From ff83d1e92b24b8913f2fea56072372dec5b8274a Mon Sep 17 00:00:00 2001 From: Dominique Makowski Date: Wed, 1 May 2024 21:09:03 +0100 Subject: [PATCH] update --- BayesianStatistics/2024/10_Models/index.html | 161 ++++++++++++------ BayesianStatistics/2024/10_Models/index.qmd | 68 ++++++-- .../figure-revealjs/unnamed-chunk-13-1.png | Bin 717889 -> 727609 bytes 3 files changed, 161 insertions(+), 68 deletions(-) diff --git a/BayesianStatistics/2024/10_Models/index.html b/BayesianStatistics/2024/10_Models/index.html index ed7aa74..7c3cce9 100644 --- a/BayesianStatistics/2024/10_Models/index.html +++ b/BayesianStatistics/2024/10_Models/index.html @@ -628,51 +628,68 @@

Exercice

  • Interpret and visualize effect of the Sepal Width on Sepal Length (depending on the Species)
  • -
    -
    -
    -
    -

    -
    -
    -
    -
    +

    Model Comparison

    -
    -

    BF and R2

    -
    -
    +
    +

    Bayes Factors

      -
    • Paradoxically, Bayes Factors are complicated to obtain to compare Bayesian models +
    • It is possible to estimate Bayes Factors to compare models
    • +
    +
    +
    +
    bayestestR::bayesfactor_models(model, model2, model3)
    +
    +
    Bayes Factors for Model Comparison
    +
    +    Model                                       BF
    +[2] 0 + Intercept + Species               3.14e+06
    +[3] 0 + Intercept + Species * Petal.Width 6.20e+17
    +
    +* Against Denominator: [1] 0 + Intercept + Petal.Width
    +*   Bayes Factor Type: marginal likelihoods (bridgesampling)
    +
    +
    +
      -
    • Not impossible, but typically computationally expensive
    • -
    • Not recommended given that good alternatives exist
    • -
    -
  • Other indices of model performance? +
  • Paradoxically, BFs are not straightforward to obtain to compare Bayesian models
      -
    • R2
    • +
    • Typically computationally expensive
    • +
    • Typically algorithmically complex
    • +
    • Typically not very reliable
  • +
  • Not necessarily the first choice as good alternatives exist
  • +
  • Other indices of model performance?
  • + +
    +
    +

    Coefficient of Determination - R2

    +
    +
    +
      +
    • R2 measures how well the model explains the data already observed, focusing on variance reduction
    • +
    • R2 can sometimes give misleading impressions in cases of overfitting; a model might appear to perform very well on the training data but poorly on new data
    • +
    • R2 is primarily applicable (and intuitive) to linear models where the relationship between variables is the primary interest
    -
    performance::r2(model)
    +
    performance::r2(model)
    # Bayesian R2 with Compatibility Interval
     
       Conditional R2: 0.927 (95% CI [0.920, 0.932])
    -
    performance::r2(model2)
    +
    performance::r2(model2)
    # Bayesian R2 with Compatibility Interval
     
       Conditional R2: 0.941 (95% CI [0.936, 0.944])
    -
    performance::r2(model3)
    +
    performance::r2(model3)
    # Bayesian R2 with Compatibility Interval
     
    @@ -682,11 +699,6 @@ 

    BF and R2

    -
      -
    • R2 measures how well the model explains the data already observed, focusing on variance reduction
    • -
    • R2 can sometimes give misleading impressions in cases of overfitting; a model might appear to perform very well on the training data but poorly on new data
    • -
    • R2 is primarily applicable (and intuitive) to linear models where the relationship between variables is the primary interest
    • -

    ELPD

    @@ -709,25 +721,25 @@

    ELPD - WAIC

    • An index of prediction error adjusted for the number of parameters
    • It provides a balance between model fit and complexity, penalizing models that have too many parameters (similar to the BIC).
    • -
    • Computationally more straightforward than LOO, but might not be as accurate (more sensitive to outliers).
    • +
    • Computationally more straightforward than LOO, but might not be as accurate (more sensitive to outliers)
    -
    loo::waic(model)
    +
    loo::waic(model)
    
    -Computed from 4000 by 150 log-likelihood matrix.
    +Computed from 4000 by 150 log-likelihood matrix
     
               Estimate   SE
     elpd_waic   -104.5  9.5
     p_waic         3.3  0.6
     waic         208.9 19.0
    -
    loo::waic(model2)
    +
    loo::waic(model2)
    
    -Computed from 4000 by 150 log-likelihood matrix.
    +Computed from 4000 by 150 log-likelihood matrix
     
               Estimate   SE
     elpd_waic    -89.1 10.6
    @@ -736,10 +748,10 @@ 

    ELPD - WAIC

    1 (0.7%) p_waic estimates greater than 0.4. We recommend trying loo instead.
    -
    loo::waic(model3)
    +
    loo::waic(model3)
    
    -Computed from 4000 by 150 log-likelihood matrix.
    +Computed from 4000 by 150 log-likelihood matrix
     
               Estimate   SE
     elpd_waic    -64.4 11.5
    @@ -759,7 +771,7 @@ 

    loo_compare()

    -
    loo::loo_compare(loo::waic(model), loo::waic(model2), loo::waic(model3))
    +
    loo::loo_compare(loo::waic(model), loo::waic(model2), loo::waic(model3))
           elpd_diff se_diff
     model3   0.0       0.0  
    @@ -777,63 +789,60 @@ 

    ELPD - LOOIC

    • Leave-One-Out (LOO) Cross-Validation is a method to assess model performance by estimating how well a model predicts each observation, one at a time, using the rest of the data
    • Instead of refitting the model n times, each time leaving out one of the n data points, approximations like PSIS (Pareto Smoothed Importance Sampling) are used to avoid extensive computation
    • -
    • Provides a robust measure of model’s predictive accuracy (without direct complexity adjustment)
    • +
    • Provides a robust measure of model’s predictive accuracy (without direct complexity adjustment - but indirect through overfitting sensitivity)
    -
    loo::loo(model)
    +
    loo::loo(model)
    
    -Computed from 4000 by 150 log-likelihood matrix.
    +Computed from 4000 by 150 log-likelihood matrix
     
              Estimate   SE
     elpd_loo   -104.5  9.5
     p_loo         3.3  0.6
     looic       209.0 19.0
     ------
    -MCSE of elpd_loo is 0.0.
    -MCSE and ESS estimates assume MCMC draws (r_eff in [0.4, 1.3]).
    +Monte Carlo SE of elpd_loo is 0.0.
     
    -All Pareto k estimates are good (k < 0.7).
    +All Pareto k estimates are good (k < 0.5).
     See help('pareto-k-diagnostic') for details.
    -
    loo::loo(model2)
    +
    loo::loo(model2)
    
    -Computed from 4000 by 150 log-likelihood matrix.
    +Computed from 4000 by 150 log-likelihood matrix
     
              Estimate   SE
     elpd_loo    -89.1 10.6
     p_loo         4.4  0.8
     looic       178.3 21.2
     ------
    -MCSE of elpd_loo is 0.0.
    -MCSE and ESS estimates assume MCMC draws (r_eff in [0.4, 1.1]).
    +Monte Carlo SE of elpd_loo is 0.0.
     
    -All Pareto k estimates are good (k < 0.7).
    +All Pareto k estimates are good (k < 0.5).
     See help('pareto-k-diagnostic') for details.
    -
    loo::loo(model3)
    +
    loo::loo(model3)
    
    -Computed from 4000 by 150 log-likelihood matrix.
    +Computed from 4000 by 150 log-likelihood matrix
     
              Estimate   SE
     elpd_loo    -64.5 11.5
     p_loo         7.0  1.2
     looic       128.9 23.1
     ------
    -MCSE of elpd_loo is 0.1.
    -MCSE and ESS estimates assume MCMC draws (r_eff in [0.4, 1.0]).
    +Monte Carlo SE of elpd_loo is 0.1.
     
    -All Pareto k estimates are good (k < 0.7).
    +All Pareto k estimates are good (k < 0.5).
     See help('pareto-k-diagnostic') for details.
    -
    loo::loo_compare(loo::loo(model), loo::loo(model2), loo::loo(model3))
    +
    loo::loo_compare(loo::loo(model), loo::loo(model2), loo::loo(model3))
           elpd_diff se_diff
     model3   0.0       0.0  
    @@ -851,26 +860,68 @@ 

    Better by how much? Interpretation rules of thumb

    • \(> |1.96|\) ~ p < .05
    • \(> |3.29|\) ~ p < .001
    • +
    • Use 1-pnorm(|z-score|) to get the one-tailed p-value
    +
  • But the standarized difference does not make much sense if the absolute is small (\(|\Delta_{ELPD}|<4\)) +
      +
    • It’s not useful to say one model is likely to provide better predictions than other if the the predictions are almost the same anyways
    • +
  • +
  • Reporting should include the raw ELPD difference and its SE (and eventually the z-score or p-value if it makes sense)
  • Exercice

      +
    • Make a conclusion about which model is better and report the analysis
    • +
    +
    +
    +

    Exercice

    +
    • A researcher is interested in the relationship between the two dimensions (verbal and quantitative) of the SAT (a standardized test for college admissions), in particular whether the quantitative score is predicted by the verbal score. A colleague suggests that gender is important.
    • -
    • Using the real dataset sat.act from the psych package, analyze the data and answer the question.
    • +
    • Using the real dataset sat.act from the psych package, analyze the data and answer the question
    • Load the dataset as follows:
    -
    df <- psych::sat.act
    -df$gender <- as.factor(df$gender)  # gender=1: males, 2: females
    +
    df <- psych::sat.act
    +df$gender <- as.factor(df$gender)  # gender=1: males, 2: females
    +
    +

    Conclusion

    +
      +
    • You have (hopefully) learned: +
        +
      • What Bayesian statistics were about, the philosophy and applications of Bayes theorem
      • +
      • The strengths and drawbacks, and the differences and similarities with other approaches
      • +
      • How to apply it using simple and less simple models
      • +
    • +
    • This background knowledge helped you gain confidence about knowing where to look to learn more
    • +
    • Bayesian statistics is a very rapidly evolving field +
        +
      • Challenge to teach & learn (lots of uncertainty)
      • +
    • +
    • Important to keep up-to-date with the latest developments +
        +
      • Follow the development of new and existing packages (e.g., brms, rstan, …)
      • +
      • Follow forums, blogs and developers (e.g., r-bloggers, Twitter/X, …)
      • +
    • +
    • Learn by engaging with the community +
        +
      • Ask for help - it will often be helpful to people
      • +
      • Help others (share tips and tricks, little successes)
      • +
    • +
    • Spread the Bayesian gospel to your fellow academics +
        +
      • But don’t be a blind zealot - think for yourselves
      • +
    • +
    +

    The End (for good)

    Thank you!

    diff --git a/BayesianStatistics/2024/10_Models/index.qmd b/BayesianStatistics/2024/10_Models/index.qmd index 0820cae..57aa39b 100644 --- a/BayesianStatistics/2024/10_Models/index.qmd +++ b/BayesianStatistics/2024/10_Models/index.qmd @@ -214,6 +214,7 @@ p + geom_line(data = pred, aes(y = Predicted), ```{r} #| echo: false +#| include: false m <- brms::brm(Sepal.Length ~ 0 + Intercept + Species * Sepal.Width, data = iris, refresh=0) @@ -235,17 +236,37 @@ ggplot(iris, aes(x = Sepal.Width, y = Sepal.Length, color = Species)) + ## Model Comparison {.center background-color="#003B49"} -## BF and R2 +## Bayes Factors + +- It is *possible* to *estimate* Bayes Factors to compare models + +::: {.fragment} + +```{r} +#| echo: true + +bayestestR::bayesfactor_models(model, model2, model3) +``` + +::: + +- Paradoxically, BFs are not straightforward to obtain to compare Bayesian models + - Typically computationally expensive + - Typically algorithmically complex + - Typically not very reliable +- Not necessarily the first choice as good alternatives exist +- Other indices of model performance? + +## Coefficient of Determination - R2 ::: {.columns} :::: {.column width=50%} -- Paradoxically, Bayes Factors are complicated to obtain to compare Bayesian models - - Not impossible, but typically computationally expensive - - Not recommended given that good alternatives exist -- Other indices of model performance? - - **R2** +- R2 measures how well the model explains the data **already observed**, focusing on variance reduction +- R2 can sometimes give misleading impressions in cases of **overfitting**; a model might appear to perform very well on the training data but poorly on new data +- R2 is primarily applicable (and intuitive) to **linear models** where the **relationship between variables** is the primary interest + :::: @@ -267,9 +288,6 @@ performance::r2(model3) ::: -- R2 measures how well the model explains the data **already observed**, focusing on variance reduction -- R2 can sometimes give misleading impressions in cases of **overfitting**; a model might appear to perform very well on the training data but poorly on new data -- R2 is primarily applicable (and intuitive) to **linear models** where the **relationship between variables** is the primary interest ## ELPD @@ -285,7 +303,7 @@ performance::r2(model3) - **WAIC** (Widely Applicable Information Criterion) - An index of prediction error adjusted for the number of parameters - It provides a **balance between model fit and complexity**, penalizing models that have too many parameters (similar to the BIC). - - Computationally more straightforward than LOO, but might not be as accurate (more sensitive to outliers). + - Computationally more straightforward than LOO, but might not be as accurate (more sensitive to outliers) ::: {.fragment} @@ -321,7 +339,7 @@ loo::loo_compare(loo::waic(model), loo::waic(model2), loo::waic(model3)) - Leave-One-Out (LOO) Cross-Validation is a method to assess model performance by estimating how well a model predicts each observation, one at a time, using the rest of the data - Instead of refitting the model *n* times, each time leaving out one of the *n* data points, approximations like PSIS *(Pareto Smoothed Importance Sampling)* are used to avoid extensive computation -- Provides a robust measure of model's predictive accuracy (without direct complexity adjustment) +- Provides a robust measure of model's predictive accuracy (without direct complexity adjustment - but indirect through overfitting sensitivity) ::: {.fragment} @@ -352,12 +370,20 @@ loo::loo_compare(loo::loo(model), loo::loo(model2), loo::loo(model3)) - The "standardized" difference ($\Delta_{ELPD} / SE$) can be interpreted as a z-score - $> |1.96|$ ~ p < .05 - $> |3.29|$ ~ p < .001 + - Use `1-pnorm(|z-score|)` to get the one-tailed p-value +- But the standarized difference does not make much sense if the absolute is small ($|\Delta_{ELPD}|<4$) + - It's not useful to say one model is likely to provide better predictions than other if the the predictions are almost the same anyways +- Reporting should include the raw ELPD difference and its SE (and eventually the z-score or *p*-value if it makes sense) + +## Exercice {background-color="#FFAB91"} + +- Make a conclusion about which model is better and report the analysis ## Exercice {background-color="#FFAB91"} - A researcher is interested in the relationship between the two dimensions (verbal and quantitative) of the SAT (a standardized test for college admissions), in particular whether the quantitative score is predicted by the verbal score. A colleague suggests that gender is important. -- Using the real dataset `sat.act` from the `psych` package, analyze the data and answer the question. +- Using the real dataset `sat.act` from the `psych` package, analyze the data and answer the question - Load the dataset as follows: ::: {.fragment} @@ -388,7 +414,23 @@ df$gender <- as.factor(df$gender) # gender=1: males, 2: females - +## Conclusion + +- You have (hopefully) learned: + - What Bayesian statistics were about, the philosophy and applications of Bayes theorem + - The strengths and drawbacks, and the differences and similarities with other approaches + - How to apply it using simple and less simple models +- This background knowledge helped you gain confidence about knowing where to look to learn more +- Bayesian statistics is a very rapidly evolving field + - Challenge to teach \& learn (lots of uncertainty) +- Important to keep up-to-date with the latest developments + - Follow the development of new and existing packages (e.g., *brms*, *rstan*, ...) + - Follow forums, blogs and developers (e.g., *r-bloggers*, Twitter/X, ...) +- Learn by engaging with the community + - Ask for help - it will often be helpful to people + - Help others (share tips and tricks, little successes) +- Spread the Bayesian gospel to your fellow academics + - But don't be a blind zealot - **think for yourselves** ## The End (for good) {.center background-color="#212121"} diff --git a/BayesianStatistics/2024/10_Models/index_files/figure-revealjs/unnamed-chunk-13-1.png b/BayesianStatistics/2024/10_Models/index_files/figure-revealjs/unnamed-chunk-13-1.png index 032ec1854b7aa5199b45e9e353d2fe6925af2327..a7127e91e6d172110702ebe201d9ef58d31a19ce 100644 GIT binary patch literal 727609 zcmeFa2UL?;7dFa#jvXr~0s=||MUWDtR}~3_Py|OT6wwHx2q;ZDjv$~A3_VnljuaIH zLE2a#fzVN!fJz4e=^g$Zk~oo>yZ-yH_5XL>d)LQWg9`6^-gEZZ``OQa_IWeDI$Ef0 zTR65bF)?jZS5whtVq&vqV%qw8(|_P6ViQgd@W*CrwF@>(Oa`iy|9)c)K6QeL=>U_u 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