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Easier VI for Covid-19 We have a workplace with AK workers, uj,--- wg, where we monitor Covid-19

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Easier VI for Covid-19

We have a workplace with AK workers, uj,--- wg, where we monitor Covid-19. Any day d each worker wy is either non-infected, infected, or has antibodies, i.e., there is a latent variable Zi with a value in {n,i,a}, with the obvious interpretation. A non-infected individual becomes with probability t infected the day after the individual has had contact with an infected individual (and though only one such contact may occur with any single infected individual during a day, an uninfected may have contact with several infected during a day). An individual that becomes infected on day d is aware of the infection, and will on day d+ 9 get antibodies with probability a. Otherwise, the individual remains/returns to the non-infected state. An infected individual stays at home with probability a. and is otherwise present at the workplace. We have access to a contact graph Gd and an absence

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table Ad for each day d € [D], A^k = 1 if worker k is home on day d and otherwise 0. Consider G = G_d

as given so the joint is P(A, Z,?OG),

where  ? = (t,a,0). There are beta priors on Bernoulli parameters i, a, and o. No other reasons than Covid-19 makes any worker stay at home. On day one w1) is infected and all other workers are non-infected. Let Z* = Z*,..Z§ and Z = Z!,...Z*. Design a VI algorithm for approximating the

posterior probability over Z and use the VI distribution

A

4(Z) = |] a(Zj).ak

Hint: extend the latent variable so that you also can keep track of how long an individual has been infected.

3.5 Hard VI for Covid-19

For the above model, i.e., in 3.4 , design a VI algorithm for approximating the posterior probability over Z, but in this case, use the VI distribution

qZ) = |II] 2").

Hint notice that Z^k has a Markov property given Z^-k, i.e., given Z~* and Z aK the variables zk d-1

are independent of zk 41:D°