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 [attachment:bayes.doc 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 [http://blog.ted.com/2006/11/statistician_pe.php here.]
- [attachment:bayes2.doc More on Bayes theorem:Illustration of priors and likelihoods]
Using statistical distributions of likelihoods and priors to obtain posterior distributions
Baguley (2012, p.393-395) gives formulae for posterior mean and variance of means (u) and variances ($$\sigmatext{2}$$ for a normal distribution, N(u, $$\sigmatext{2}$$), with an assumed prior distribution of form N($$u_text{p}$$, $$\sigmatext{2}_text{p}$$) and an obtained likelihood distribution (obtained using sample data) equal to a N($$\hat{u}_text{lik}$$, $$\hat{\sigma}text{2}_text{lik}$$). In particular
$$ sigmatext{2}_text{post} = ( \frac{1}{sigmatext{2}_text{lik}} + \frac{1}{sigmatext{2}_text{p}})text{-1}$$
$$ u_text{post} = (\frac{sigmatext{2}_text{post}}{sigmatext{2}_{lik}}) u_text{lik} + (\frac{sigmatext{2}_text{post}}{sigmatext{2}_{p}}) u_text{p} $$
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.