Notice that sampling methods could in general be used even when

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Notice that sampling methods could in general be
used even when we dont 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
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Generating a Sample from the Network
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(RainWg), then we need
P(SpRain,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 blanketwhich is not in causal
ordersince MB contains children and their
parents.
ltC, S, R, Wgt
Network ?Samples ?Joint distribution
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That is, the rejection sampling method doesnt
really use the bayes network that much
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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
cant get fractional samples)
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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
cant get fractional samples)
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Note that the other parents of zj are part of
the markov blanket
P(raincl,sp,wg) P(raincl) P(wgsp,rain)
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First-order Logic
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Facts Objects relations
FOPC
Prob FOPC
Ontological commitment
Prob prop logic
Prop logic
facts
t/f/u
Deg belief
Epistemological commitment
Assertions t/f
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Atomic?Propositional?Relational?First order
Expressiveness of Representations

• 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

higher-order representations
can (sometimes) be compiled to lower order
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Why FOPC

• If your thesis is utter vacuous
• Use first-order predicate calculus.
• With sufficient formality
• The sheerest banality
• Will be hailed by the critics "Miraculous!"

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?general object referent
Cant have predicates of predicates.. thus
first-order
Connection to propositional logic
Think of atomic sentences as propositions
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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
• cant quantify on predicates cant say
• Forall P Reflexive(P) ?? forall x,y P(x,y) gt
  P(y,x) you have to write it once per relation)
• Order of quantifiers matters

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Family ValuesFalwell vs. Mahabharata

• According to a recent CTC study,
• .90 of the men surveyed said they
  will marry the same woman..
• Jessica Alba.

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Caveat Order of quantifiers matters
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)
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
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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

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More on writing sentences
Everyone at ASU is smart Someone at UA is
smart

• Forall usually goes with implications (rarely
  with conjunctive sentences)
• There-exists usually goes with conjunctionsrarely
  with implications

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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

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Notes on encoding English statements to FOPC

• 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

• 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
  doesnt hold and the English one holds and vice
  versa

The Semantic Web Connection
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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 tooThe real World
assertion corresponding to a proposition
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Connection to propositional logic Think of
atomic sentences as propositions
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Herbrand Interpretations
Let us think of interpretations for FOPC that are
more like interpretations for prop logic

• 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)Fr
  iend(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

If there are n constants and p k-ary predicates,
then --Size of HU n --Size of HB
pnk 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
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