Bayesian Statistics

Prior distributions
describing a prior

where is the mode what is the range of possible values identify any particularly unlikely or zero values for the prior estimate is the distribution skew

priors with similar mode, but different variance (precision) agree on the most likely value, but represents less confidence in the estimate.

noninformative prior
When a prior is flat across the complete range of possible values for theta, then it is said to be a non-informative prior

Beta(1,1) is a non-informative prior because it is flat across the range 0<x<1

uninformative priors
The spread of the prior reflects the degree of confidence in the beliefs

tall and narrow, confidence

wide and flatter for less confidence

strong prior mean of the prior is known as the prior mean

weak prior

Precision
precision is the reciprocal of the variance

When the variance increases, the precision decrease

variance and precision of a prior are known as the prior variance and prior precisio

= Posterior =

$$f(\theta|x) \approx L(\theta)f(\theta)$$