... proportionality¹

The variable $y$ is said to be proportional to the variable $x$, if there exists a non-zero number $k$ such that $y = kx$, the relation is often denoted $y \propto x$ and the constant ratio $k = y/x$, is called the constant of proportionality.

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... dimension²

A quantity ${\mbox{\boldmath $\theta$}}$ is said to be of dimension $d$ if it may equally well be written $\{\theta_1, \theta_2, \ldots, \theta_d\}$, i.e. it consists of $d$ distinct items.

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... statistic³

Let ${\mbox{\boldmath $x$}} = (x_1, \ldots, x_n)^T$ be a vector of observations from a distribution with parameters ${\mbox{\boldmath $\theta$}} = (\theta_1,\ldots, \theta_k)$. Let ${\mbox{\boldmath $t$}} = t_1,\ldots, t_q)$ be $q$ functions of ${\mbox{\boldmath $x$}}$. Then the set of statistics ${\mbox{\boldmath $t$}}$ is said to be sufficient for ${\mbox{\boldmath $\theta$}}$ if the likelihood function ${\it l}({\mbox{\boldmath $\theta$}}\vert {\mbox{\boldmath $x$}})$ can be expressed in the form

$${\it l}({\mbox{\boldmath $\theta$}}\vert {\mbox{\boldmath $... ...t g}({\mbox{\boldmath $\theta$}}\vert {\mbox{\boldmath $t$}})$$

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... candidate.⁴

If there exists a generated sequence of values and the current value is $X_i$, to generate $X_{i+1}$, the process is to generate a possible value for $X_{i+1}$, for example $y$, and then decide whether the chain moves to that value or stays in the current value. The value $y$ is called the candidate.

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... distribution⁵

The current value of the chain is called the state and the state space of the distribution is the space consisting of all possible values the distribution might take.

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... 0.01)⁶

rnorm(n, m, s) is the S-Plus command that generates $n$ random numbers mean $m$, standard deviation $s$ so this generates $1000$ samples using the MCMC sample from the posterior distribution of the mean as a means vector and the known parameter $\sigma$ as standard deviation.

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