How does Gibbs sampling produce values for a variable using the univariate conditional probability? I have a question about Gibbs sampling for generating samples. The Gibbs sampling algorithm is often stated.


*

*$x^0 = (x_1^0, x_2^0, \ldots, x_n^0)$ //initialize random values

*for $t=1$ in $T$ //iterate to generat T samples


*

*$x_1^t \sim P(X_1 | x_2^{t-1}, x_3^{t-1}, \ldots, x_n^{t-1})$ //compute univariate conditional probability

*$x_2^t \sim P(X_2 | x_1^{t}, x_3^{t-1}, \ldots, x_n^{t-1})$

*$\vdots$

*$x_n^t \sim P(X_n | x_1^t, x_2^t, \ldots, x_{n-1}^{t})$



How do we actually compute the univariate conditional probability? If we assume a multivariate gaussian probability distribution for the variables, would this not be an expensive operation?
For example, if we wanted to compute the univariate conditional probability for $x_1$, that is defined by the following.
$
\begin{eqnarray*}
x_1^t & = & P(X_1 | x_2^{t-1}, x_3^{t-1}, \ldots, x_n^{t-1}) 
 & = & f(x_2,\ldots,x_n) 
 & = &
\frac{1}{\sqrt{(2\pi)^{n-1}|\boldsymbol\Sigma|}}
\exp\left(-\frac{1}{2}({\mathbf x}-{\boldsymbol\mu})^\mathrm{T}{\boldsymbol\Sigma}^{-1}({\mathbf x}-{\boldsymbol\mu})
\right)
\end{eqnarray*}
$
Now, $\Sigma$ is the covariance matrix between $X_2, \ldots, X_n$ (if I understand correctly, it does not include covariance with $X_1$). This covariance matrix will be different for each $X_i$ (it will always be between the variables in the set, $X \setminus X_i$.
And then I also noticed $|\boldsymbol\Sigma|$, which is the determinant of the covariance matrix. If I understand correctly, this determinant too have to be computed for each $\Sigma$ associated with a $X_i$. 
Is my understanding about the covariance matrix and its determinant being unique for each $X_i$? 
Lastly, when I am computing the univariate conditional probability (e.g. using a multivariate gaussian distribution), I get back a value $x_i^t$, which is the value for $X_i$ in the next iteration. But what I have a problem understanding is that $x_i^t$ is a probability of $X_i$ and not a value. It doesn't seem that $x_i^t$ will be in the domain of $X_i$ since it is a probability value in the space [0, 1], and $X_i$ might have a domain $[\infty,-\infty]$.
Am I misunderstanding Gibbs sampling for variables with a multivariate gaussian distribution with respect the covariance matrix and univariate conditional probability? 
If I understand correctly, for $\Sigma$, I suppose I can compute it once for $X_1, \ldots, X_n$ and then create the $\Sigma_i$ and pre-compute $|\Sigma_i|$ once. 
 A: You raised two broad questions


*

*How expensive is Gibbs sampling?

*In your multivariate Gaussian example, the domains don't seem to match up, since the values obtained via Gibbs sampling seem to be probabilities.


I will address the second question first.
Your interpretation of how the Gibbs sampling works is incorrect. Consider the simple example a random variable having a standard Gaussian distribution. That is $X \sim N(0,1)$. Then the density of $X$ is
$$f_X(x) = \dfrac{1}{\sqrt{2\pi}} e^{-x^2/2}. $$
When you draw an observation $x$ from the normal distribution, you don't have $x = f_X(x)$, but rather the $x$ drawn satisfies that density equation. So $x$ will not be in $[0,1]$, but rather since the normal variable is defined on all $\mathbb{R}$, $x$ could theoretically lie anywhere on the real line. A draw from a Normal distribution (multivariate or not) can be made using existing software (for example in R using rnorm). The density of the standard normal random variable looks life.

This density means that most sampled points are going to be between -2 and 2, even though the height of the function does not surpass $.5$. (Another thing, for continuous random variables, it is possible for the density function $f_X(x)$ to be larger than one, for example, the density of the N$(0, .2)$ as shown below, where the density function reaches 2.)

So basically, drawing points from a distribution is different from evaluating the density function. Gibbs draws points from a distribution. Once you have a closed form expression of the conditional distribution, you can use existing software in most case. Often the full conditionals are Normal, Gamma, Inverse Gamma, $\chi^2$, Inverse Gaussian, Beta etc.
To address your first question about how computationally intensive a Gibbs sampler can be, the answer would change from problem to problem. Sometimes a full conditional distribution requires an update from say a normal distribution that looks like
$$X_1|X_{-1}, \sigma^2,y \sim N(y^TA^{-1}y, \sigma^2), $$
where say $A$ is a large dimensional square matrix. Then every Gibbs update update requires a large matrix inverse, which will be expensive. However, in most such cases a Gibbs update will still be cheaper than a Metropolis-Hastings algorithm (since that requires 2 likelihood evaluations). In addition, some cases Gibbs updates can also be fairly cheap, so it depends from case to case really.

EDIT:
In your example, to draw from $f(x_1^T|x_2^{t-1}, \dots, x_{n}^{t-1})$ you can use existing functions in programming languages (R, python, C, C++, Java) that let you draw from known distributions like Normal, Gamma, Inverse Gamma, Beta, Poisson, Binomial, etc. Basically, the way the algorithm works is after drawing $x_0$ you have observed the first $n$ $x$s. For $t = 1$, when you update $x_1^{1}$ you use the other $x_0$ values to get $x_1^{1}$, using the $f(x_1^T|x_2^{t-1}, \dots, x_{n}^{t-1})$ distribution. So for example you can initialize
$$x_0 \sim N(0, I_n), $$
and then to update you have the conditional density for $x_1$ as
$$x_1^{1} \sim N\left( \left(\begin{array}{c}x_2^0 \\x_3^0 \\ \vdots \\x_n^0   \end{array}\right)  , \Sigma \right)$$
You can draw from this distribution just from the programs since you know $(x_2^0, x_3^0, \dots, x_n^0)$.
A: When you use a Gibbs sampler you draw from the conditional distribution. The symbol '$\sim$' means 'distributed as'. So your sampling scheme means that you draw $x_1 ^t$ from it's conditional distribution given all the other variables, then you draw $x_2 ^t$ from its distribution conditional on the other variables (including the $x_t ^t$ which we just drew). Also even if you did have to calculate a lot of multivariate normal densities as long as you don't have a very large number of variables the computational cost is probably not going to be that high.
