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    "topic": "Model selection",
    "year": 1978,
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        "description": "The model-selection penalty is an asymptotic approximation under the stated finite-dimensional regular model and prior assumptions. It is not an exact finite-sample evidence value or a universal rule for singular, high-dimensional or misspecified models. Prior and parameter-count conventions matter. Later generalized criteria retain separate credit.",
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      "statement": "# Schwarz: regular-interior Bayesian dimension penalty\n\nSchwarz derived a dimension-dependent log-sample penalty for asymptotic Bayesian model comparison in regular finite-dimensional settings, giving the criterion now called BIC.\n\nFitting more parameters usually improves likelihood, but it also spreads prior probability over more possible parameter values. Schwarz compared models through integrated likelihood. Near a regular interior best fit, a k-dimensional neighborhood contracts at the scale n to the power minus k over two. Its logarithm contributes a penalty of k log n over two to the fitted log likelihood. This connects model dimension to Bayesian evidence rather than to the predictive information objective of Akaike’s criterion. The original analysis focuses on independent, identically distributed observations and linear submodels of an exponential family with proper, locally positive priors. The scope needs care: its printed fixed-statistic proposition also invokes an attained interior maximum. A Bernoulli sample containing only zeros has its best fit at a boundary; an exactly integrable proper prior then produces a different log-sample penalty. This boundary example restricts the proposition’s literal scope, while leaving the familiar regular-interior construction intact. Singular or growing-dimensional models require separate analysis.\n\n## Scientific scope\n\nThe model-selection penalty is an asymptotic approximation under the stated finite-dimensional regular model and prior assumptions. It is not an exact finite-sample evidence value or a universal rule for singular, high-dimensional or misspecified models. Prior and parameter-count conventions matter. Later generalized criteria retain separate credit.\n\n## People and systems\n\nGideon Schwarz\n\n\n\n## Original sources\n\n- [Gideon Schwarz, Estimating the Dimension of a Model, original Annals paper, CMU copy](https://www.andrew.cmu.edu/user/kk3n/simplicity/schwarzbic.pdf)\n- [IMS Crossref deposit for Schwarz1978 dimension-estimation paper](https://api.crossref.org/works/10.1214%2Faos%2F1176344136)\n",
      "summary": "Schwarz derived a dimension-dependent log-sample penalty for asymptotic Bayesian model comparison in regular finite-dimensional settings, giving the criterion now called BIC.\n\nFitting more parameters usually improves likelihood, but it also spreads prior probability over more possible parameter values. Schwarz compared models through integrated likelihood. Near a regular interior best fit, a k-dimensional neighborhood contracts at the scale n to the power minus k over two. Its logarithm contributes a penalty of k log n over two to the fitted log likelihood. This connects model dimension to Bayesian evidence rather than to the predictive information objective of Akaike’s criterion. The original analysis focuses on independent, identically distributed observations and linear submodels of an exponential family with proper, locally positive priors. The scope needs care: its printed fixed-statistic proposition also invokes an attained interior maximum. A Bernoulli sample containing only zeros has its best fit at a boundary; an exactly integrable proper prior then produces a different log-sample penalty. This boundary example restricts the proposition’s literal scope, while leaving the familiar regular-interior construction intact. Singular or growing-dimensional models require separate analysis.",
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