## Tokyo Probability Seminar

Seminar information archive ～09/18｜Next seminar｜Future seminars 09/19～

Date, time & place | Monday 16:00 - 17:30 126Room #126 (Graduate School of Math. Sci. Bldg.) |
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Organizer(s) | Makiko Sasada, Shuta Nakajima |

### 2023/08/07

17:00-18:30 Room #123 (Graduate School of Math. Sci. Bldg.)

Approximation of Random Variables by Elements that are independent of a given sigma algebra (English)

**Freddy Delbaen**(Professor emeritus at ETH Zurich)Approximation of Random Variables by Elements that are independent of a given sigma algebra (English)

[ Abstract ]

Given a square integrable m-dimensional random variable $X$ on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and a sub sigma algebra $\mathcal{A}$, we show that there exists another m-dimensional random variable $Y$, independent of $\mathcal{A}$ and minimising the $L^2$ distance to $X$. Such results have an importance to fairness and bias reduction in Artificial Intelligence, Machine Learning and Network Theory. The proof needs elements from transportation theory, a parametric version due to Dudley and Blackwell of the Skorohod theorem, selection theorems, … The problem also triggers other approximation problems. (joint work with C. Majumdar)

Given a square integrable m-dimensional random variable $X$ on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and a sub sigma algebra $\mathcal{A}$, we show that there exists another m-dimensional random variable $Y$, independent of $\mathcal{A}$ and minimising the $L^2$ distance to $X$. Such results have an importance to fairness and bias reduction in Artificial Intelligence, Machine Learning and Network Theory. The proof needs elements from transportation theory, a parametric version due to Dudley and Blackwell of the Skorohod theorem, selection theorems, … The problem also triggers other approximation problems. (joint work with C. Majumdar)