Conformal Predictive System

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July 31, 2017, at 06:29 PM by Vovk - added universally consistent predictive distributions
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There are universally consistent predictive distributions (Vovk, 2017).
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Conformal predictive systems can be applied for the purpose of [[conformal decision making | decision making]].
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Conformal predictive systems can be applied for the purpose of [[conformal decision making | decision making]].  Universally consistent predictive distributions can be used for making asymptotically efficient decisions.
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* Vladimir Vovk, Jieli Shen, Valery Manokhin, and Min-ge Xie (2017).  [[http://www.alrw.net/articles/17.pdf | Nonparametric predictive distributions based on conformal prediction]].  On-line Compression Modelling Project (New Series), Working Paper 17, April 2017.
to:
* Vladimir Vovk, Jieli Shen, Valery Manokhin, and Min-ge Xie (2017).  [[http://www.alrw.net/articles/17.pdf | Nonparametric predictive distributions based on conformal prediction]].  On-line Compression Modelling Project (New Series), Working Paper 17, April 2017.
* Vladimir Vovk (2017).  [[http://www.alrw.net/articles/18.pdf | Universally consistent predictive distributions]].  On-line Compression Modelling Project (New Series), Working Paper 18
, April 2017.
July 31, 2017, at 05:04 PM by Vovk - added another class of conformal predictive distributions and decision making
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where $\hat y$ is the prediction for the label of $x$ based on the training sequence $z_1,\ldots,z_n$.
The width $p^y(1)-p^y(0)$ of such conformal predictive distributions is typically equal to $1/(n+1)$, where $n$ is the length of the training sequence, except for at most $n$ values of $y$.
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where $\hat y$ is the prediction for the label of $x$ based on the training sequence $z_1,\ldots,z_n$ and $(x,y)$.
An even wider class is
\[
  A((z_1,\ldots,z_n),(x,y)) := (y - \hat y)/\sigma_y,
\]
where $\sigma_y > 0$ is an estimate of the variability or difficulty of $y$ computed from the training sequence and $(x,y)$.  (The methods for computing $\hat y$ and $\sigma_y$ are supposed invariant with respect to permutations of $z_1,\ldots,z_n$.) 
The width $p^y(1)-p^y(0)$ of such conformal predictive distributions is typically equal to $1/(n+1)$, where $n$ is the length of the training sequence, except for at most $n$ values of $y$.
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!!Conformal decision making

Conformal predictive systems can be applied for the purpose of [[conformal decision making | decision making]].
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Conformal predictive systems are introduced in the recent technical report Vovk et al. (2017).  Essentially, these are [[conformal predictor | conformal transducers]] that, for each training sequence and each test object, output [[p-values]] $p^y$ that are increasing as a function of the label $y$, assumed to be a real number.  The function $y\mapsto p^y$ is then called a ''predictive distribution''.
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Conformal predictive systems are introduced in the recent technical report Vovk et al. (2017).  Essentially, these are [[conformal predictor | conformal transducers]] that, for each training sequence and each test object, output [[p-value]]s $p^y$ that are increasing as a function of the label $y$, assumed to be a real number.  The function $y\mapsto p^y$ is then called a ''predictive distribution''.
May 17, 2017, at 07:14 AM by Vovk - added two remarks
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Conformal predictive systems are introduced in the recent technical report Vovk et al. (2017).  Essentially, these are [[conformal predictor | conformal transducers]] that are increasing as function of the label $y$, assumed to be a real number.
to:
Conformal predictive systems are introduced in the recent technical report Vovk et al. (2017).  Essentially, these are [[conformal predictor | conformal transducers]] that, for each training sequence and each test object, output [[p-values]] $p^y$ that are increasing as a function of the label $y$, assumed to be a real number.  The function $y\mapsto p^y$ is then called a ''predictive distribution''.

A wide class of [[conformal predictor | conformity measures]] that often lead to conformal predictive systems is
\[
  A((z_1,\ldots,z_n),(x,y)) := y - \hat y,
\]
where $\hat y$ is the prediction for the label of $x$ based on the training sequence $z_1,\ldots,z_n$.
The width $p^y(1)-p^y(0)$ of such conformal predictive distributions is typically equal to $1/(n+1)$, where $n$ is the length of the training sequence, except for at most $n$ values of $y$.

The formal definition of conformal predictive systems takes account of the fact that, in the case of [[conformal predictor | smoothed conformal predictors]], $p^y$ also depends on the random number $\eta\in[0,1]$, and a fuller notation is $p^y(\eta)$.  It is also required that $p^y(0)\to0$ as $y\to-\infty$ and $p^y(1)\to1$ as $y\to\infty$.

Notice that in the context of conformal predictive systems the p-values acquire properties of probabilities.  Besides, they have some weak properties of [[conditionality | object conditionality]]: e.g., the ''central prediction regions'' $\{y\mid\epsilon/2\le p^y\le 1-\epsilon/2\}$ are not empty, except in very pathological cases
.
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Conformal predictive systems are introduced in the recent technical report Vovk et al. (2017).  Essentially, these are [[conformal prediction | conformal transducers]] that are increasing as function of the label $y$, assumed to be a real number.
to:
Conformal predictive systems are introduced in the recent technical report Vovk et al. (2017).  Essentially, these are [[conformal predictor | conformal transducers]] that are increasing as function of the label $y$, assumed to be a real number.
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* Vladimir Vovk, Jieli Shen, Valery Manokhin, and Min-ge Xie (2017).  [[http://www.alrw.net/articles/17.pdf | Nonparametric predictive distributions based on conformal prediction]].  On-line
Compression Modelling Project (New Series), Working Paper 17, April 2017.
to:
* Vladimir Vovk, Jieli Shen, Valery Manokhin, and Min-ge Xie (2017).  [[http://www.alrw.net/articles/17.pdf | Nonparametric predictive distributions based on conformal prediction]].  On-line Compression Modelling Project (New Series), Working Paper 17, April 2017.
May 13, 2017, at 11:20 AM by Vovk - created the page
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Conformal predictive systems are introduced in the recent technical report Vovk et al. (2017).  Essentially, these are [[conformal prediction | conformal transducers]] that are increasing as function of the label $y$, assumed to be a real number.

'''Bibliography'''
* Vladimir Vovk, Jieli Shen, Valery Manokhin, and Min-ge Xie (2017).  [[http://www.alrw.net/articles/17.pdf | Nonparametric predictive distributions based on conformal prediction]].  On-line
Compression Modelling Project (New Series), Working Paper 17, April 2017.