Let me throw in a few points in addition to Bogdanovist's answer

As you say, you train $k$ different models. They differ in that 1/(k-1)th of the training data is exchanged against other cases. These models are sometimes called surrogate models because the (average) performance measured for these models is taken as a surrogate of the performance of the model trained on all cases.

Now, there are some assumptions in this process.

• Assumption 1: the surrogate models are equivalent to the "whole data" model.
  It is quite common that this assumption breaks down, and the symptom is the well-known pessimistic bias of $k$-fold cross validation (or other resampling based validation schemes). The performance of the surrogate models is on average worse than the performance of the "whole data" model if the learning curve has still a positive slope (i.e. less training samples lead to worse models).

• Assumption 2 is a weaker version of assumption 1: even if the surrogate models are on average worse than the whole data model, we assume them to be equivalent to each other. This allows summarizing the test results for $k$ surrogate models as one average performance.
  Model instability leads to the breakdown of this assumption: the true performance of models trained on $N \frac{k - 1}{k}$ training cases varies a lot. You can measure this by doing iterations/repetitions of the $k$-fold cross validation (new random assignments to the $k$ subsets) and looking at the variance (random differences) between the predictions of different surrogate models for the same case.

• The finite numer of cases means the performance measurement will be subject to a random error (variance) due to the finite number of test cases. This source of variance is different (and thus adds to) from the model instablilty variance.

The differences in the observed performance are due to these two sources of variance.

The "selection" you think about is a data set selection: selecting one of the surrogate models means selecting a subset of training samples and claiming that this subset of training samples leads to a superior model. While this may be truely the case, usually the "superiority" is spurious. In any case, as picking "the best" of the surrogate models is a data-driven optimization, you'd need to validate (measure performance) this picked model with new unknown data. The test set within this cross validation is not independent as it was used to select the surrogate model.

You may want to look at our paper, it is about classification where things are usually worse than for regression. However, it shows how these sources of variance and bias add up.
Beleites, C. and Neugebauer, U. and Bocklitz, T. and Krafft, C. and Popp, J.: Sample size planning for classification models. Anal Chim Acta, 2013, 760, 25-33.
DOI: 10.1016/j.aca.2012.11.007
accepted manuscript on arXiv: 1211.1323

Let me throw in a few points in addition to Bogdanovist's answer

As you say, you train $k$ different models. They differ in that 1/(k-1)th of the training data is exchanged against other cases. These models are sometimes called surrogate models because the (average) performance measured for these models is taken as a surrogate of the performance of the model trained on all cases.

Now, there are some assumptions in this process.

• Assumption 1: the surrogate models are equivalent to the "whole data" model.
  It is quite common that this assumption breaks down, and the symptom is the well-known pessimistic bias of $k$-fold cross validation (or other resampling based validation schemes). The performance of the surrogate models is on average worse than the performance of the "whole data" model if the learning curve has still a positive slope (i.e. less training samples lead to worse models).

• Assumption 2 is a weaker version of assumption 1: even if the surrogate models are on average worse than the whole data model, we assume them to be equivalent to each other. This allows summarizing the test results for $k$ surrogate models as one average performance.
  Model instability leads to the breakdown of this assumption: the true performance of models trained on $N \frac{k - 1}{k}$ training cases varies a lot. You can measure this by doing iterations/repetitions of the $k$-fold cross validation (new random assignments to the $k$ subsets) and looking at the variance (random differences) between the predictions of different surrogate models for the same case.

• The finite number of cases means the performance measurement will be subject to a random error (variance) due to the finite number of test cases. This source of variance is different from (and thus adds to) the model instablilty variance.

The differences in the observed performance are due to these two sources of variance.

The "selection" you think about is a data set selection: selecting one of the surrogate models means selecting a subset of training samples and claiming that this subset of training samples leads to a superior model. While this may be truely the case, usually the "superiority" is spurious. In any case, as picking "the best" of the surrogate models is a data-driven optimization, you'd need to validate (measure performance) this picked model with new unknown data. The test set within this cross validation is not independent as it was used to select the surrogate model.

You may want to look at our paper, it is about classification where things are usually worse than for regression. However, it shows how these sources of variance and bias add up.
Beleites, C. and Neugebauer, U. and Bocklitz, T. and Krafft, C. and Popp, J.: Sample size planning for classification models. Anal Chim Acta, 2013, 760, 25-33.
DOI: 10.1016/j.aca.2012.11.007
accepted manuscript on arXiv: 1211.1323

Let me throw in a few points in addition to Bogdanovist's answer

As you say, you train $k$ different models. They differ in that 1/(k-1)th of the training data is exchanged against other cases. These models are sometimes called surrogate models because the (average) performance measured for these models is taken as a surrogate of the performance of the model trained on all cases.

Now, there are some assumptions in this process.

• Assumption 1: the surrogate models are equivalent to the "whole data" model.
  It is quite common that this assumption breaks down, and the symptom is the well-known pessimistic bias of $k$-fold cross validation (or other resampling based validation schemes). The performance of the surrogate models is on average worse than the performance of the "whole data" model if the learning curve has still a positive slope (i.e. less training samples lead to worse models).

• Assumption 2 is a weaker version of assumption 1: even if the surrogate models are on average worse than the whole data model, we assume them to be equivalent to each other. This allows summarizing the test results for $k$ surrogate models as one average performance.
  Model instability leads to the breakdown of this assumption: the true performance of models trained on $N \frac{k - 1}{k}$ training cases varies a lot. You can measure this by doing iterations/repetitions of the $k$-fold cross validation (new random assignments to the $k$ subsets) and looking at the variance (random differences) between the predictions of different surrogate models for the same case.

• The finite numer of cases means the performance measurement will be subject to a random error (variance) due to the finite number of test cases. This source of variance is different (and thus adds to) from the model instablilty variance.

The differences in the observed performance are due to these two sources of variance.

The "selection" you think about is a data set selection: selecting one of the surrogate models means selecting a subset of training samples and claiming that this subset of training samples leads to a superior model. While this may be truely the case, usually the "superiority" is spurious. In any case, as picking "the best" of the surrogate models is a data-driven optimization, you'd need to validate (measure performance) this picked model with new unknown data. The test set within this cross validation is not independent as it was used to select the surrogate model.

You may want to look at our paper, it is about classification where things are usually worse than for regression. However, it shows how these sources of variance and bias add up.
Beleites, C. and Neugebauer, U. and Bocklitz, T. and Krafft, C. and Popp, J.: Sample size planning for classification models. Anal Chim Acta, 2013, 760, 25-33.
DOI: 10.1016/j.aca.2012.11.007
accepted manuscript on arXiv: 1211.1323

Let me throw in a few points in addition to Bogdanovist's answer

As you say, you train $k$ different models. They differ in that 1/(k-1)th of the training data is exchanged against other cases. These models are sometimes called surrogate models because the (average) performance measured for these models is taken as a surrogate of the performance of the model trained on all cases.

Now, there are some assumptions in this process.

• Assumption 1: the surrogate models are equivalent to the "whole data" model.
  It is quite common that this assumption breaks down, and the symptom is the well-known pessimistic bias of $k$-fold cross validation (or other resampling based validation schemes). The performance of the surrogate models is on average worse than the performance of the "whole data" model if the learning curve has still a positive slope (i.e. less training samples lead to worse models).

• Assumption 2 is a weaker version of assumption 1: even if the surrogate models are on average worse than the whole data model, we assume them to be equivalent to each other. This allows summarizing the test results for $k$ surrogate models as one average performance.
  Model instability leads to the breakdown of this assumption: the true performance of models trained on $N \frac{k - 1}{k}$ training cases varies a lot. You can measure this by doing iterations/repetitions of the $k$-fold cross validation (new random assignments to the $k$ subsets) and looking at the variance (random differences) between the predictions of different surrogate models for the same case.

• The finite numer of cases means the performance measurement will be subject to a random error (variance) due to the finite number of test cases. This source of variance is different (and thus adds to) from the model instablilty variance.

The differences in the observed performance are due to these two sources of variance.

The "selection" you think about is a data set selection: selecting one of the surrogate models means selecting a subset of training samples and claiming that this subset of training samples leads to a superior model. While this may be truely the case, usually the "superiority" is spurious. In any case, as picking "the best" of the surrogate models is a data-driven optimization, you'd need to validate (measure performance) this picked model with new unknown data. The test set within this cross validation is not independent as it was used to select the surrogate model.

You may want to look at our paper, it is about classification where things are usually worse than for regression. However, it shows how these sources of variance and bias add up.
Beleites, C. and Neugebauer, U. and Bocklitz, T. and Krafft, C. and Popp, J.: Sample size planning for classification models. Anal Chim Acta, 2013, 760, 25-33.
DOI: 10.1016/j.aca.2012.11.007
accepted manuscript on arXiv: 1211.1323