Conjugate Prior.ipynb

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   "source": [
    "# Posterior probability using Normal distribution in conjugate bayesian framework\n",
    "\n",
    "- For more information, check http://en.wikipedia.org/wiki/Conjugate_prior"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Load data"
   ]
  },
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   "metadata": {},
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     "name": "stdout",
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      "\n"
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   "source": [
    "load('DOAdata.mat','DOAs')"
   ]
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      "\n",
      "p_test =\n",
      "\n",
      "    0.0627\n",
      "\n",
      "\n"
     ]
    }
   ],
   "source": [
    "n = size(DOAs,1);   % number of observations\n",
    "d = size(DOAs,2);   % number of dimension\n",
    "\n",
    "mu0 = [27.4 62.1];\n",
    "sigma0 = [2.47 0.97;0.97 0.75];\n",
    "\n",
    "sigma = cov(DOAs);  % estimated covariance matrix from data\n",
    "xx = mean(DOAs,1);  % sample mean\n",
    "\n",
    "mu_post = inv(inv(sigma0)+n*inv(sigma))*(inv(sigma0)*mu0'+n*inv(sigma)*xx');\n",
    "sigma_post = inv(inv(sigma0)+n*inv(sigma)) + sigma;\n",
    "\n",
    "doa_test = [26.0 61.0]'; \n",
    "nu0 = 1000;     % large number\n",
    "df = nu0 + n - d + 1;\n",
    "alpha = df;\n",
    "lambda = inv(sigma_post);\n",
    "\n",
    "% multivariate student distribution (Eqn. 3.20 Webb-Copsey page:81))\n",
    "p_test = gammaln(0.5*(alpha+d)) - gammaln(alpha/2) - (d/2)*log(alpha*pi) + ...\n",
    "    0.5*log(det(lambda)) - ((alpha+d)/2)* log(1+(1/alpha)*(doa_test-mu_post)'*lambda*(doa_test-mu_post));\n",
    "\n",
    "p_test = exp(p_test)   % p_test = 0.0627"
   ]
  }
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