)
theobservedsequence
8h : P (h) = 1/|H| = .25
P (BBBBCBB|p) = p ⇤ p ⇤ p ⇤ p ⇤ (1� p) ⇤ p ⇤ p
![Page 49: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/49.jpg)
approximate inference
Monte Carlo approximation
(rejection sampling)
1. repeat many times: a. choose h according to prior, simulate
predictions b. accept h iff simulated e is equal to observed e
2. plot/analyze accepted samples
![Page 50: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/50.jpg)
Demo!
![Page 51: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/51.jpg)
Today’s highlights
• Probability as an intensional logic – Linguistic application: epistemic modality
• Problems of tractability => generative models
• Sampling is a useful way to think of inference in generative models
Do generative models and sampling have interesting linguistic applications?
![Page 52: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/52.jpg)
Linguistic applications: next 3 lectures
1. indicative conditionals
2. causal models & counterfactuals
3. reasoning about impossibilia
![Page 53: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/53.jpg)
Indicative conditionals
Conditional reasoning as rejection sampling – enforces Stalnaker’s thesis
Background semantics is trivalent – define a sampler over trivalent sentences
Linguistic advantages: – avoids Lewis-style triviality results – semantic treatment of conditional restriction
Connections w/ other ways to avoid triviality
![Page 54: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/54.jpg)
Causal models & counterfactuals
Parenthood in gen. models naturally thought of as causal influence Counterfactual reasoning as intervention
– connections to Lewis/Stalnaker semantics – reasons to prefer the causal models approach
Filling a major gap: treatment of complex, quantified antecedents
![Page 55: Modals, conditionals, and probabilistic generative models · 2019. 11. 29. · 4 lectures: The plan 1. probability, generative models, a bit on epistemic modals 2. indicative conditionals](https://reader035.fdocuments.in/reader035/viewer/2022071604/613f0082c500cf75ab363efa/html5/thumbnails/55.jpg)
Reasoning about impossibilia
What if 2 weren’t prime? – doesn’t make sense in possible-worlds semantics – but people understand the question …
Generative models can represent non-causal information, e.g., a theory of arithmetic
– probabilistic programs support interventions – lazy computation means we only compute
partial representations
Connections to hyperintensionality