Direct comparison
Prior vs Posterior: What Changes After the Data
Prior vs posterior: the difference between them, how the data update one into the other, and when a prior pulls the result. Sourced side-by-side table.
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How do Prior, Posterior compare side by side?
The table below compares Prior, Posterior across 9 procurement-relevant dimensions, from what it is through source.
Side-by-side comparison
| Dimension | Prior | Posterior |
|---|---|---|
| What it is | The distribution that reflects any information we have about the parameter from a source external to the observed data (Sinclair et al., CMAJ Open 2016). | The result of a Bayesian analysis, expressed as the distribution of the unknown parameters (Sinclair et al., CMAJ Open 2016). |
| Where its information comes from | Outside the data being analysed. | The prior and the likelihood, the information from the observed data, combined in accordance with Bayes' theorem. |
| Role of the observed data | None: it is specified from external information. | Central: the observed data enter through the likelihood. |
| Conditioning in the predictive form | Prior predictive distribution: conditions on no data (Stan Users Guide). | Posterior predictive distribution: conditions on the observed data (Stan Users Guide). |
| Predictive form | Like the posterior predictive distribution, but with no observed data. | The distribution over new observations given previous observations: p(ỹ | y) = ∫ p(ỹ | θ) p(θ | y) dθ. |
| When the prior is more informative than the likelihood | Is the reference the posterior sits closer to. | Is located closer to the prior distribution (Sinclair et al.). |
| When the prior is vague | Has little or no influence over the final results (Sinclair et al.). | Is driven mainly by the observed data. |
| When to look at each | To see what was assumed before the data, and to run a prior predictive check. | To see what is concluded after the data, and to run a posterior predictive check. |
| Source | Sinclair A, Xie X, Saab L, Dendukuri N. CMAJ Open 2016;4(4):E706-E718, https://pmc.ncbi.nlm.nih.gov/articles/PMC5173486/; Stan Users Guide, Posterior and Prior Predictive Checks, https://mc-stan.org/docs/stan-users-guide/posterior-predictive-checks.html (read 2026-10-29). | Same two sources, plus Stan Users Guide, Posterior Prediction, https://mc-stan.org/docs/stan-users-guide/posterior-prediction.html (read 2026-10-29). |
Common questions
Common questions about Prior vs Posterior
What is the difference between prior and posterior?
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The prior reflects information about a parameter from a source external to the observed data. The posterior is the updated distribution after that prior is combined with the likelihood, the information from the observed data, in accordance with Bayes' theorem.
Does the prior affect the posterior?
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Yes, to a degree that depends on how informative it is. When the prior is more informative than the likelihood, the posterior is located closer to the prior; a vague prior has little or no influence over the final results (Sinclair et al., CMAJ Open 2016).
Going deeper
