Problem 1: Descendants of Effects
\nWe will investigate the absence of conditional independence guarantees between two random variables when an arbitrary descendant of a common effect is observed. We will consider the simple
\ncase of a causal chain of descendants:
\nSuppose that all random variables are binary. The marginal distributions of A and B are both
\nuniform (0.5, 0.5), and the CPTs of the common effect D0 and its descendants are as follows:
\nA B Pr(+d0 | A, B)
\n+a +b 1.0
\n+a \u2212b 0.5
\n\u2212a +b 0.5
\n\u2212a \u2212b 0.0
\nDi\u22121 Pr(+di
\n| Di\u22121)
\n+di\u22121 1.0
\n\u2212di\u22121 0.0
\n(a) Give an analytical expression for the joint distribution Pr(D0, D1, \u00b7 \u00b7 \u00b7 , Dn). Your expression
\nshould only contain CPTs from the Bayes net parameters. What is the size of the full joint
\ndistribution, and how many entries are nonzero?
\n(b) Suppose we observe Dn = +dn. Numerically compute the CPT Pr(+dn|D0). Please show how
\nyou can solve for it using the joint distribution in (a), even if you do not actually use it.
\n(c) Let\u2019s turn our attention to A and B. Give a minimal analytical expression for Pr(A, B, D0, +dn).
\nYour expression should only contain CPTs from the Bayes net parameters or the CPT you found
\nin part (b) above.
\n(d) Lastly, compute Pr(A, B | +dn). Show that A and B are not independent conditioned on Dn.
\nProblem 2: Bayes Net 1
\nThe following Bayes net is the \u201cFire Alarm Belief Network\u201d from the Sample Problems of the Belief
\nand Decision Networks tool on AIspace. All variables are binary.
\n(a) Which pair(s) of nodes are guaranteed to be independent given no observations in the Bayes
\nnet? Now suppose Alarm is observed. Identify and briefly explain the nodes whose conditional
\nindependence guarantees, given Alarm, are different from their independence guarantees, given
\nno observations.
\n(b) We are interested in computing the conditional distribution Pr(Smoke | report). Give an
\nanalytical expression in terms of the Bayes net CPTs that computes this distribution (or its
\nunnormalized version). What is the maximum size of the resultant table if all marginalization
\nis done at the end?
\n(c) We employ variable elimination to solve for the Question Assignment above. Identify a variable ordering that i)
\nyields the greatest number of operations possible, and ii) yields the fewest number of operations
\npossible. Also give the max table sizes in each case.
\n(d) Following your second variable ordering above, numerically solve for Pr(Smoke | report) using
\nthe default parameters in the applet example. You may check your answer using the applet,
\nbut you should work it out yourself and show your work.
\n2
\n(a) We can describe all guaranteed independences in the Bayes net by defining two or more subsets
\nof nodes Si
\n, such that all nodes in Si are independent of all nodes in Sj for i \u0338= j. For
\nexample, we can define S1 = {Smokes} and S2 = {Influenza, Sore Throat, Fever} given no
\nobservations. Do the same for conditionally independent nodes i) given Influenza, ii) given
\nBronchitis, and iii) given both Influenza and Bronchitis. Make sure your answers capture all
\nguaranteed independences.
\n(b) Consider using likelihood weighting to solve two queries, one in which Influenza and Smokes
\nare observed, and one in which Coughing and Wheezing are observed. Explain how the two
\ncases differ in the distribution of the resulting samples, as well as the weights that are applied
\nto the samples.
\n(c) We perform Gibbs sampling and would like to resample the Influenza variable conditioned on
\nthe current sample (+s, +st, \u2212f, \u2212b, +c, \u2212w). Give a minimal analytical expression for the
\nsampling distribution Pr(Influenza | sample) (or its unnormalized form). What is the maximum
\nsize of the table that has to be constructed?
\n(d) Numerically solve for the sampling distribution Pr(Influenza | sample) using the default parameters in the applet example. You may check your answer using the applet, but you should
\nwork it out yourself and show your work.
\nProblem 3: Bayes Net 2
\nThe following Bayes net is the \u201cSimple Diagnostic Example\u201d from the Sample Problems of the
\nBelief and Decision Networks tool on AIspace. All variables are binary.<\/p>\n
———<\/p>\n
Problem 1: Effects’ Descendants
\nWhen an arbitrary descendent of a common effect is seen, we will look into the lack of conditional independence guarantees between two random variables. We’ll look at a simple example.<\/p>\n
The following is an example of a causal chain of descendants:<\/p>\n
Assume that all random variables are of the binary type. A and B’s marginal distributions are identical.<\/p>\n
The CPTs of the common effect D0 and its descendants are as follows: uniform (0.5, 0.5), and the CPTs of the common effect D0 and its descendants are as follows:<\/p>\n
Pr(+d0 | A, B) A B<\/p>\n
1.0 +a +b<\/p>\n
+a b 0.5+a b 0.5+a b 0.5+a<\/p>\n
\u2212a +b 0.5<\/p>\n
\u2212a \u2212b 0.0<\/p>\n
Di\u22121 Pr(+di<\/p>\n
| Di\u22121)<\/p>\n
+di\u22121 1.0<\/p>\n
\u2212di\u22121 0.0<\/p>\n
(a) Give an analytical expression for the joint distribution Pr(D0, D1, \u00b7 \u00b7 \u00b7 , Dn). Your expression<\/p>\n
should only contain CPTs from the Bayes net parameters. What is the size of<\/p>\n","protected":false},"excerpt":{"rendered":"
Problem 1: Descendants of Effects We will investigate the absence of conditional independence guarantees between two random variables when an arbitrary descendant of a common\u2026<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[9033,7586,1386,7589,9034,9031,7587,7588,7584,7585,9032],"tags":[8809,7838,8811,8808,8813,8807,8806,8814,8810,8812],"class_list":["post-106880","post","type-post","status-publish","format-standard","hentry","category-assignment-help-uae","category-best-dissertation-writing-service-in-australia","category-best-research-paper-writing-services-in-usa","category-capstone-project-writing-service","category-cheap-essay-writing-services-china","category-essay-writing-service-usa","category-legitimate-phd-dissertation-writing-services-usa","category-term-paper-writing-service","category-thesis-writing-service-in-canada","category-uae-best-dissertation-writing-help-service","category-write-my-essay-australia","tag-best-master-thesis-writing-service","tag-best-thesis-writing-services","tag-cheap-thesis-writing-services","tag-master-thesis-help","tag-master-thesis-writing-service","tag-thesis-help-online-free","tag-thesis-writing-help-online","tag-thesis-writing-services-cost","tag-thesis-writing-services-near-me","tag-thesis-writing-services-reviews"],"_links":{"self":[{"href":"https:\/\/www.studyproessays.com\/help\/wp-json\/wp\/v2\/posts\/106880","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.studyproessays.com\/help\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.studyproessays.com\/help\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.studyproessays.com\/help\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/www.studyproessays.com\/help\/wp-json\/wp\/v2\/comments?post=106880"}],"version-history":[{"count":0,"href":"https:\/\/www.studyproessays.com\/help\/wp-json\/wp\/v2\/posts\/106880\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.studyproessays.com\/help\/wp-json\/wp\/v2\/media?parent=106880"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.studyproessays.com\/help\/wp-json\/wp\/v2\/categories?post=106880"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.studyproessays.com\/help\/wp-json\/wp\/v2\/tags?post=106880"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}