The mixture target distribution is a weighted sum of multivariate normal distributions,<\/p>\n\n\n\\pi(x) = \\sum_{k=1}^n w_k N(\\mu_k,\\Sigma_k).<\/span>\n\n\n\n The weights are such that \\sum_{k=1}^n w_k =1<\/span>. For example, one can take w_k = \\frac{1}{n}<\/span>. Now I would like each of the covariance matrices to be oriented such that the principal eigenvector points towards the origin. We can achieve this by means of a change of basis,<\/p>\n\n\n\\Sigma_k = \\begin{pmatrix}\\cos \\left( \\frac{2\\pi k}{n}\\right) & -\\sin \\left( \\frac{2\\pi k}{n}\\right) \\\\ \\sin \\left( \\frac{2\\pi k}{n}\\right) & \\cos \\left( \\frac{2\\pi k}{n}\\right)\\end{pmatrix}\\begin{pmatrix} \\sigma_1^2 & 0 \\\\ 0 & \\sigma_2^2 \\end{pmatrix}\\begin{pmatrix}\\cos \\left( \\frac{2\\pi k}{n}\\right) & -\\sin \\left( \\frac{2\\pi k}{n}\\right) \\\\ \\sin \\left( \\frac{2\\pi k}{n}\\right) & \\cos \\left( \\frac{2\\pi k}{n}\\right)\\end{pmatrix}^{-1}.<\/span>\n\n\n\n Here’s an example:<\/p>\n\n\n\n I used this distribution to study the effectiveness of different MCMC tempering methods with multi-modal target distributions, which I wrote up as a short report in my first term at STOR-i. See below for the report, and I hope you find it as interesting as I did exploring these methods. <\/p>\n\n\n\n![\"\"](\"https:\/\/www.lancaster.ac.uk\/stor-i-student-sites\/malcolm-connolly\/wp-content\/uploads\/sites\/64\/2024\/12\/Rplot02-1024x688.png\")
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