{
  "annotation_id": "58e5e07f-3ff1-478c-83c2-3a396ce740f9",
  "date_created": "2026-03-02T18:00:36.385000Z",
  "date_modified": "2026-03-02T18:00:36.385000Z",
  "file_hash": "a360a88df00bb72d1acff2efc3f504a6d0975a819f5bdd96388f4634f748c33d",
  "private": false,
  "record": {
    "abstract": "The main object of this paper is to present some general concepts of Bayesian\ninference and more specifically the estimation of the hyperparameters in\ninverse problems. We consider a general linear situation where we are given\nsome data $\\yb$ related to the unknown parameters $\\xb$ by $\\yb=\\Ab \\xb+\\nb$\nand where we can assign the probability laws $p(\\xb|\\thetab)$,\n$p(\\yb|\\xb,\\betab)$, $p(\\betab)$ and $p(\\thetab)$. The main discussion is then\nhow to infer $\\xb$, $\\thetab$ and $\\betab$ either individually or any\ncombinations of them. Different situations are considered and discussed. As an\nimportant example, we consider the case where $\\theta$ and $\\beta$ are the\nprecision parameters of the Gaussian laws to whom we assign Gamma priors and we\npropose some new and practical algorithms to estimate them simultaneously.\nComparisons and links with other classical methods such as maximum likelihood\nare presented. Keywords: Bayesian inference, Hyperparameter estimation, Inverse\nproblems, Maximum likelihood.",
    "arxiv_id": "physics/0111123",
    "authors": [
      "A. Mohammad-Djafari"
    ],
    "categories": [
      "physics.data-an"
    ],
    "title": "A full Bayesian approach for inverse problems",
    "url": "https://arxiv.org/abs/physics/0111123"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "0c2eff28-1a5b-46d8-906f-f83c724f450d",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}