Again, I run into the question that “what’s the difference between GP and GPs”? Here, GP is Gaussian process while GPs is Gaussian processes. So you see the difference?

In this post, I decided to solve this confusion (a better words? puzzle? bewilderment? perplexity?) once and for all.

According to Wikipedia:Gaussian_process:

… a Gaussian process is a stochastic process …

So, first, we can infer that GP is countable. (This is not silly.)

According to Wikipedia:Stochastic_process:

The term random function is also used to refer to a stochastic or random process,[26][27] because a stochastic process can also be interpreted as a random element in a function space.[28][29]

So, roughly speaking, a GP is a random function, such that every finite collection of those random variables has a multivariate normal distribution.

According to Wikipedia:Multivariate_normal_distribution:

multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution

结论（待进一步验证）：

随机变量 | GP |
---|---|

随机变量们 | GPs |

对一个确定值$x$进行$n$次观测，每次观测得到一个随机变量$\tilde{x}_i$ | 对一个函数$f$进行$N$次采样，每次采样得到是随机函数在某个点$x$的latent value+noise即$\tilde{f}_i=f_i+\epsilon_i$ |