title: ?????????-???????????????-week2.2
tags: note
notebook: 6- ????????????-9-Probabilistic Graphical Models 1: Representation
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?????????-???????????????-week2.2
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Markov Assumption.
If a dynamic system X satisfies the Markov assumption for all time t???0, which of the following statements must be true? You may select 1 or more options.
(X(t+1)???X(0:(t?1))|X(t))
?????? (X(t+1)???X(0:(t?1)))
???????????????????????? P(X(t+1))=P(X(t?1)) for all possible values of X
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Independencies in DBNs.
In the following DBN, which of the following independence assumptions are true? You may select 1 or more options.
(O(t)???O(t?1))
???????????????????????? (O(t)???X(t?1)???X(t))
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When X(t) is known, there is no active trail from O(t) to any other node in the network.(X(t+1)???X(t)???X(t?1))
???????????????????????? (X(t)???X(t?1))
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Applications of DBNs.
For which of the following applications might one use a DBN (i.e. the Markov assumption is satisfied)? You may select 1 or more options.
Modeling data taken at different locations along a road, where the data at each location is influenced by only the data at the same location and at the location directly to the East
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Consider each location to be a time slice, and order the locations from East to West. Viewed in this way, this data satisfies the Markov assumption.Modeling time-series data, where the events at each time-point are influenced by only the events at the one time-point directly before it
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This perfectly satisfies the Markov assumption.Predicting the probability that today will be a snow day (school will be closed because of the snow), when this probability depends only on whether yesterday was a snow day.
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Let each day be a time slice, and order the days in chronological order. Viewed in this way, this data satisfies the Markov assumption.Modeling the behavior of people, where a person???s behavior is influenced by only the behavior of people in the same generation and the people in his/her parents??? generation.
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Consider each generation to be a time-slice, and this data satisifes the Markov assumption.??? 4 ?????????
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Plate Semantics.
"Let A and B be random variables inside a common plate indexed by i. Which of the following statements must be true? You may select 1 or more options.
For each i, A(i) and B(i) are not independent.
???????????????????????? For each i, A(i) and B(i) are independent.
???????????????????????? There is an instance of A and an instance of B for every i.
?????? For each i, A(i) and B(i) have edges connecting them to the same variables outside of the plate.
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*Plate Interpretation.
Consider the plate model below (with edges removed). Which of the following might a given instance of X possibly represent in the grounded model? (You may select 1 or more options. Keep in mind that this question addresses the variable???s semantics, not its CPD.)
Whether a specific teacher T is a tough grader
This model does not have any information about how hard of a grader the teacher is, but it does have information about classes and schools.Whether someone with expertise E taught something of difficulty D at school S
???????????????????????? Whether a teacher with expertise E taught a course of difficulty D
???????????????????????? Whether a specific teacher T taught a specific course C at school S
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None of these options can represent X in the grounded model
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Grounded Plates.
Using the same plate model, now assume that there are s schools, t teachers in each school, and c courses taught by each teacher. How many instances of the Expertise variable are there?
ct
st
Not enough information given to know
?????????????????????????????? st
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Template Models. Consider the plate model shown below. Assume we are given K Markets, L Products, M Consumers and N Locations. What is the total number of instances of the variable P in the grounded BN?
K?L?M
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There will be one grounded instance of P for each combination of Market, Consumer, and Product. There will be K?L?M of these combinations.K?L?M?N
(L?M)K
K?(N+(L?M))
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Template Models. Consider the plate model from the previous question. What might P represent?
Whether a specific product PROD was consumed by consumer C in market M
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In the grounded model, there will be an instance of P for each combination of Product and Consumer, and there is a combination like this for each Market. Thus, we are looking at a random variable that will say something about a specific product, market, and consumer combination. The correct answer is the only one that does this.Whether a specific product PROD was consumed by consumer C in all markets
Whether a specific product of brand q was consumed by a consumer with age t in a market of type m that is in location a
Whether a specific product PROD was consumed by consumer C in market M in location L
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Time-Series Graphs. Which of the time-series graphs satisfies the Markov assumption? You may select 1 or more options.
(a)
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(b) is a time-series graph in which all variables in each time slice are independent of all variables in time slices at least 2 time slices before, given all variables in the previous time slice (X(t+1),Y(t+1),Z(t+1)???X(t?1),Y(t?1),Z(t?1)|X(t),Y(t),Z(t)).(c)
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*Unrolling DBNs. Below are 2-TBNs that could be unrolled into DBNs. Consider these unrolled DBNs (note that there are no edges within the first time-point). In which of them will (X(t)???Z(t)???Y(t)) hold for all t, assuming Obs(t) is observed for all t and X(t) and Z(t) are never observed? You may select 1 or more options.
Hint: Unroll these 2-TBNs into DBNs that are at least 3 time steps long (i.e., involving variables from t?1,t,t+1).
(a)
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The independence assumption holds in this network because knowing Y(t) blocks what was the only active trail from X(t) to Z(t).(c)
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