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Take the coin flipping example. When we decide to use the Bernoulli distribution to model a coin flip, of course with and without a conjugate prior would make some difference for estimation.

Would it be possible to pick a different model to model the coin flip probability so that without any prior, we can still achieve the same estimation?

I'm thinking of using the cdf of beta distribution as the model. to be more specific.

Model 1. using Bernoulli distribution for a single trial

probability mass function ${\displaystyle f(k;p)={\begin{cases}p&{\text{if }}k=1,\\q=1-p&{\text{if }}k=0.\end{cases}}}$

Model 2. using the cumulative beta distribution for a single trial

this mass function ${\displaystyle g(k;p, \alpha, \beta)={\begin{cases}\frac{B(p; \alpha, \beta)}{B(\alpha, \beta)} &{\text{if }}k=1,\\q=1-\frac{B(p; \alpha, \beta)}{B(\alpha, \beta)} &{\text{if }}k=0.\end{cases}}}$

Here $B$ is beta function for 2 variables and Incomplete Beta Function for 3 variables. Note $\frac{B(p; \alpha, \beta)}{B(\alpha, \beta)}$ is the cumulative distribution function for beta distribution.

Would MAP with model 1 yield the same result comparing to MLE with model 2 ?

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  • $\begingroup$ This is very confusing. If one observes one single coin flip, it is equal to zero with a certain probability $p$ and to one with the complement $1-p$. There is no escaping the Bernoulli model. $\endgroup$ – Xi'an Jan 3 '20 at 8:17
  • $\begingroup$ thanks for the comment, maybe this is not a good example, but my question is around for a set of events whether can we choose a different likelihood model that produces the same model with posterior. $\endgroup$ – yupbank Jan 3 '20 at 14:41
  • $\begingroup$ This is my point. When observing either 0 or 1, there is a single model that fits. $\endgroup$ – Xi'an Jan 3 '20 at 14:45
  • $\begingroup$ it doesn't have to be a coin flip example though.if could be something like temperature observation. $\endgroup$ – yupbank Jan 3 '20 at 15:03

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