Lecture 16. F) "Counterexample"

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Consider now a case where conjugate priors are not used.

  • Likelihood: [math]f_{\left.X\right|\mu}=N\left(\mu,1\right)[/math]
  • Prior: [math]f_{\mu}=Beta\left(4,5\right)[/math]

In this case,

[math]f_{\left.\mu\right|X}\propto f_{\left.X\right|\mu}.f_{\mu}=\exp\left(-\frac{1}{2}\left(x-\mu\right)^{2}\right)\mu^{3}\left(1-\mu\right)^{4}1\left(\mu\in\left(0,1\right)\right).[/math]

This expression is already complex for a single observation of [math]x[/math], and not similar to any classical distribution. As more observations are added, the expression of the posterior will complicate even further. On the other hand, when conjugate priors are used, only the parameters evolve. The use of conjugate priors will be clear in the next section.