How do I calculate and interpret conditional probabilities?
Gigerenzer (2002) suggests a way to obtain conditional probabilities using frequencies in a decision tree.
Cortina and Dunlap (1997) give an example evaluating the detection rate of a test (positive/negative result) to detect schizophrenia (disorder).
To do this one fixes the following:
The base rate of schizophrenia in adults (2%)
The test will correctly identify schizophrenia (give a positive result) on 95% of people with schizophrenia
The test will correctly identify normal individuals (give a negative result) on 97% of normal people.
Despite this we can show the test is unreliable.
This is a more intuitive way of illustrating the equivalent Bayesian equation:
$$\mbox{P(No disorder|+ result) = }\frac{\mbox{P(No disorder) * P(+ result | No disorder)}}{\mbox{P(No disorder) * P(+ result | No disorder) + P(Disorder) * P(- result | Disorder)}}$$
A talk with subtitles further illustrating aspects of conditional probabilities given by Ted Donnelly (Oxford), a geneticist, is available for viewing here.
Using statistical distributions of likelihoods and priors to obtain posterior distributions
Baguley (2012, p.393-395) gives formulae for the posterior mean ($$u_text{post}$$) and variance ($$\sigma_text{post}^text{2}$$)
for a normal distribution, of form
N(u, $$\sigma^text{2}$$), with an assumed prior distribution of form N($$u_text{p}, \sigma_text{p}^text{2}$$) and an obtained likelihood distribution (obtained using sample data) equal to a N($$\hat{u}_text{lik}, \hat{\sigma}_text{lik}^text{2}$$). In particular
$$sigma_text{post}2$$ =
$$ [ 1 /(\hat{sigma}_\mbox{lik}2 $$ $$ + 1 /(\sigma_\mbox{p}2 ] -1$$
$$ u_text{post} = $$
$$(\frac{sigma_\mbox{post}2}{\hat{sigma}_\mbox{lik}2})$$ $$ \hat{u}_\mbox{lik} + (\frac{sigma_text{post}2}{sigma_\mbox{p}2}) u_text{p} $$
Baguley also gives references for obtaining posterior distributions for data having a binomial distribution which assumes a beta distribution as its prior distribution. For this reason the posterior distribution, in this case, is called a beta-binomial distribution.
References
Baguley T (2012) Serious Stats. A guide to advanced statistics for the behavioral sciences. Palgrave Macmillan:New York.
Cortina JM, Dunlap WP (1997) On the logic and purpose of significance testing Psychological methods 2(2) 161-172.
Gigerenzer G (2002) Reckoning with risk: learning to live with uncertainty. London: Penguin.
Krushchk JK (2011) Doing bayesian data analysis: a tutorial using R and BUGS. Academic Press:Elsevier. For further reading: genuinely accessible to beginners illustrating using prior and posterior probabilities in inference for ANOVAs and other regression models.