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 ?
