MAP Bernoulli Estimate

Compute the maximum a posteriori (MAP) estimate of a Bernoulli probability under a Beta$(a,b)(a,\; b)$ prior, given 0/1 observations.

Implement solve(x, a, b).

Returns. Return a float. The posterior mode is only interior when both posterior parameters exceed 1; use $a,b\ge 1a,\; b\; \backslash ge\; 1$. Observations must be 0/1 and $a,b\ge 1a,b\backslash ge1$, otherwise ValueError is raised; with $a=b=1a=b=1$ and no observations every value is equally likely, so that case raises ValueError too.

Examples

Example 1

Input:

solve([1, 1, 0, 1], 2.0, 2.0)

Output:

0.666667

Example 2

Input:

solve([0, 0], 2.0, 2.0)

Output:

0.25