# Record
**Author:** @cameron.stream (`did:plc:gfrmhdmjvxn2sjedzboeudef`)

## `knowledge-bayesian-inference`
**Collection:** `site.standard.document`
**AT URI:** `at://did:plc:gfrmhdmjvxn2sjedzboeudef/site.standard.document/knowledge-bayesian-inference`

**Title:** Bayesian Inference
**Published:** Tue, 21 Jul 2026 00:14:00 GMT
**Updated:** Tue, 21 Jul 2026 00:14:00 GMT
**Description:** Updating uncertainty with evidence while keeping assumptions and prior information visible.
**Publication:** `at://did:plc:gfrmhdmjvxn2sjedzboeudef/site.standard.publication/3mr4py6clps2f`
**Path:** /bayesian-inference
**Tags:** knowledge, concept, economics, statistics, bayesian-inference, uncertainty

**Content:**
```json
{
  "text": "Bayesian inference is a statistical method for updating uncertainty with observed evidence. It represents uncertain quantities with probability distributions, combines a prior distribution with a likelihood for the observed data, and produces a posterior distribution. The posterior records what the model implies after the evidence has been incorporated.\n\n## Prior, likelihood, and posterior\n\nThe **prior distribution** describes uncertainty before the current data are considered. It can encode substantive knowledge, regularize weakly identified parameters, or state broad uncertainty. A prior is part of the model and should be tested for how strongly it shapes the result.\n\nThe **likelihood** describes how probable the observed data would be under different parameter values or latent states. Bayes' rule combines the prior and likelihood into the **posterior distribution**.\n\nThis does not remove judgment from analysis. Choices about variables, data-generating processes, measurement, and functional form remain. Bayesian methods make some of those assumptions explicit and propagate them through the resulting uncertainty.\n\n## Prediction and model checking\n\nA posterior distribution describes uncertainty conditional on the model. It does not automatically describe uncertainty about whether the model itself is appropriate.\n\nPosterior predictive checking addresses that limitation by generating data from the fitted model and comparing those simulated observations with the real data. A model can estimate its own parameters precisely while still failing to reproduce important features of the world. Predictive discrepancies expose where the model's assumptions are inadequate.\n\n## Computation\n\nMost useful Bayesian models cannot be solved in closed form. Their posterior distributions are approximated with methods such as Markov chain Monte Carlo, variational inference, sequential Monte Carlo, or enumeration. [Probabilistic programming](https://cameron.stream/knowledge/probabilistic-programming) systems separate the model description from much of this inference machinery.\n\n[Cameron](https://cameron.stream/knowledge/cameron) has used Bayesian methods in economics and contributed to Turing.jl, a probabilistic programming system in Julia.\n\n## Sources\n\n- [Bayesian Data Analysis](https://sites.stat.columbia.edu/gelman/book/)\n- [Stan User's Guide](https://mc-stan.org/docs/stan-users-guide/)\n- [Turing.jl documentation](https://turinglang.org/docs/)",
  "$type": "site.standard.content.markdown",
  "version": "1.0"
}
```

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