Can I perform typical Bayesian contrast analysis, if my data takes the form of quantiles?

I'm working with some data from a cell analyzer, which takes millions of measurements and returns mean, median and 5% / 95% quantiles for the distribution. Preliminary model comparisons have indicated good fit to the data for a gamma distribution.

The experimental design is a straightforward ANOVA-type hierarchical model:

$$y_{ij} \sim Gamma(Sh, Sh/\mu_{ij})$$ $$log(\mu_{ij}) = \beta_{0} + \beta^{i}_{1}, i \in \{ 1, \dots 3 \} +$$ $$\beta^{j}_{2}, j \in \{ 1, \dots 3 \}, + \beta^{i}_{1} * \beta^{j}_{2}$$

In the Bayesian context the $\beta$ and Sh variables would have some sort of uninformative hyperparameters.

I want to investigate the contrasts between groups. I would prefer to do this in a Bayesian manner as I prefer my answer in the form of a probability mass.

What I would like to do is

• Fit each group's mean and quantiles to a gamma distribution using, for example, get.gamma.par
• Draw samples from the fitted distribution for each group
• Evaluate contrasts by subtracting samples of contrast groups as described in, for example, the Kruschke textbook

I am not sure if this procedure is legit because I am not generating posterior distributions using MCMC, but rather have essentially been given the posterior parametrization by this cell analyzer.

Can I still run a Bayesian contrast analysis given data in this form?

• So summarizing: instead of raw data you are given summary statistics (mean, median, 5%, 95% quantiles) and want to perform Bayesian analysis of such data, do I understand you correctly? What kind of model do you assume for this data (ignoring the fact that you have only summary statistics)? – Tim Nov 28 '17 at 10:20
• Yes you are correct. I'm not really assuming, but choosing a model via model comparison. It is all-positive data with some right skew (putting mean well higher than median). A gamma is quite a good fit and makes sense for all-positive measurements. – barnhillec Nov 28 '17 at 10:31
• Still, can you describe in greater details what is exactly the model you want to estimate? You've written that it is hierarchical and ANOVA-like, but could you give us more details? – Tim Nov 28 '17 at 10:33
• Post edited to contain a quantitative model specification, apologies for not including before. – barnhillec Nov 28 '17 at 10:53

4. compare the simulated summary statistics $\hat S_\text{sim}$ with real summary statistics $S$ using some distance measure $\rho$, and when $\rho(S, \hat S_\text{sim}) < \varepsilon$ then accept the parameters, otherwise reject them.