How does the posterior of a dirac prior look like?

Question

Edit for the Moderators: Should this question migrate to stats.stackexchange?

I have a very basic question concerning updating from a prior to a posterior in bayesian statistics.

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Setting:

• I enter in a casino and I have a degenerate prior over an american roulette wheel, say $\delta_{\{0\}}$.

• The outcome of the first roll I witness is not $0$.

[Notation: $\delta_{\{0\}}$ denotes the dirac measure associated with the belief that the roulette is unfair and completely bias towards $0$]

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Questions:

1. How do I update my degenerate prior?
2. Is my posterior the uniform over $\{ 0, 1 , \dots , 36 \}$?.
3. What kind of prior is it?
   To what kind of posterior does it lead?
   [I assume there are some technical names for these things, that I am unaware of]

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Of course the questions come from the fact that it is not possible to update anything with such a prior, if the hard evidence I can obtain does not support it.

Any feedback is most welcome.
Thank you for your time.

Answer 1

1. Using Bayes' rule, for each value $x$ you have $$P(x | data) \propto P(data | x) prior(x)$$ Therefore, $P(x|data)$ is zero if $x \neq 0$ and so $P(0|data) = 1$, so your posterior is the same as the prior.

2. No, the posterior is the same as the prior. Using this prior says that you have decided that the answer is zero and no evidence can convince you otherwise.

3. This could be called a point prior or Dirac prior. It should never be used because one of the basic principles of statistics is that a prior should not assign zero probability to anything. This is called Cromwell's Principle.

Edit: another name for a mixture of a Dirac prior and a uniform distirbution is spike and slab prior. You can find out more about it by searching for this term.