# Bayesian statistics

IBM® SPSS® Statistics provides support for the following Bayesian statistics.

One Sample and Paired Sample T-tests
The Bayesian One Sample Inference procedure provides options for making Bayesian inference on one-sample and two-sample paired t-test by characterizing posterior distributions. When you have normal data, you can use a normal prior to obtain a normal posterior.
Binomial Proportion tests
The Bayesian One Sample Inference: Binomial procedure provides options for executing Bayesian one-sample inference on Binomial distribution. The parameter of interest is π, which denotes the probability of success in a fixed number of trials that may lead to either success or failure. Note that each trial is independent of each other, and the probability π remains the same in each trial. A binomial random variable can be seen as the sum of a fixed number of independent Bernoulli trials.
Poisson Distribution Analysis
The Bayesian One Sample Inference: Poisson procedure provides options for executing Bayesian one-sample inference on Poisson distribution. Poisson distribution, a useful model for rare events, assumes that within small time intervals, the probability of an event to occur is proportional to the length of waiting time. A conjugate prior within the Gamma distribution family is used when drawing Bayesian statistical inference on Poisson distribution.
Related Samples
The Bayesian related sample inference design is quite similar to the Bayesian one-sample inference in terms of handling paired samples. You can specify the variable names in pairs, and run the Bayesian analysis on the mean difference.
Independent Samples T-tests
The Bayesian independent sample inference procedure provides options for using a group variable to define two unrelated groups, and make Bayesian inference on the difference of the two group means. You can estimate the Bayes factors by using different approaches, and also characterize the desired posterior distribution either assuming the variances are known or unknown.
Pairwise Correlation (Pearson)
The Bayesian inference about Pearson correlation coefficient measures the linear relation between two scale variables jointly following a bivariate normal distribution. The conventional statistical inference about the correlation coefficient has been broadly discussed, and its practice has long been offered in IBM SPSS Statistics. The design of the Bayesian inference about Pearson correlation coefficient allows you to draw Bayesian inference by estimating Bayes factors and characterizing posterior distributions.
Linear Regression
Bayesian inference about Linear Regression is a statistical method that is broadly used in quantitative modeling. Linear regression is a basic and standard approach in which researchers use the values of several variables to explain or predict values of a scale outcome. Bayesian univariate linear regression is an approach to Linear Regression where the statistical analysis is undertaken within the context of Bayesian inference.
One-way ANOVA
The Bayesian One-Way ANOVA procedure produces a one-way analysis of variance for a quantitative dependent variable by a single factor (independent) variable. Analysis of variance is used to test the hypothesis that several means are equal. SPSS Statistics supports Bayes-factors, conjugate priors, and non-informative priors.
Log-Linear Regression Models
The design for testing the independence of two factors requires two categorical variables for the construction of a contingency table, and makes Bayesian inference on the row-column association. You can estimate the Bayes factors by assuming different models, and characterize the desired posterior distribution by simulating the simultaneous credible interval for the interaction terms.
One-way Repeated Measures ANOVA
The Bayesian One-way Repeated Measures ANOVA procedure measures one factor from the same subject at each distinct time point or condition, and allows subjects to be crossed within the levels. It is assumed that each subject has a single observation for each time point or condition (as such, the subject-treatment interaction is not accounted for).