# Bayesian update with continuous prior and likelihood

Prior
Density function: $$f_0(\theta) = P(\Theta=\theta) = {\frac {1}{\sigma_ 0 {\sqrt {2\pi }}}}\exp \left(-{\frac {1}{2}}\left({\frac {\theta-\mu_ 0 }{\sigma_ 0 }}\right)^{2} \right)$$
Density function: $$f_0(\theta) = P(\Theta=\theta) = {\frac {1}{\theta\sigma_ 0 {\sqrt {2\pi }}}}\ \exp \left(-{\frac {\left(\ln \theta-\mu_ 0 \right)^{2}}{2\sigma_ 0 ^{2}}}\right)$$
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.)
Likelihood
Likelihood function: $$f_1(\theta) = P(E \mid \Theta=\theta) = {\frac {1}{\sigma_ 1 {\sqrt {2\pi }}}}\exp \left(-{\frac {1}{2}}\left({\frac {\theta-\mu_ 1 }{\sigma_ 1 }}\right)^{2} \right)$$
Likelihood function: $$f_1(\theta) = P(E \mid \Theta=\theta) = \binom{s+f}{s} \theta^{s}(1-\theta)^{f}$$
Click here to see an example.
One use case that may be of particular interest is updating a prior on a parameter B based on b, an a statistical estimate of B (for example from a study you conducted or are reading about).
This tool may be helpful for converting between 95% confidence intervals, standard errors, and p-values.