# Bayesian One Sample Inference: Binomial

This feature requires SPSS® Statistics Standard Edition or the Advanced Statistics option.

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.

Although it is not necessary, a prior from the Beta distribution family is normally chosen when estimating a binomial parameter. The Beta family is conjugate for the binomial family, and as such leads to the posterior distribution with a closed form still in the Beta distribution family.

- From the menus choose:
- Select the appropriate Test Variables from the
Available Variables list. At least one variable must be selected.Note: The available variables list provides all variables except for Date and String variables.
- Select the desired Bayesian Analysis:
- Characterize Posterior Distribution: When selected, the Bayesian inference is made from a perspective that is approached by characterizing posterior distributions. You can investigate the marginal posterior distribution of the parameter(s) of interest by integrating out the other nuisance parameters, and further construct credible intervals to draw direct inference. This is the default setting.
- Estimate Bayes Factor: When selected, estimating Bayes factors (one of
the notable methodologies in Bayesian inference) constitutes a natural ratio to compare the marginal
likelihoods between a null and an alternative hypothesis.
Table 1. Commonly used thresholds to define significance of evidence Bayes Factor Evidence Category Bayes Factor Evidence Category Bayes Factor Evidence Category >100 Extreme Evidence for H1 1-3 Anecdotal Evidence for H1 1/30-1/10 Strong Evidence for H0 30-100 Very Strong Evidence for H1 1 No Evidence 1/100-1/30 Very Strong Evidence for H0 10-30 Strong Evidence for H1 1/3-1 Anecdotal Evidence for H0 1/100 Extreme Evidence for H0 3-10 Moderate Evidence for H1 1/10-1/3 Moderate Evidence for H0 *H0: Null Hypothesis**H1: Alternative Hypothesis* - Use Both Methods: When selected, both the Characterize Posterior Distribution and Estimate Bayes Factor inference methods as used.

- Select and/or enter the appropriate Success Categories and Hypothesis
Values settings. The table reflects the variables that are currently in the
Test Variables list. As variables are added or removed from the
Test Variables, the table automatically adds or removes the same variables
from its variable pair columns.
- When Characterize Posterior Distribution is selected as the Bayesian Analysis, the Success Categories column is enabled.
- When Estimate Bayes Factor or Use Both Methods are selected as the Bayesian Analysis, all editable columns are enabled.

- Point Null
- Enables and disables the Null Proportion option. When the setting is enabled, both the Null Prior Shape and Null Prior Scale options are disabled.
- Null Prior Shape
- Specifies the shape parameter
*a*_{0}under the null hypothesis of Binomial inference. - Null Prior Scale
- Specifies the scale parameter
*b*_{0}under the null hypothesis of Binomial inference. - Null Proportion
- Specifies the shape parameter
*a*_{0}and the scale parameter*b*_{0}under the null hypothesis for a conjugate prior distribution (to accommodate the Beta and the Haldane’s priors). The valid range is numeric values between 0 and 1. - Alternate Prior Shape
- A required parameter to specify
*a*_{0}under the alternative hypothesis of Binomial inference if Bayes factor is to be estimated. - Alternate Prior Scale
- A required parameter to specify
*b*_{0}under the alternative hypothesis of Binomial inference if Bayes factor is to be estimated. - Success Categories
- Provides options for defining conjugate prior distributions. The provided options specify how
success is defined, for numerical and string variables, when the data value(s) are tested against
the test value.
- Last Category
- The default setting that performs the binomial test using the last numerical value found in the category after it is sorted in an ascending order.
- First Category
- Performs the binomial test using the first numerical value found in the category after it is sorted in an ascending order.
- Midpoint
- Uses the numerical values ≥ the midpoint as cases. A midpoint value is the average of the minimum and maximum sample data.
- Cutpoint
- Uses the numerical values ≥ a specified cutoff value as cases. The setting must be a single numeric value.
- Level
- Treats user specified string values (can be more than 1) as cases. Use commas to separate the different values.

- You can optionally click Criteria to specify Bayesian One Sample Inference: Criteria settings (credible interval percentage, missing values options, and numerical method settings), or click Priors to specify Bayesian One Sample Inference: Binomial/Poisson Priors settings (conjugate or custom prior distributions).

^{1}Lee, M.D., and Wagenmakers, E.-J. 2013.

*Bayesian Modeling for Cognitive Science: A Practical Course*. Cambridge University Press.

^{2}Jeffreys, H. 1961.

*Theory of probability*. Oxford University Press.