Bayesian update with continuous prior and likelihood

\( \mu_0\)
\( \sigma_0\)

Density function: $$ f_0(\theta) = P(\Theta=\theta) = {\frac {1}{\sigma_ 0 {\sqrt {2\pi }}}}\exp \left(-{\frac {1}{2}}\left({\frac {\theta-\mu_ 0 }{\sigma_ 0 }}\right)^{2} \right) $$

\( \mu_0\)
\( \sigma_0\)

Density function: $$ f_0(\theta) = P(\Theta=\theta) = {\frac {1}{\theta\sigma_ 0 {\sqrt {2\pi }}}}\ \exp \left(-{\frac {\left(\ln \theta-\mu_ 0 \right)^{2}}{2\sigma_ 0 ^{2}}}\right) $$

\( \alpha_0\)
\( \beta_0\)

Density function: $$ f_0(\theta) = P(\Theta=\theta) = \frac {\theta^{\alpha_ 0 -1}(1-\theta)^{\beta_ 0 -1}}{\mathrm {B} (\alpha_ 0 ,\beta_ 0 )} $$ (See Wikipedia for the definition of the function \(\mathrm {B} \) in the normalization constant.)

Param1
Param2

Use 95% interval mode?

\( \mu_1\)
\( \sigma_1\)

2.5%
97.5%

Likelihood function: $$ f_1(\theta) = P(E \mid \Theta=\theta) = {\frac {1}{\sigma_ 1 {\sqrt {2\pi }}}}\exp \left(-{\frac {1}{2}}\left({\frac {\theta-\mu_ 1 }{\sigma_ 1 }}\right)^{2} \right) $$

successes \(s\)
failures \(f\)

Likelihood function: $$ f_1(\theta) = P(E \mid \Theta=\theta) = \binom{s+f}{s} \theta^{s}(1-\theta)^{f} $$