3/31. Notice that sampling methods could in general be used even when we don’t know the bayes net (and are just observing the world)! We should strive.

3/31

Notice that sampling methods could in general be used even when we don’t know the bayes net (and are just observing the world)! We should strive to make the sampling more efficient given that we know the bayes net

Generating a Sample from the Network

<C, ~S, R, W>

Network Samples Joint distribution

Note that the sample is generated in the causal order so all the required probabilities can be read off from CPTs! (To see how useful this is, consider generating the sample in the order of wetgrass, rain, sprinkler … To sample WG, we will first need P(WG), then we need P(Rain|Wg), then we need P(Sp|Rain,Wg)—NONE of these are directly given in CPTs –and have to be computed… Note that in MCMC we do (re)sample a node given its markov blanket—which is not in causal order—since MB contains children and their parents.

That is, the rejection sampling method doesn’t really use the bayes network that much…

Notice that to attach the likelihood to the evidence, we are using the CPTs in the bayes net. (Model-free empirical observation, in contrast, either gives you a sample or not; we can’t get fractional samples)

Notice that to attach the likelihood to the evidence, we are using the CPTs in the bayes net. (Model-free empirical observation, in contrast, either gives you a sample or not; we can’t get fractional samples)

Note that the other parents of zj are part of the markov blanket

P(rain|cl,sp,wg) = P(rain|cl) * P(wg|sp,rain)

First-order Logic

Prop logic

First order predicate logic(FOPC)

Prob. Prop. logic

Objects,relations

Degree ofbelief

First order Prob. logic

Objects,relations

Degree ofbelief

Degree oftruth

Fuzzy Logic

Time

First order Temporal logic(FOPC)

Assertions;t/f

Epistemological commitment

Ontological commitment

t/f/u Degbelief

facts

FactsObjectsrelations

Proplogic

Probproplogic

FOPC ProbFOPC

AtomicPropositionalRelationalFirst order

• Atomic representations: States as blackboxes..

• Propositional representations: States as made up of state variables

• Relational representations: States made up of objects and relations between them– First-order: there are

functions which “produce” objects.. (so essentially an infinite set of objects

• Propositional can be compiled to atomic (with exponential blow-up)

• Relational can be compiled to propositional (with exponential blo-up) if there are no functions– With functions, we

cannot compile relational representations into any finite propositional representation

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ns

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Expressiveness of Representations

4/2

Connection to propositional logic: Think of “atomic sentences” as propositions…

general object referent

Can’t have predicates of predicates.. thus first-order

Important facts about quantifiers

• Forall and There-exists are related through negation..– ~[forall x P(x)] = Exists x ~P(x)

– ~[exists x P(x)] = forall x ~P(x)

• Quantification is allowed only on variables – can’t quantify on predicates; can’t say

– [Forall P Reflexive(P) forall x,y P(x,y) => P(y,x) —you have to write it once per relation)

• Order of quantifiers matters

Family Values:Falwell vs. Mahabharata

• According to a recent CTC study,

“….90% of the men surveyed said they will marry the same woman..”

“…Jessica Alba.”

Caveat: Order of quantifiers matters

),( yxlovesyx),( yxlovesxy

)],(),([

)],(),([

)],(),([),(

)],(),([

)],(),([

)],(),([),(

TweetyTweetylovesTweetyFidoloves

FidoTweetylovesFidoFidoloves

yTweetylovesyFidolovesyyxlovesxy

TweetyTweetylovesFidoTweetyloves

TweetyFidolovesFidoFidoloves

TweetyxlovesFidoxlovesxyxlovesyx

TweetyandFidowithworldaConsider

“either Fido loves both Fido and Tweety; or Tweety loves both Fido and Tweety”

“ Fido or Tweety loves Fido; and Fido or Tweety loves Tweety”

Loves(x,y) means x loves y

Intuitively, x depends on y as it is in the scope of the quantification on y (foreshadowing Skolemization)

Caveat: Decide whether a symbol is predicate, constant or function…

• Make sure you decide what are your constants, what are your predicates and what are your functions

• Once you decide something is a predicate, you cannot use it in a place where a predicate is not expected! In the previous example, you cannot say

)(DogSmall

More on writing sentences

• Forall usually goes with implications (rarely with conjunctive sentences)

• There-exists usually goes with conjunctions—rarely with implications

Everyone at ASU is smart

Someone at UA is smart

)(),(

)(),(

xSmartASUxxAt

xSmartASUxAtx

)(),(

)(),(

xSmartUAxAtx

xSmartUAxAtx

Apt-pet

• An apartment pet is a pet that is small

• Dog is a pet

• Cat is a pet

• Elephant is a pet

• Dogs and cats are small.

• Some dogs are cute

• Each dog hates some cat

• Fido is a dog )(

),()()(

)()(

)()(

)()(

)()(

)()(

)()(

)()()(

fidodog

yxhatesycatyxdogx

xcutexdogx

xsmallxcatx

xsmallxdogx

xpetxelephantx

xpetxcatx

xpetxdogx

xaptPetxpetxsmallx

Notes on encoding English statements to FOPC

• You get to decide what your predicates, functions, constants etc. are. All you are required to do is be consistent in their usage.

• When you write an English sentence into FOPC sentence, you can “double check” by asking yourself if there are worlds where FOPC sentence doesn’t hold and the English one holds and vice versa

• Since you are allowed to make your own predicate and function names, it is quite possible that two people FOPCizing the same KB may wind up writing two syntactically different KBs

• If each of the KBs is used in isolation, there is no problem. However, if the knowledge written in one KB is supposed to be used in conjunction with that in another KB, you will need “Mapping axioms” which relate the “vocabulary” in one KB to the vocabulary in the other KB.

• This problem is PRETTY important in the context of Semantic Web

The “Semantic Web” Connection

Two different Tarskian Interpretations

This is the same as the one on The left except we have green guy for Richard

Problem: There are too darned many Tarskian interpretations. Given one, you can change it by just substituting new real-world objects Substitution-equivalent Tarskian interpretations give same valuations to the FOPC statements (and thus do not change entailment) Think in terms of equivalent classes of Tarskian Interpretations (Herbrand Interpretations)

We had this in prop logic too—The realWorld assertion corresponding to a proposition

Connection to propositional logic: Think of “atomic sentences” as propositions…

Herbrand Interpretations• Herbrand Universe

– All constants• Rao,Pat

– All “ground” functional terms • Son-of(Rao);Son-of(Pat);• Son-of(Son-of(…(Rao)))….

• Herbrand Base– All ground atomic sentences made with

terms in Herbrand universe• Friend(Rao,Pat);Friend(Pat,Rao);Friend(

Pat,Pat);Friend(Rao,Rao)• Friend(Rao,Son-of(Rao));• Friend(son-of(son-of(Rao),son-of(son-

of(son-of(Pat))– We can think of elements of HB as

propositions; interpretations give T/F values to these. Given the interpretation, we can compute the value of the FOPC database sentences

))(,(

),(

),(),(,

RaoofsonPatFriend

PatRaoFriend

yxLikesyxFriendyx

If there are n constants; andp k-ary predicates, then --Size of HU = n --Size of HB = p*nk

But if there is even one function, then |HU| is infinity and so is |HB|. --So, when there are no function symbols, FOPC is really just syntactic sugaring for a (possibly much larger) propositional database

Let us think of interpretations for FOPC that are more like interpretations for prop logic