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Consider a set of iid samples <math>\{X_i\}_{i=1}^N</math> where <math>X_i \in\mathbb{R}^n</math> is such that <math>X_i \sim N(\mu,\Sigma)</math>. Suppose we know <math>\Sigma</math>, but wish to estimate <math>\mu</math> using BPE. If we assume a prior distribution for the unknown mean to be distributed as a Gaussian random variable, we will obtain a posterior distribution for the mean which is also Gaussian, i.e. <math>p(\mu|X_1,X_2,\ldots,X_N) = N(\mu_N,\Sigma_N)</math>, where <math>\mu_N</math> and <math>\Sigma_N</math> are calculated to utilize both our prior knowledge of <math>\mu</math> and the samples <math>\{X_i\}_{i=1}^N</math>. Fukunaga p. 391 derives that the parameters <math>\mu_N</math> and <math>\Sigma_N</math> are calculated as follows: | Consider a set of iid samples <math>\{X_i\}_{i=1}^N</math> where <math>X_i \in\mathbb{R}^n</math> is such that <math>X_i \sim N(\mu,\Sigma)</math>. Suppose we know <math>\Sigma</math>, but wish to estimate <math>\mu</math> using BPE. If we assume a prior distribution for the unknown mean to be distributed as a Gaussian random variable, we will obtain a posterior distribution for the mean which is also Gaussian, i.e. <math>p(\mu|X_1,X_2,\ldots,X_N) = N(\mu_N,\Sigma_N)</math>, where <math>\mu_N</math> and <math>\Sigma_N</math> are calculated to utilize both our prior knowledge of <math>\mu</math> and the samples <math>\{X_i\}_{i=1}^N</math>. Fukunaga p. 391 derives that the parameters <math>\mu_N</math> and <math>\Sigma_N</math> are calculated as follows: | ||
| − | + | <math>\mu_N = \frac{\Sigma}{N}(\Sigma_\mu + \frac{\Sigma}{N})^{-1}\mu_0 + \Sigma_\mu(\Sigma_\mu + \frac{\Sigma}{N})^{-1}\left(\frac1N\sum_{i=1}^NX_i\right)</math>, | |
| − | where | + | where <math>\mu_0</math> is the initial "guess" for the mean <math>\mu</math>, and <math>\Sigma_\mu</math> is the "confidence" in that guess. In other words, we can consider that <math>N(\mu_0,\Sigma_\mu)</math> is the prior distribution for <math>\mu</math> that we would assume without seeing any samples. For the covariance parameter, we have |
| − | + | <math>\Sigma_N = \Sigma_0(\Sigma_0+\frac{\Sigma}{N})^{-1}\frac{\Sigma}{N}</math>. | |
| − | We find that as the number of samples increases, that the effect of the prior knowledge ( | + | We find that as the number of samples increases, that the effect of the prior knowledge (<math>\mu_0</math>,<math>\Sigma_\mu</math>) decreases so that |
| − | + | <math>\lim_{N\rightarrow\infty}\mu_N = \frac1N\sum_{i=1}^NX_i</math>, and <math>\lim_{N\rightarrow\infty}\Sigma_N = 0</math>. | |
- Estimation of covariance, given a known mean | - Estimation of covariance, given a known mean | ||
| − | Again, given iid samples | + | Again, given iid samples <math>\{X_i\}_{i=1}^N</math>, <math>X_i \in\mathbb{R}^n</math>, <math>X_i \sim N(\mu,\Sigma)</math>, let us now estimate <math>\Sigma</math> with <math>\mu</math> known. As in Fukinaga p. 392, we assume that both the posterior distribution of <math>\Sigma</math> is normal (i.e. <math>p(X|\Sigma) = N(\mu,\Sigma)</math>), and it can be shown that the sample covariance matrix follows a Wishart Distribution. Fukinaga p.392 shows the distribution <math>p(K|\Sigma_0,N_0)</math>, where <math>K = \Sigma^{-1}</math>, and parameter <math>\Sigma_0</math> represents the initial "guess" for <math>\Sigma</math> and <math>N_0</math> represents "how many samples were used to compute <math>\Sigma_0</math>". Note that we compute the distribution for <math>K = \Sigma^{-1}</math> instead of <math>\Sigma</math> directly, since the inverse covariance matrix is used in the definition for a normal distribution. It can be shown, then, that |
| − | | | + | <math>p(K|\Sigma_0,N_0) = c(n,N_0)\left|\frac12N_0\Sigma_0\right|^{(N_0-1)/2}|K|^{(N_0-n-2)/2}\exp(-\frac12\mathrm{trace}(N_0\Sigma_0K))</math>, |
| − | where | + | where <math>c(n,N_0) = \left\{\pi^{n(n-1)/4}\prod_{i=1}^n\Gamma\left(\frac{N_0-i}{2}\right)\right\}^{-1}</math>. |
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Revision as of 12:01, 6 March 2008
`BPE - Bayesian Parameter Estimation from Lecture 7 <https://engineering.purdue.edu/people/mireille.boutin.1/ECE301kiwi/Lecture7>`_
BPE FOR MULTIVARIATE GAUSSIAN : - Estimation of mean, given a known covariance Consider a set of iid samples $ \{X_i\}_{i=1}^N $ where $ X_i \in\mathbb{R}^n $ is such that $ X_i \sim N(\mu,\Sigma) $. Suppose we know $ \Sigma $, but wish to estimate $ \mu $ using BPE. If we assume a prior distribution for the unknown mean to be distributed as a Gaussian random variable, we will obtain a posterior distribution for the mean which is also Gaussian, i.e. $ p(\mu|X_1,X_2,\ldots,X_N) = N(\mu_N,\Sigma_N) $, where $ \mu_N $ and $ \Sigma_N $ are calculated to utilize both our prior knowledge of $ \mu $ and the samples $ \{X_i\}_{i=1}^N $. Fukunaga p. 391 derives that the parameters $ \mu_N $ and $ \Sigma_N $ are calculated as follows:
$ \mu_N = \frac{\Sigma}{N}(\Sigma_\mu + \frac{\Sigma}{N})^{-1}\mu_0 + \Sigma_\mu(\Sigma_\mu + \frac{\Sigma}{N})^{-1}\left(\frac1N\sum_{i=1}^NX_i\right) $,
where $ \mu_0 $ is the initial "guess" for the mean $ \mu $, and $ \Sigma_\mu $ is the "confidence" in that guess. In other words, we can consider that $ N(\mu_0,\Sigma_\mu) $ is the prior distribution for $ \mu $ that we would assume without seeing any samples. For the covariance parameter, we have
$ \Sigma_N = \Sigma_0(\Sigma_0+\frac{\Sigma}{N})^{-1}\frac{\Sigma}{N} $.
We find that as the number of samples increases, that the effect of the prior knowledge ($ \mu_0 $,$ \Sigma_\mu $) decreases so that
$ \lim_{N\rightarrow\infty}\mu_N = \frac1N\sum_{i=1}^NX_i $, and $ \lim_{N\rightarrow\infty}\Sigma_N = 0 $.
- Estimation of covariance, given a known mean Again, given iid samples $ \{X_i\}_{i=1}^N $, $ X_i \in\mathbb{R}^n $, $ X_i \sim N(\mu,\Sigma) $, let us now estimate $ \Sigma $ with $ \mu $ known. As in Fukinaga p. 392, we assume that both the posterior distribution of $ \Sigma $ is normal (i.e. $ p(X|\Sigma) = N(\mu,\Sigma) $), and it can be shown that the sample covariance matrix follows a Wishart Distribution. Fukinaga p.392 shows the distribution $ p(K|\Sigma_0,N_0) $, where $ K = \Sigma^{-1} $, and parameter $ \Sigma_0 $ represents the initial "guess" for $ \Sigma $ and $ N_0 $ represents "how many samples were used to compute $ \Sigma_0 $". Note that we compute the distribution for $ K = \Sigma^{-1} $ instead of $ \Sigma $ directly, since the inverse covariance matrix is used in the definition for a normal distribution. It can be shown, then, that
$ p(K|\Sigma_0,N_0) = c(n,N_0)\left|\frac12N_0\Sigma_0\right|^{(N_0-1)/2}|K|^{(N_0-n-2)/2}\exp(-\frac12\mathrm{trace}(N_0\Sigma_0K)) $,
where $ c(n,N_0) = \left\{\pi^{n(n-1)/4}\prod_{i=1}^n\Gamma\left(\frac{N_0-i}{2}\right)\right\}^{-1} $.
