1. Show that the counting numbers defined by equation (11.26) are convex.

2. In this exercise, we will derive another message passing algorithm, called the two-way algorithm, for finding fixed points of region-based energy functionals; this algorithm allows for more general region graphs than in exercise. It uses two messages along each link r → r’: one from r to r’ and one from r’ to r. Consider a region-based free energy as in equation (11.27). For any region r, let pr = |Up(r)| be the number of regions that are directly upward of r. Assume that for any top region (so that pr = 0), we have that κr = 1. We now define qr = (1 − κr)/pregion, taking qr = 1 when pr = 0 (so that κr = 1, as per our assumption). Assume furthermore that qr ≠ 2, and define βr = 1/(2 − qr).

The following equalities define the messages and the potentials in this algorithm:

Note that the messages  are as we would expect: the message sent from r to r’ is simply the product of r’s initial potential with all of its incoming messages except the one from r’ (and similarly for the message sent from r’ to r). However, as we discussed in our derivation of the region graph algorithm, this computation will double-count the information that arrived at r from r’ via an indirect path. The final computation of the messages δ  is intended to correct for that double-counting. In this exercise, you will show that the fixed points are precisely the same as the fixed points of the update equations equation (11.71)–equation (11.76).

a. We begin by defining the messages in terms of the beliefs and the Lagrange multipliers:

Show that these messages satisfy the fixed-point equations in equation (11.33).

b. Show that

Conclude that equation (11.76) holds.

c. Show that

Show that the only solution to these equations is given by equation (11.73) and equation (11.74).

d. Show by direct derivation that theorem 11.6 holds for the potentials defined by equation (11.71)–11.76.

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1. Show that if Q = i Q(Xi|Ui), then

2. Develop a variational approximation using Bayesian networks. Assume that Q is represented by a Bayesian network of a given structure G. Derive the fixed-point characterization of parameters that maximize the energy functional for this type of approximation.

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1 Submission of coursework must be undertaken according to the relevant procedure – whether online or paper-based. Lecturers will give information as to which procedure must be followed, and details of submission procedures and penalty fees can be obtained from Academic Administration or the general student handbook. 2 All coursework must be submitted to the Academic Admin Office and a receipt must be obtained. Under no circumstances can other College staff accept them. Please check the Academic Admin Office opening hours. 3 Late coursework will be accepted by Academic Admin Office and marked according to the guidelines given in your Student Handbook for this year. 4 If you need an extension (even for one day) for a valid reason, you must request one. Collect a coursework extension request form from the Academic Admin Office. Then take the form to your lecturer, along with evidence to back up your request. The completed form must be accompanied by evidence such as a medical certificate in the event of you being sick. The completed form must then be returned to Academic Admin for processing. This is the only way to get an extension. 5 General guidelines for submission of coursework: a) All work must be word-processed and must be of “good” standard. b) Document margins shall not be more than 2.5cm or less than 1.5cm c) Font size in the range of 11 to 14 points distributed to including headings and body text. Preferred typeface to be of a common standard such as Arial or Times New Roman for the main text. d) Any computer files generated such as program code (software), graphic files that form part of the course work must be submitted either online with the documentation or on a CD for paper submissions. e) The copy of the course work submitted may not be returned to you after marking and you are advised to have your personal copy for your reference. f) All work completed, including any software constructed may not be used for any purpose other than the purpose of intended study without prior written permission from St Patrick’s International College. Learning Outcomes and Assessment Criteria Learning Outcomes Assessment requirements. To achieve each outcome a learner must demonstrate the ability to LO1 Be able to use a variety of sources for the collection of data, both primary and secondary 1. create a plan for the collection of primary and secondary data for a given business problem 2. present the survey methodology and sampling frame used 3. design a questionnaire for a given business problem

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1. Students are required to cover all stated requirements.2. Your answer must be both uploaded to Moodle in word file and handed over aprinted copy.3. You need to support your answers with appropriate APA style references where necessary.4. Only include information in the body of your document.your appendixes that has been directly referred to in5. Include a title/cover page containing the subject title and code and the name, student id numbers.6. Please save the document as MGT101AT2_first name_Surename_Student Number Eg MGT101AT2_John_Smith_201400001Topic: Learning OrganisationYou are required to write a 1500 words essay about your understanding of learning organisation. You need to refer to a well-known company which is deemed to be a learning organisation. You must provide your argument on why a learning organisation is important for company growth. No less than 6 reference are requested. You are suggested to use ProQuest Research Library that is listed on Elite Education Institute website for literature review.Note: Your assignment will be online submitted through Turnitin for checking plagiarism.

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