Bayesian One Sample Inference: Poisson

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

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.

1. From the menus choose:

   Analyze > Bayesian Statistics > One Sample Poisson

2. 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.
3. 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

     ¹

     ²

   • Use Both Methods: When selected, both the Characterize Posterior Distribution and Estimate Bayes Factor inference methods as used.
4. Select and/or enter the appropriate 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, none of the columns are 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 Rate 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 a₀ under the null hypothesis of Poisson inference.

   Null Prior Scale

   Specifies the scale parameter b₀ under the null hypothesis of Poisson inference.

   Null Rate

   Specifies the shape parameter a₀ and the scale parameter b₀ under the null hypothesis for a conjugate prior distribution (to accommodate the Poisson-Gamma relationship). The minimum value must be an numeric value grater than 0; the maximum value must be a max double value.

   Alternate Prior Shape

   A required parameter to specify a₁ under the alternative hypothesis of Poisson inference if Bayes factor is to be estimated.

   Alternate Prior Scale

   A required parameter to specify b₁ under the alternative hypothesis of Poisson inference if Bayes factor is to be estimated.

5. 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).