Frequentist vs Bayesian Statistics

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## Frequentist vs Bayesian Statistics

#### Opening Talk of MAC Seminar of Statistics

### Sothea .textsc[Has], PhD

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## What's statistics?
- Statistics is a branch of mathematics that deals with .stress[collecting], .stress[organizing], .stress[analyzing], .stress[interpreting], .stress[presenting] data, .stress[designing experiments/surveys], and .stress[drawing valid conclusions from data].

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## What's statistics?
- Statistics is a branch of mathematics that deals with .stress[collecting], .stress[organizing], .stress[analyzing], .stress[interpreting], .stress[presenting] data, .stress[designing experiments/surveys], and .stress[drawing valid conclusions from data]. <h0br>

## Why do we need it? <hbr>

- Decision making

- Controlling risks

- Maximizing profits

- Comprehension...

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<img src="https://d1jnx9ba8s6j9r.cloudfront.net/blog/wp-content/uploads/2019/02/Statistics-Applications-Math-And-Statistics-For-Data-Science-Edureka-768x484.png" align="middle" height="230">
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## Consider an example <hbr>

🪙 Coin toss: `$n=100$` and observed `$55$` Heads & `$45$` Tails.
--

- What's your estimate of `$p=\mathbb{P}(\text{Head})$`? <hbr>

> 🔑 There exists a .stress[fixed] parameter `$p$` generating the observations.

- Statistical model: triple `$(S, {\cal A}, {\cal P})$`
 - `$S=\{0,1\}$`: sample space
 - `${\cal A}=\{S,\{1\},\{0\},\emptyset\}$`: a `$\sigma$`-algebra or tribu on `$S$`
 - `${\cal P}=\{\mathbb{P}_{p}:p\in [0,1]\}$`: a family of probability distributions ([Bernoulli](https://en.wikipedia.org/wiki/Bernoulli_distribution)): <hbr>

`$$\forall p \in [0,1]: \mathbb{P}_p(X=1)=p=1-\mathbb{P}_p(X=0).$$`
> ⚠️ This model is .stress[identifiable] i.e., `$p\to\mathbb{P}_p$` is one-to-one.

<hbr>

- Observations: `$X_1,...,X_n\overset{\text{iid}}{\sim} \mathbb{P}_p$` for some .stress[fixed] `$p\in [0,1]$`.
--

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## Consider an example <hbr>

<hbr>

### Likelihood method <hbr>

- Likelihood: probab of observing `$H=55$` and `$T=45$`.

`$$\begin{align}L(p)&=\mathbb{P}_p(X_1=x_1,...,X_{100}=x_{100})\\
&\overset{\text{iid}}{=}\prod_{i=1}^n\mathbb{P}_p(X_i=x_i)\\
&=\left(\prod_{i:x_i=1}\mathbb{P}_p(X_i=x_i)\right)\left(\prod_{i:x_i=0}\mathbb{P}_p(X_i=x_i)\right)\\
&=p^{55}(1-p)^{45}\end{align}$$`

]

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## Consider an example <hbr>

<hbr>

### Likelihood method <hbr>

- Likelihood: probab of observing `$H=55$` and `$T=45$`.

`$$\begin{align}L(p)&=p^{55}(1-p)^{45}\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\ \ \ { }\end{align}$$`
]

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## Consider an example <hbr>

<hbr>

### Likelihood method <hbr>

- Likelihood: probab of observing `$H=55$` and `$T=45$`.

`$$\begin{align}L(p)&=p^{55}(1-p)^{45}\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\ \ \ { }\end{align}$$`

]

- `$L(p)$` maximized at `$\hat{p}=0.55$` (.stress[Maximum Likelihood Estimator]). <hbr>

<hbr>

### How confident are we? <hbr>

- Confidence interval at `$\beta=0.95$` i.e., `$\mathbb{P}[p\in\text{CI}(\beta)]=\beta$`. Estimated by: <hbr>

`$$\text{CI}(0.95)=\left[\hat{p}-\sqrt{\hat{p}(1-\hat{p})}\frac{1.96}{\sqrt{n}},\hat{p}+\sqrt{\hat{p}(1-\hat{p})}\frac{1.96}{\sqrt{n}}\right]=[0.452, 0.648]$$`
<hbr>

----

<h1br>

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## Consider an example <hbr>
.pull-left-60[ <hbr>
- Likelihood estimator: `$\hat{p}=0.55$`.

- Estimated `$\text{CI}(0.95)=[0.452,0.648]$`.
]

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## Consider an example <hbr>
.pull-left-60[ <hbr>
- Likelihood estimator: `$\hat{p}=0.55$`.

- Estimated `$\text{CI}(0.95)=[0.452,0.648]$`.
]
.pull-right-40[ <h1br>
💡 .stress[If we repeat this ] `$100$` .stress[times, we expect that] `$p\in\text{CI}(0.95)$` .stress[around 95 times].

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## Consider an example <hbr>
.pull-left-60[ <hbr>
- Likelihood estimator: `$\hat{p}=0.55$`.

- Estimated `$\text{CI}(0.95)=[0.452,0.648]$`.
]
.pull-right-40[ <h1br>
💡 .stress[If we repeat this ] `$100$` .stress[times, we expect that] `$p\in\text{CI}(0.95)$` .stress[around 95 times].

⚠️ The only source of uncertainty is the data `$X_j$`s.
]

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# Frequentist approach <hbr>

- Key points: 
> - There's a fixed parameter producing the observations.
  - Suitable for long run with large observations.
  - The uncertainty is due to sampling error only.
--

- Statistical model: `$(S, {\cal A}, {\cal P})$` 
- Estimation: maximum likelihood, moment method,...
- Uncertainty analysis: test or confidence interval,...
- Interpretation, inference, conclusion. <hbr>

### Challenge <hbr>

>  Propose a statistical model for **hatched eggs** 🐣, **birth weights** 👶 and **dice game** 🎲.

???

- `$S=(0,\infty)$`
- `${\cal A}={\cal B}(S)$`
- `$\mathbb{P}_{\theta}={\cal N}$` or `$\chi^2$`.

### 🤓 What a statistician might do

- Probablity: `$X_1,...,X_N\sim^{iid}{\cal B}(p), i.e., X_i=\begin{cases}1&\text{Head, prob }p\\ 0&\text{Tail, prob }1-p\end{cases}$`

- Likelihood: `$L(p|x_1,...,x_N):=\mathbb{P}_p(\text{Observing such observations})$`

- Find `$\hat{p}:=\arg\max_{p}L(p|X_1,...,X_N)$`

- Confidence interval `$\text{CI}(\alpha)$`: `$\mathbb{P}(p\in \text{CI}(\alpha))\geq \alpha$`.

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## Recall Bayes's formula <hbr>
- If `$O,H$` are two events with `$\mathbb{P}(O)\times\mathbb{P}(H)>0$`, then
.pull-left[ <hbr>
`$$\underbrace{\mathbb{P}(H|O)}_{\text{Posterior}}=\frac{\overbrace{\mathbb{P}(O|H)}^{\text{Likelihood}}\times\overbrace{\mathbb{P}(H)}^{\text{Prior}}}{\underbrace{\mathbb{P}(O)}_{\text{Marginal}}}$$`

.right[[<img src="https://upload.wikimedia.org/wikipedia/commons/e/e1/ThomasBayes.png" align="middle" height="250">](https://upload.wikimedia.org/wikipedia/commons/e/e1/ThomasBayes.png)]

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.center[<img src="img/Bayes.jpg" align="middle" height="150">] <hbr>

.left[[<img src="img/Bayes_info.png" align="middle" height="250">](https://en.wikipedia.org/wiki/Thomas_Bayes)]
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## Recall Bayes's formula <hbr>
- If `$O,H$` are two events with `$\mathbb{P}(O)\times\mathbb{P}(H)>0$`, then
.pull-left[ <hbr>
`$$\underbrace{\mathbb{P}(H|O)}_{\text{Posterior}}=\frac{\overbrace{\mathbb{P}(O|H)}^{\text{Likelihood}}\times\overbrace{\mathbb{P}(H)}^{\text{Prior}}}{\underbrace{\mathbb{P}(O)}_{\text{Marginal}}}$$`
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<h1br>

### Explanation: <hbr>

> - Prior `$\mathbb{P}(H)$`: our initial belief in the .stress[Hypothesis] `$H$`. <vbr>
- Likelihood `$\mathbb{P}(O|H)$`: given `$H$`, how likely for `$O$` to occur. <vbr>
- Marginal `$\mathbb{P}(O)$`: chance that `$O$` would occur in general. <vbr>
- Posterior `$\mathbb{P}(H|O)$`: `$O$` is observed, how likely for `$H$` to occur?

---
## Covid test accuracy during the peak <hbr>
.pull-left[
- During the .stress[peak]: Cov+ `$=2\%$`, how one is likely to be infected if he/she was tested positive?]
.pull-right[
.center[<img src="img/covid1.jpg" align="middle" height="140" width="300">]
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`$$\begin{align}\mathbb{P}(\text{Cov}+|T+)&=\frac{\mathbb{P}(T+|\text{Cov}+)\mathbb{P}(\text{Cov}+)}{\mathbb{P}(T+)}\\
&=\frac{\mathbb{P}(T+|\text{Cov}+)\mathbb{P}(\text{Cov}+)}{\mathbb{P}(T+|\text{Cov}+)\mathbb{P}(\text{Cov}+)+\mathbb{P}(T+|\text{Cov}-)\mathbb{P}(\text{Cov}-)}\\
&=\frac{1\times 0.02}{1\times 0.02+0.001\times 0.98}\approx 0.95\end{align}$$`

- I think we can trust the test! <img src="https://t4.ftcdn.net/jpg/03/23/71/91/240_F_323719173_LcDYRnQuNiaBrRmRzsY4u6JWjj4IjvRv.jpg" align="middle" width="40">

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## Covid test during quarantine <hbr>
.pull-left[
- During .stress[quarantine]: Cov+ `$=0.2\%$`, how one is likely to be infected if he/she was tested positive?]
.pull-right[
.center[<img src="img/covid2.jpg" align="middle" height="140" width="300">]
]

`$$\begin{align}\mathbb{P}(\text{Cov}+|T+)&=\frac{1\times 0.002}{1\times 0.002+0.001\times 0.998}\approx 0.667\end{align}$$`
--

- .stress[False positive] `$\mathbb{P}(\text{Cov}-|T+)\approx 1/3$`.

- Wait! What! 🤔 I'm having another test!

> 💡 Prior information can alter the posterior!

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## Bayesian statistics <hbr>

> 🔑 Parameter is a .stress[random variable] conditioned on observations. <hbr>

### Back to coin toss example <h0br>
.pull-left[
- Same model except for `${\cal P}$`:

- Prior belief: `$p\sim \mathbb{P}_{\text{pr}}$` on `$[0,1]$`

- `$O=\{$`H `$=55$` & T `$=45\}$`,
]

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## Bayesian statistics <hbr>

> 🔑 Parameter is a .stress[random variable] conditioned on observations. <hbr>

### Back to coin toss example <h0br>
.pull-left[
- Same model except for `${\cal P}$`:

- Prior belief: `$p\sim \mathbb{P}_{\text{pr}}$` on `$[0,1]$`

- `$O=\{$`H `$=55$` & T `$=45\}$`,

`$$\begin{align}\overbrace{\mathbb{P}(p|O)}^{\text{Posterior}}&=\frac{\overbrace{\mathbb{P}(O|p)}^{\text{Likelihood}}\times\overbrace{\mathbb{P}_{\text{pr}}(p)}^{\text{Prior}}}{\mathbb{P}(O)}\\
&\propto \mathbb{P}(O|p)\times \mathbb{P}_{\text{pr}}(p)\\
&\propto p^{55}(1-p)^{45}\times\mathbb{P}_{\text{pr}}(p)\end{align}$$`
]

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## Bayesian statistics <hbr>

> 🔑 Parameter is a .stress[random variable] conditioned on observations. <hbr>

### Back to coin toss example <h0br>
.pull-left[
- Same model except for `${\cal P}$`:

- Prior belief: `$p\sim \mathbb{P}_{\text{pr}}$` on `$[0,1]$`

- `$O=\{$`H `$=55$` & T `$=45\}$`,
 
`$$\begin{align}\overbrace{\mathbb{P}(p|O)}^{\text{Posterior}}&=\frac{\overbrace{\mathbb{P}(O|p)}^{\text{Likelihood}}\times\overbrace{\mathbb{P}_{\text{pr}}(p)}^{\text{Prior}}}{\mathbb{P}(O)}\\
&\propto \mathbb{P}(O|p)\times \mathbb{P}_{\text{pr}}(p)\\
&\propto p^{55}(1-p)^{45}\times\mathbb{P}_{\text{pr}}(p)\end{align}$$`
]
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???

https://mspeekenbrink.github.io/sdam-book/ch-Bayes-factors.html

---
## Bayesian statistics <hbr>

> 🔑 Parameter is a .stress[random variable] conditioned on observations. <hbr>

### Credible interval <h0br>
.pull-left[
- For `$\beta\in[0,1]$`, `$\text{CrI}(\beta)$` satisfies: `$$\underbrace{\mathbb{P}(p\in\text{CrI}(\beta)|O)}_{\text{Posterior}}=\beta.$$`

]
.pull-right[ <br-4>
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---
count: false
## Bayesian statistics <hbr>

> 🔑 Parameter is a .stress[random variable] conditioned on observations. <hbr>

### Credible interval <h0br>
.pull-left[
- For `$\beta\in[0,1]$`, `$\text{CrI}(\beta)$` satisfies: `$$\underbrace{\mathbb{P}(p\in\text{CrI}(\beta)|O)}_{\text{Posterior}}=\beta.$$`

- Many ways to define it:  
  - Below is as likely as above.
  - Minimal size.
  - Centered at the mean.

]
.pull-right[ <br-4>
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]

---
count: false
## Bayesian statistics <hbr>

> 🔑 Parameter is a .stress[random variable] conditioned on observations. <hbr>

### Credible interval <h0br>
.pull-left[
- For `$\beta\in[0,1]$`, `$\text{CrI}(\beta)$` satisfies: `$$\underbrace{\mathbb{P}(p\in\text{CrI}(\beta)|O)}_{\text{Posterior}}=\beta.$$`

- Many ways to define it:  
  - Below is as likely as above.
  - Minimal size.
  - Centered at the mean.

> .purple[Uniform]: [0.453,0.641].

]
.pull-right[ <br-4>
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]

---
count: false
## Bayesian statistics <hbr>

> 🔑 Parameter is a .stress[random variable] conditioned on observations. <hbr>

### Credible interval <h0br>
.pull-left[
- For `$\beta\in[0,1]$`, `$\text{CrI}(\beta)$` satisfies: `$$\underbrace{\mathbb{P}(p\in\text{CrI}(\beta)|O)}_{\text{Posterior}}=\beta.$$`

- Many ways to define it:  
  - Below is as likely as above.
  - Minimal size.
  - Centered at the mean.
  
> .purple[Uniform]: [0.453,0.641].

]
.pull-right[ <br-4>
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<h0br>

💡 For the chosen prior and the observed observations, there is a `$\beta %$` chance that `$p\in\text{CrI}(\beta)$`.

]

---
## Bayesian statistics <hbr>

> 🔑 Parameter is a .stress[random variable] conditioned on observations. <hbr>

### Credible interval <h0br>
.pull-left[
- For `$\beta\in[0,1]$`, `$\text{CrI}(\beta)$` satisfies: `$$\underbrace{\mathbb{P}(p\in\text{CrI}(\beta)|O)}_{\text{Posterior}}=\beta.$$`

- Many ways to define it:  
  - Below is as likely as above.
  - Minimal size.
  - Centered at the mean.

> .purple[Uniform]: [0.453,0.641].

]
.pull-right[ <br-4>
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<h0br>

💡 For the chosen prior and the observed observations, there is a `$\beta %$` chance that `$p\in\text{CrI}(\beta)$`.

⚠️ This uncertainty is due to the parameter itself.
]

---
# Bayesian statistics <hbr>

- Key points: 
> - The parameter is modeled by a random variable.
  - It is updated through iteration of data collection process.
  - The uncertainty is due to the parameter itself.
--

- Statistical model: `$(S, {\cal A}, {\cal P})$` with prior
`$\theta\sim\mathbb{P}_{\text{pr}}.$` 
- Computation of .stress[posterior] through Bayes's rule.
- Uncertainty analysis: tests or credible intervals,...
- Interpretation, inference, conclusion (according to prior/observations).

### Important remark

> ⚠️ Influence of the prior must be taken into account!

???

- `$S=(0,\infty)$`
- `${\cal A}={\cal B}(S)$`
- `$\mathbb{P}_{\theta}={\cal N}$` or `$\chi^2$`.

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.pull-left[ <hbr>
## Frequentist <hbr>
- Parameter is fixed, data is random: i.e., `$\mathbb{P}(x_1,..,x_n|p)$`.

- Uncertainty analysis (**test/CI**) is interpreted w.r.t data.

- For long run and large observations.

]

.pull-right[
## Bayesian <hbr>
- Data is fixed, parameter is random: i.e., `$\mathbb{P}(p|x_1,..,x_n)$`.

- Uncertainty analysis (**test/CrI**) is interpreted w.r.t parameter conditioned on data & prior.

- Works well for small observations and can be iteratively updated.

]

.pull-left-60[<br-4>
## Do and don't <hbr>
- .stress[Do] define data and questions then choose the appropriate statistical analysis.

- .stress[Do] investigate the influence of the prior when performing Bayesian analysis.

- .stress[Do] ensure that your interpretation is correct with `$p$`-value, **CI** or **CrI**.
]
.pull-right-40[ 
- .stress[Don't] assume that Bayesian approach will solve insufficient or poor-quality data! Data quality is the most important thing in every analysis.
]

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## 📚 Further reading

.center[
[<img src="img/ref1.png" align="middle" height="270">](https://www.redjournal.org/action/showPdf?pii=S0360-3016%2821%2903256-9)
[<img src="img/ref3.jpg" align="middle" height="270">](http://ndl.ethernet.edu.et/bitstream/123456789/39363/1/Leonhard%20Held.pdf)
]

## 🖥️ Slides

- Link: [https://hassothea.github.io/files/teaching/MAC/Freq_Bayes_Stat.html](https://hassothea.github.io/files/teaching/MAC/Freq_Bayes_Stat.html)
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background-image: url('img/MAC_no_bg.jpg')
background-size: cover

<h1br>
### 📣 Coming up next... <h0br>
.pull-left[ <h0br>
.center[<img src="img/Samnang.jpg" align="middle" height="100">]

**Samnang**: .stress[Multidimensional Gaussian Distribution] (**June, 2024**)

**Samai**: .stress[Linear & Quadratic Discriminant Analysis] (**July, 2024**)

]
.pull-right[ <h0br>
.center[<img src="img/sreyneath.jpg" align="middle" height="100">]

**Sreyneath**: .stress[Logistic Regression] (**August, 2024**)

**Kimsia**: .stress[Gradient Descent Algorithm] (**September, 2024**)
]
<h0br>
.center[<img src="img/Samreth.jpg" align="middle" height="100">

**Samreth**: .stress[Principal Component Analysis] (**October, 2024**)
]
---
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name: appendix1
### Appendix 1 : Confidence Interval Approximation <hbr>
- .stress[Objective]: if `$X_1,...,X_n\overset{\text{iid}}{\sim} \mathcal{B}(p)$` and for `$n$` large enough, one has `$$\mathbb{P}\left(p\in\text{CI}\right)\underset{n\to\infty}{\to}1-\alpha$$`
with 
`$$\text{CI}=\left[\sin^2\left(\arcsin(\sqrt{\hat{p}})-\frac{q_{1-\alpha/2}}{2\sqrt{n}}\right),\sin^2\left(\arcsin(\sqrt{\hat{p}})+\frac{q_{1-\alpha/2}}{2\sqrt{n}}\right)\right]$$`
--

- .stress[In practice], we can approximate using Student's distribution:

`$$\frac{\sqrt{n}(\hat{p}-p)}{\hat{\sigma}}\underset{n\to\infty}{\leadsto} T(n-1)\underset{n\text{ large}}{\approx} {\cal N}(0,1)$$`
where `$\hat{\sigma}^2=\frac{1}{n-1}\sum_{i=1}(X_i-\overline{X}_n)^2=\hat{p}(1-\hat{p})$`. This leads to our result!

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### Appendix 1 : Confidence Interval Approximation <hbr>
- .stress[Sketch of proof of objective]:
 - **Central Limit Theorem**: 
 
 `$X_1,...X_n$` are iid r.v with `$\mathbb{E}(X_1)=\mu, \mathbb{V}(X_1)=\sigma^2<\infty$` then `$$\frac{\sqrt{n}(\overline{X}_n-\mu)}{\sigma}\overset{\cal L}{\to}{\cal N}(0,1).$$`

- `$\Delta$`**-method**: if `$R_n\overset{\cal L}{\to} \theta_0$` with rate `$1/r_n$` such that `$$r_n(R_n-\theta_0)\overset{\cal L}{\to}{\cal N}(0,g(\theta_0))$$` and if `$f:\Theta\to \mathbb{R}$` is differentiable at `$\theta_0\in\Theta$` then `$$r_n\left(f(R_n)-f(\theta_0)\right)\overset{\cal L}{\to}{\cal N}(0,g(\theta_0).[f'(\theta_0)]^2).$$`

- The proof is completed using `$f(x)=2\arcsin(\sqrt{x}).$` .small[👉  [Back to main slide](#CI)]

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&#128218; [Linder, T. (2002). Learning-Theoretic Methods in Vector Quantization. In: Györfi, L. (eds) Principles of Nonparametric Learning. International Centre for Mechanical Sciences, vol 434. Springer, Vienna.](https://link.springer.com/chapter/10.1007/978-3-7091-2568-7_4)

&#128218; [Banerjee, S. Merugu, I.S. Dhillon, and J. Ghosh.  Clustering with Bregman divergences. Journal of Machine Learning Research, 6:1705–1749, 2005](chrome-extension://oemmndcbldboiebfnladdacbdfmadadm/https://jmlr.org/papers/volume6/banerjee05b/banerjee05b.pdf)

&#128218; [A. Fischer.  Quantization and clustering with Bregman divergences. Journal of Multivariate Analysis, 101(10):2207–2221, 2010.](chrome-extension://oemmndcbldboiebfnladdacbdfmadadm/https://www.lpsm.paris/_media/users/fischer/quantization_and_clustering_with_bregman_divergences.pdf)

&#128218; [S. Has, A. Fischer, and M. Mougeot. Kfc: A clusterwise supervised learning procedure based on the aggregation of distances. Journal of Statistical Computation and Simulation, 0(0):1–21, 2021. doi: 10.1080/00949655.2021.  1891539.](https://www.tandfonline.com/doi/abs/10.1080/00949655.2021.1891539)

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