Bayes

Unidentifiable model

Recall the unidentifiable model 2-parameters model introduced in the 'Consistency' topic :

ex.id.Unid
package ex.id model Unid { random RealVar p1 ?: latentReal, p2 ?: latentReal random IntVar launch1 ?: 0, launch2 ?: 0, launch3 ?: 0, nextLaunch ?: latentInt laws { launch1 | p1, p2 ~ Bernoulli(p1 * p2) launch2 | p1, p2 ~ Bernoulli(p1 * p2) launch3 | p1, p2 ~ Bernoulli(p1 * p2) nextLaunch | p1, p2 ~ Bernoulli(p1 * p2) p1 ~ ContinuousUniform(0, 1) p2 ~ ContinuousUniform(0, 1) } }

See output Reproduce

Now let us compare it with the following single parameter identifiable model:

ex.id.Id
package ex.id import ex.id.ProductOfUnifs model Id { random RealVar p ?: latentReal random IntVar launch1 ?: 0, launch2 ?: 0, launch3 ?: 0, nextLaunch ?: latentInt laws { launch1 | p ~ Bernoulli(p) launch2 | p ~ Bernoulli(p) launch3 | p ~ Bernoulli(p) nextLaunch | p ~ Bernoulli(p) p ~ ProductOfUnifs() } }

See output Reproduce
Exercise 1

Notice that the two approximations of the marginal likelihood are very close. Establish if the true marginal likelihoods are actually equal and reflect on the implication on model selection under unidentifiability.