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I found this question without any context while preparing for a job interview.

The only reason I can think of is that the Normal distribution's support are the real numbers.

Why wouldn't we use a uniform distribution? Because there is no uniform distribution over the real numbers from -infinity to infinity?

Thanks!

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    $\begingroup$ Partly because the support of a normal distribution is the real line. Also, normal distributions can express a wide variety of subjective information for a prior by adjusting the mean and SD. And if you need a relatively non-informative prior, you can choose a normal prior with a very large variance (very small precision). $\endgroup$ – BruceET Jan 6 '19 at 20:28
  • $\begingroup$ @BruceET, I think this is a fine answer (perhaps with some minor editing), rather than a comment. $\endgroup$ – StatsStudent Jan 6 '19 at 20:59
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There is no reason to use the Normal distribution as a default prior for a real parameter and there are many instances in the Bayesian literature where another prior is used. For instance, when testing for a zero mean in a Normal sample, Jeffreys (1939) uses a standard Cauchy distribution as a prior. The "uniform" prior does not exist as a probability distribution but it can be used as an "improper:" prior, provided the posterior is well defined. This is also a choice found in Jeffreys (1939). Note that using a Normal prior with a large variance is a poor idea, as large may not be large enough.

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The first question you need to think about is why you consider that you need a totally uninformative prior. Do you really know absolutely nothing about the situation? Some allegedly uninformative priors place unrealistically high densities on regions in the sample space that a moment's reflection will tell you is unlikely.

Moreover, you do not need to restrict yourself to a single choice of prior.

In short, beware of default choices of prior.

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