From eb41c5e5af6eec13c2ef8c6666cdbbc17d46c6a7 Mon Sep 17 00:00:00 2001 From: Josh Cook Date: Fri, 19 Nov 2021 06:04:02 -0500 Subject: [PATCH 1/5] chore: setup section 11 --- _site.yml | 2 + index.Rmd | 1 + ..._normal-approx-freq-properties_bda3-04.Rmd | 42 ++++++++++++++++++ .../BDA3_ch04_reading-instructions.pdf | Bin 0 -> 133607 bytes slides/slides_ch4.pdf | Bin 0 -> 616149 bytes 5 files changed, 45 insertions(+) create mode 100644 notes/11_normal-approx-freq-properties_bda3-04.Rmd create mode 100644 reading-instructions/BDA3_ch04_reading-instructions.pdf create mode 100644 slides/slides_ch4.pdf diff --git a/_site.yml b/_site.yml index e0b5776..2c5bfcb 100644 --- a/_site.yml +++ b/_site.yml @@ -35,6 +35,8 @@ navbar: href: notes/09_model-selection_bda3-7.html - text: "Section 10. Decision analysis" href: notes/10_decision-analysis_bda3-9.html + - text: "Section 11. Normal approximation & Frequency properties" + href: notes/11_normal-approx-freq-properties_bda3-04.html - text: "Exercises" menu: - text: "Chapter 1" diff --git a/index.Rmd b/index.Rmd index 04b4a86..9fd2848 100644 --- a/index.Rmd +++ b/index.Rmd @@ -54,6 +54,7 @@ The recommended way to go through the material is: | **8. Model checking & Cross-validation** | [notes](notes/08_model-checking-and-cv_bda3-6-7.html) | (none) | (none) | | **9. Model comparison and selection** | [notes](notes/09_model-selection_bda3-7.html) | (none) | [assignment 8](assignments/jhcook-assignment-08.html) | | **10. Decision analysis** | [notes](notes/10_decision-analysis_bda3-9.html) | (none) | [assignment 9](assignments/jhcook-assignment-09.html) | +| **11. Normal approximation & Frequency properties** | [notes](notes/11_normal-approx-freq-properties_bda3-04.html) | (none) | (none) | ## Stan models diff --git a/notes/11_normal-approx-freq-properties_bda3-04.Rmd b/notes/11_normal-approx-freq-properties_bda3-04.Rmd new file mode 100644 index 0000000..f650abd --- /dev/null +++ b/notes/11_normal-approx-freq-properties_bda3-04.Rmd @@ -0,0 +1,42 @@ +--- +title: "11. Normal approximation & Frequency properties" +date: "2021-11-19" +output: distill::distill_article +--- + +```{r setup, include=FALSE} +knitr::opts_chunk$set(echo = TRUE, dpi = 300, comment = "#>") +``` + +## Resources + +- reading: + - BDA3 ch 4. *Asymptotics and connections to non-Bayesian approaches* + - [reading instructions](../reading-instructions/BDA3_ch04_reading-instructions.pdf) +- lectures: + - [Lecture 11.1. 'Normal approximation (Laplace approximation)'](https://aalto.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=e22fedc7-9fd3-4d1e-8318-ab1000ca45a4) + - [Lecture 11.2. 'Large sample theory and counter examples'](https://aalto.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=a8e38a95-a944-4f3d-bf95-ab1000dbdf73) +- [slides](../slides/slides_ch04.pdf) + +## Notes + +### Reading instructions + +- chapter outline: + - 4.1 Normal approximation (Laplace’s method) + - 4.2 Large-sample theory + - 4.3 Counter examples + - 4.4 Frequency evaluation (not part of the course, but interesting) + - 4.5 Other statistical methods (not part of the course, but interesting) + +### Chapter 4. Asymptotics and connections to non-Bayesian approaches + +- notes from the book + +### Lecture notes + +#### Lecture 11.1. 'Normal approximation (Laplace approximation)' + + + +#### Lecture 11.2. 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zG9@^fSKUNeSerHYDo>_*f9#Di?uVQL3PAHb!vsIzE9EA>M@8dyt@dl>&G4_(5lW-~HoC^qIUmzY~{QfMC) z62Kb|39HS$;^9^QCr<*f{qIpgcRi){YoX1eU*D}@HBf5b3GEi$)Q#h*tzXBb=@aUK z^%lMKd!=;OQ)=H2y%zm>o&vh-DYYMhjTZfFT)_ZPYERdE(V~BRUcn}ygr2VXw1ofq zivmjMDWRw6J}u#!eF`YLr|5|WN+*5V$^U&rDV_AYfjauMj_Zzxyko721F=%i=Cr19~)VhYp}{#ZFicUC<3Z&l2!v2 zKlEX(1}(n$U#*5L{^_(HDdXLpVMlK z36n_}S;Bwu5WN%_>3UhQ}a&&ldWo8OCF*GnVGzujpMNdWwJvCwL literal 0 HcmV?d00001 From ab30a2ce1300288e943a817a82a195480319c49d Mon Sep 17 00:00:00 2001 From: Josh Cook Date: Wed, 1 Dec 2021 06:20:08 -0500 Subject: [PATCH 2/5] feat: notes on BDA3 ch 4 --- ..._normal-approx-freq-properties_bda3-04.Rmd | 47 ++++++++++++++++++- 1 file changed, 45 insertions(+), 2 deletions(-) diff --git a/notes/11_normal-approx-freq-properties_bda3-04.Rmd b/notes/11_normal-approx-freq-properties_bda3-04.Rmd index f650abd..a58775b 100644 --- a/notes/11_normal-approx-freq-properties_bda3-04.Rmd +++ b/notes/11_normal-approx-freq-properties_bda3-04.Rmd @@ -25,13 +25,56 @@ knitr::opts_chunk$set(echo = TRUE, dpi = 300, comment = "#>") - chapter outline: - 4.1 Normal approximation (Laplace’s method) - 4.2 Large-sample theory - - 4.3 Counter examples + - 4.3 Counterexamples - 4.4 Frequency evaluation (not part of the course, but interesting) - 4.5 Other statistical methods (not part of the course, but interesting) ### Chapter 4. Asymptotics and connections to non-Bayesian approaches -- notes from the book +- *asymptotic theory*: as sample size increased, the influence of the prior on the posteior decreases + - used as the justification of non-informative priors in many cases + +#### 4.1 Normal approximations of the posterior distribution + +- if the posterior distribution $p(\theta|y)$ is unimodal and symmetric, it is useful to approximate it as a normal + - therefore, the log of the posterior is a quadratic function of $\theta$ +- "For a finite sample size $n$, the normal approximation is typically more accurate for conditional and marginal distributions of components of $\theta$ than for the full joint distribution." (pg. 85) +- common to use the normal approximations to quickly debug or sanity-check a model's code + +#### 4.2 Large-sample theory + +*asymptotic normality of the posterior distribution*: with more data from the same underlying process, the posterior distribution of the parameter vector approaches multivariate normality even if the true distribution of the data is not within the parametric family under consideration (pg. 87) + - particularly for independent samples from the data-generating process +- summary at the limit of large $n$: + - the posterior mode $\hat\theta$ approaches the true $\theta_0$ + - the likelihood dominates the prior distribution + +#### 4.3 Counterexamples to the theorems + +- there are many instances where large amounts of data do not allow for the normal approximation: +- **underidentified models and nonidentified parameters** + - "the model is *underidentified* given data $y$ if the likelihood $p(\theta|y)$ is equal for a range of $\theta$" (pg. 89) + - there is no single point $\theta_0$ to which the posterior distribution can converge given infinite data + - a parameter can be nonidentified if there is no supply of information about it + - results in its posterior being identical to its prior +- **number of parameters increasing with sample size** + - in complicated problems, the number of parameters can scale with the amount of data + - e.g. Gaussian processes or hierarchical models +- **aliasing** + - the same likelihood function repeats at a discrete set of points + - a special case of underidentified parameters + - e.g. a mixture model with two mixed distributions with the same parameters +- **unbounded likelihoods** + - if the likelihood is unbounded, there might not be any posterior mode with the parameter space + - invalidates bot the consistency results and the normal approximation +- **improper posterior distributions** + - an improper posterior integrates to infinity, not to 1 as is required by the normal approximation theory + - an improper posterior can only occur with an improper prior +- **prior distributions that exclude the point of convergence** +- **convergence to the edge of parameter space** + - if $\theta_0$ is at the edge of the parameter space, the distribution cannot be symmetric +- **tails of the distribution** + - the normal approximation can be true for almost all of the mass of the posterior but not be true at the tails ### Lecture notes From 81440295104e0fe365d5e9f7ada0f95c9085b496 Mon Sep 17 00:00:00 2001 From: Josh Cook Date: Wed, 1 Dec 2021 07:26:40 -0500 Subject: [PATCH 3/5] feat: Lecture 11.1 and begin 11.2 --- ..._normal-approx-freq-properties_bda3-04.Rmd | 11 +- ...normal-approx-freq-properties_bda3-04.html | 2625 +++++++++++++++++ slides/slides_ch4.pdf | Bin 616149 -> 994311 bytes 3 files changed, 2635 insertions(+), 1 deletion(-) create mode 100644 notes/11_normal-approx-freq-properties_bda3-04.html diff --git a/notes/11_normal-approx-freq-properties_bda3-04.Rmd b/notes/11_normal-approx-freq-properties_bda3-04.Rmd index a58775b..e6a574c 100644 --- a/notes/11_normal-approx-freq-properties_bda3-04.Rmd +++ b/notes/11_normal-approx-freq-properties_bda3-04.Rmd @@ -80,6 +80,15 @@ knitr::opts_chunk$set(echo = TRUE, dpi = 300, comment = "#>") #### Lecture 11.1. 'Normal approximation (Laplace approximation)' - +(no additional notes) #### Lecture 11.2. 'Large sample theory and counter examples' + +- *large sample theory*: + - *consistency*: if the true distribution is included in the parametric family then the posterior converges to a point $\theta_0$ when $n \rightarrow \infty$ + - "included in the parametric family": $f(y) = p(y|\theta_0)$ for some $\theta_0$ + - the point does not have uncertainty + - same result as MLE + - if the true distribution is not included in the parameteric family, then there is no true $\theta_0$, so replace it with the $\theta_0$ that minimized the KL divergence + +**paused at 26:32** diff --git a/notes/11_normal-approx-freq-properties_bda3-04.html b/notes/11_normal-approx-freq-properties_bda3-04.html new file mode 100644 index 0000000..b1766cf --- /dev/null +++ b/notes/11_normal-approx-freq-properties_bda3-04.html @@ -0,0 +1,2625 @@ + + + + + + + + + + + + + + + + + + + + 11. Normal approximation & Frequency properties + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

+

11. Normal approximation & Frequency properties

+ + + +
+ + +
+

Resources

+
+

Notes

+

Reading instructions

+
    +
  • chapter outline: +
      +
    • 4.1 Normal approximation (Laplace’s method)
    • +
    • 4.2 Large-sample theory
    • +
    • 4.3 Counterexamples
    • +
    • 4.4 Frequency evaluation (not part of the course, but interesting)
    • +
    • 4.5 Other statistical methods (not part of the course, but interesting)
    • +
  • +
+

Chapter 4. Asymptotics and connections to non-Bayesian approaches

+
    +
  • asymptotic theory: as sample size increased, the influence of the prior on the posteior decreases +
      +
    • used as the justification of non-informative priors in many cases
    • +
  • +
+

4.1 Normal approximations of the posterior distribution

+
    +
  • if the posterior distribution \(p(\theta|y)\) is unimodal and symmetric, it is useful to approximate it as a normal +
      +
    • therefore, the log of the posterior is a quadratic function of \(\theta\)
    • +
  • +
  • “For a finite sample size \(n\), the normal approximation is typically more accurate for conditional and marginal distributions of components of \(\theta\) than for the full joint distribution.” (pg. 85)
  • +
  • common to use the normal approximations to quickly debug or sanity-check a model’s code
  • +
+

4.2 Large-sample theory

+

asymptotic normality of the posterior distribution: with more data from the same underlying process, the posterior distribution of the parameter vector approaches multivariate normality even if the true distribution of the data is not within the parametric family under consideration (pg. 87) - particularly for independent samples from the data-generating process - summary at the limit of large \(n\): - the posterior mode \(\hat\theta\) approaches the true \(\theta_0\) - the likelihood dominates the prior distribution

+

4.3 Counterexamples to the theorems

+
    +
  • there are many instances where large amounts of data do not allow for the normal approximation:
  • +
  • underidentified models and nonidentified parameters +
      +
    • “the model is underidentified given data \(y\) if the likelihood \(p(\theta|y)\) is equal for a range of \(\theta\)” (pg. 89)
    • +
    • there is no single point \(\theta_0\) to which the posterior distribution can converge given infinite data
    • +
    • a parameter can be nonidentified if there is no supply of information about it +
        +
      • results in its posterior being identical to its prior
      • +
    • +
  • +
  • number of parameters increasing with sample size +
      +
    • in complicated problems, the number of parameters can scale with the amount of data
    • +
    • e.g. Gaussian processes or hierarchical models
    • +
  • +
  • aliasing +
      +
    • the same likelihood function repeats at a discrete set of points +
        +
      • a special case of underidentified parameters
      • +
    • +
    • e.g. a mixture model with two mixed distributions with the same parameters
    • +
  • +
  • unbounded likelihoods +
      +
    • if the likelihood is unbounded, there might not be any posterior mode with the parameter space +
        +
      • invalidates bot the consistency results and the normal approximation
      • +
    • +
  • +
  • improper posterior distributions +
      +
    • an improper posterior integrates to infinity, not to 1 as is required by the normal approximation theory
    • +
    • an improper posterior can only occur with an improper prior
    • +
  • +
  • prior distributions that exclude the point of convergence
  • +
  • convergence to the edge of parameter space +
      +
    • if \(\theta_0\) is at the edge of the parameter space, the distribution cannot be symmetric
    • +
  • +
  • tails of the distribution +
      +
    • the normal approximation can be true for almost all of the mass of the posterior but not be true at the tails
    • +
  • +
+

Lecture notes

+

Lecture 11.1. ‘Normal approximation (Laplace approximation)’

+

(no additional notes)

+

Lecture 11.2. ‘Large sample theory and counter examples’

+
    +
  • large sample theory: +
      +
    • consistency: if the true distribution is included in the parametric family then the posterior converges to a point \(\theta_0\) when \(n \rightarrow \infty\) +
        +
      • “included in the parametric family”: \(f(y) = p(y|\theta_0)\) for some \(\theta_0\)
      • +
      • the point does not have uncertainty
      • +
      • same result as MLE
      • +
    • +
    • if the true distribution is not included in the parameteric family, then there is no true \(\theta_0\), so replace it with the \(\theta_0\) that minimized the KL divergence
    • +
    +
  • +
+ + +
+ +
+
+ + + + + + + +
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zG9@^fSKUNeSerHYDo>_*f9#Di?uVQL3PAHb!vsIzE9EA>M@8dyt@dl>&G4_(5lW-~HoC^qIUmzY~{QfMC) z62Kb|39HS$;^9^QCr<*f{qIpgcRi){YoX1eU*D}@HBf5b3GEi$)Q#h*tzXBb=@aUK z^%lMKd!=;OQ)=H2y%zm>o&vh-DYYMhjTZfFT)_ZPYERdE(V~BRUcn}ygr2VXw1ofq zivmjMDWRw6J}u#!eF`YLr|5|WN+*5V$^U&rDV_AYfjauMj_Zzxyko721F=%i=Cr19~)VhYp}{#ZFicUC<3Z&l2!v2 zKlEX(1}(n$U#*5L{^_(HDdXLpVMlK z36n_}S;Bwu5WN%_>3UhQ}a&&ldWo8OCF*GnVGzujpMNdWwJvCwL From a40e3be6c129b510ad782ec93acbcbcf4c4b9939 Mon Sep 17 00:00:00 2001 From: Josh Cook Date: Thu, 2 Dec 2021 06:27:45 -0500 Subject: [PATCH 4/5] notes: lecture 11.2 notes --- index.Rmd | 1 - notes/11_normal-approx-freq-properties_bda3-04.Rmd | 2 -- 2 files changed, 3 deletions(-) diff --git a/index.Rmd b/index.Rmd index 9fd2848..5c2b0c9 100644 --- a/index.Rmd +++ b/index.Rmd @@ -15,7 +15,6 @@ site: distill::distill_website knitr::opts_chunk$set(echo = FALSE, dpi = 300) ``` - ## Resources - [Course website](https://avehtari.github.io/BDA_course_Aalto/) diff --git a/notes/11_normal-approx-freq-properties_bda3-04.Rmd b/notes/11_normal-approx-freq-properties_bda3-04.Rmd index e6a574c..11ac09a 100644 --- a/notes/11_normal-approx-freq-properties_bda3-04.Rmd +++ b/notes/11_normal-approx-freq-properties_bda3-04.Rmd @@ -90,5 +90,3 @@ knitr::opts_chunk$set(echo = TRUE, dpi = 300, comment = "#>") - the point does not have uncertainty - same result as MLE - if the true distribution is not included in the parameteric family, then there is no true $\theta_0$, so replace it with the $\theta_0$ that minimized the KL divergence - -**paused at 26:32** From 89c8b664e5d3f6e122d14644951765e55432aff5 Mon Sep 17 00:00:00 2001 From: Josh Cook Date: Thu, 2 Dec 2021 06:28:15 -0500 Subject: [PATCH 5/5] chore: render site --- docs/about.html | 1 + docs/index.html | 7 + ..._normal-approx-freq-properties_bda3-04.Rmd | 92 + ...normal-approx-freq-properties_bda3-04.html | 2625 +++++++++++++++++ docs/project-guidelines.html | 1 + .../BDA3_ch04_reading-instructions.pdf | Bin 0 -> 133607 bytes docs/search.json | 8 +- docs/sitemap.xml | 4 +- docs/slides/slides_ch4.pdf | Bin 0 -> 994311 bytes 9 files changed, 2732 insertions(+), 6 deletions(-) create mode 100644 docs/notes/11_normal-approx-freq-properties_bda3-04.Rmd create mode 100644 docs/notes/11_normal-approx-freq-properties_bda3-04.html create mode 100644 docs/reading-instructions/BDA3_ch04_reading-instructions.pdf create mode 100644 docs/slides/slides_ch4.pdf diff --git a/docs/about.html b/docs/about.html index f884857..8ced4e7 100644 --- a/docs/about.html +++ b/docs/about.html @@ -2394,6 +2394,7 @@

${suggestion.title}

Section 8. Model checking & Cross-validation Section 9. Model comparison and selection Section 10. Decision analysis +Section 11. Normal approximation & Frequency properties

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