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    "Bayesian Inference\n",
    "==================\n",
    "\n",
    "Meaning of Probability in Bayesian Inference\n",
    "--------------------------------------------\n",
    "\n",
    "In the frequentist approach, probability is a limiting frequency. Thus,\n",
    "probabilities are always associated with random events. In Bayesian\n",
    "inference, on the other hand, probability is used to quantify your\n",
    "belief that an unobservable variable has a particular value. For example\n",
    "a Bayesian can ask questions such as:\n",
    "\n",
    "-   What is the probability that heritability for milk yield is larger\n",
    "    than 0.5?\n",
    "\n",
    "-   What is the probability that variability in milk yield is due to\n",
    "    more than 100 loci?\n",
    "\n",
    "These Bayesian probabilities are not necessarily associated with a\n",
    "random experiment that assigns values to the variables in question.\n",
    "\n",
    "Essential Elements of Bayesian Inference\n",
    "----------------------------------------\n",
    "\n",
    "-   Bayesian inference starts by specifying what you believe about the\n",
    "    parameters or unknowns through prior probabilities. In whole-genome\n",
    "    analyses, we will use a prior probability density to quantify our\n",
    "    belief that the effect of most marker covariates is zero or close to\n",
    "    zero and only a few covariates have effects that deviate from zero.\n",
    "\n",
    "-   These parameters are related to the data through the model or\n",
    "    “likelihood”, which are conditional probabilities for the data given\n",
    "    the parameters. In whole-genome analyses, this is usually a multiple\n",
    "    regression model with normally distributed residuals.\n",
    "\n",
    "-   The prior and the likelihood are combined using Bayes theorem to\n",
    "    obtain posterior probabilities, which are conditional probabilities\n",
    "    for the parameters given the data.\n",
    "\n",
    "-   Inferences about the parameters are based on the posterior.\n",
    "\n",
    "Use of Bayes Theorem\n",
    "--------------------\n",
    "\n",
    "-   Let $f(\\mathbf{\\theta)}$ denote the prior probability density\n",
    "    for $\\mathbf{\\theta}$.\n",
    "\n",
    "-   Let\n",
    "    $f(\\mathbf{y}|{\\mathbf{\\theta}})$\n",
    "    denote the likelihood\n",
    "\n",
    "-   Then, the posterior probability of $\\mathbf{\\theta}$ is:  \n",
    "\n",
    "\n",
    "$$\\begin{eqnarray}    \n",
    "f({\\mathbf \\theta}|{\\mathbf y}) \n",
    "& =\\frac{f({\\mathbf y}|{\\mathbf \\theta})f({\\mathbf \\theta})}{f({\\mathbf y})}\\\\\n",
    "& \\propto f({\\mathbf y}|{\\mathbf \\theta})f({\\mathbf \\theta})\n",
    "\\end{eqnarray}$$ \n"
   ]
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