{
  "schema": "ml-prep/item@1",
  "item": {
    "id": "original-ml-gmm",
    "area": "machine-learning",
    "topic": "unsupervised-learning",
    "origin": "original",
    "title": "Mixture responsibilities and EM",
    "skills": [
      "gaussian-mixtures",
      "expectation-maximization",
      "clustering-objective"
    ],
    "priority": "core",
    "difficulty": "medium",
    "estimated_minutes": 7,
    "prerequisites": [
      "bayes-rule"
    ],
    "prompt": "At a data point $x$, two Gaussian-mixture components have prior weights $0.4$ and $0.6$ and component densities $p(x\\mid z=1)=0.3$ and $p(x\\mid z=2)=0.1$. Compute posterior responsibilities and state the E- and M-step roles.\n",
    "answer": "The unnormalized weights are $0.12$ and $0.06$, so the responsibilities are $\\frac{2}{3}$ and $\\frac{1}{3}$. The E-step computes responsibilities; the M-step updates parameters using those expected assignments.\n",
    "check": {
      "kind": "numeric",
      "id": "original-ml-gmm",
      "values": [
        0.6666666666666666,
        0.3333333333333333
      ]
    }
  }
}