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I'm using approximate Bayesian computation to find the true value of a parameter. My prior distribution is uniform over $(0, 1)$.

I was watching this video on Bayesian learning and the lecturer states (around 36:00) that this is making a huge assumption. That is, we are assuming the value of the unknown variable has the same mean as the random variable (in my case $0.5$) and also has a variance that we can compute (I assume based off the number of observations?). He goes on to say that if you want to really model a prior like this, you should use a delta function centered around a value, $a$, which is unknown.

My questions are these:

  1. What is wrong with assuming that the mean and variance are the same as the random variable?
  2. How can I use a delta function to generate probabilities for my prior?
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    $\begingroup$ Not sure that I understand everything, but I don't see how you can use a delta function as a prior: it is like fixing your parameter at $a$. One possible problem is that the mean of your prior is equal to the true parameter, so if your likelihood has no information about the parameter, the posterior mean will be very close to the truth. So you might end up thinking that your estimation procedure is working well, when actually you are just sampling the prior. $\endgroup$
    – Matteo Fasiolo
    Commented Dec 3, 2013 at 13:38
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    $\begingroup$ Also: unless I have misunderstood the lecturer, he is not saying that you should substitute the prior with a delta function. He is saying that the belief "my parameter takes a certain value, which I don't know" can be translated into a delta function centred around an unknown point $a$. In practice you can use a proper prior (such as your uniform), but you have to be aware that this is not equivalent to the above statement of ignorance, but you are making some assumptions. $\endgroup$
    – Matteo Fasiolo
    Commented Dec 3, 2013 at 13:50
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    $\begingroup$ The question and the comment remain impenetrable to me... $\endgroup$
    – Xi'an
    Commented Nov 19, 2015 at 20:06
  • $\begingroup$ While I had thought that the question "can the delta function be used as a prior" made sense given (for example) the approach presented by Pishro-Nik especially for mixed random variables, I probably was "out of my depth" and have accordingly deleted my post here. Nobody cares and I couldn't care less. ;-) $\endgroup$
    – gwr
    Commented Nov 27, 2015 at 12:13
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    $\begingroup$ ohblahitsme: As Matteo said, a point mass (Dirac delta) prior says "I know the value of the parameter" -- in which case the data have no impact. Clearly given what you say the lecturer said about the other prior, a point prior cannot have been the intended prior. One is then left to wonder what is really meant. Can you clarify? $\endgroup$
    – Glen_b
    Commented Nov 29, 2015 at 23:59

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