Rethink2m2

Aufgabe

This question is taken from McElreath, R. (2020). Statistical rethinking: A Bayesian course with examples in R and Stan (2. Ed.). Taylor and Francis, CRC Press.

Recall the globe tossing model from the chapter (also see exercise globus1). Compute and plot the grid approximate posterior distribution for each of the following sets of observations. In each case, assume a uniform prior for p.

Data:

1. WWW
2. WWWL
3. LWWLWWW

Now assume a prior for p that is equal to zero when p < 0.5 and is a positive constant when p ≥ 0.5. Again compute and plot the grid approximate posterior distribution for each of the sets of observations in the problem just above.

NB:

• Consider 21 different values for p such that \(p = (0, .05, 1., .15, \ldots, 1)\).
• Round to 2 decimal places.

Lösung

The solution is taken from this source.

library(tidyverse)

dist <- 
  tibble(
    # Gridwerte bestimmen:
    p_grid = seq(from = 0, to = 1, length.out = 21),
    # Priori-Wskt bestimmen:
    prior  = case_when(
      p_grid < 0.5 ~ 0,  # Null, wenn p < 0.5 
      p_grid >= 0.5 ~ 1)) %>%  # 1, wenn p >= 0.5
  mutate(
    # Likelihood berechnen:
    likelihood_1 = dbinom(3, size = 3, prob = p_grid),
    likelihood_2 = dbinom(3, size = 4, prob = p_grid),
    likelihood_3 = dbinom(5, size = 7, prob = p_grid),
    # unstand. Posterior-Wskt:
    unstand_post_1 = likelihood_1 * prior,
    unstand_post_2 = likelihood_2 * prior,
    unstand_post_3 = likelihood_3 * prior,
    # stand. Post-Wskt:
    std_post_1 = unstand_post_1 / sum(unstand_post_1),
    std_post_2 = unstand_post_2 / sum(unstand_post_2),
    std_post_3 = unstand_post_3 / sum(unstand_post_3)
    ) 

Here is the Bayes Box:

p_grid	prior	likelihood_1	likelihood_2	likelihood_3	unstand_post_1	unstand_post_2	unstand_post_3	std_post_1	std_post_2	std_post_3
0.00	0	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
0.05	0	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
0.10	0	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
0.15	0	0.00	0.01	0.00	0.00	0.00	0.00	0.00	0.00	0.00
0.20	0	0.01	0.03	0.00	0.00	0.00	0.00	0.00	0.00	0.00
0.25	0	0.02	0.05	0.01	0.00	0.00	0.00	0.00	0.00	0.00
0.30	0	0.03	0.08	0.03	0.00	0.00	0.00	0.00	0.00	0.00
0.35	0	0.04	0.11	0.05	0.00	0.00	0.00	0.00	0.00	0.00
0.40	0	0.06	0.15	0.08	0.00	0.00	0.00	0.00	0.00	0.00
0.45	0	0.09	0.20	0.12	0.00	0.00	0.00	0.00	0.00	0.00
0.50	1	0.13	0.25	0.16	0.13	0.25	0.16	0.02	0.07	0.07
0.55	1	0.17	0.30	0.21	0.17	0.30	0.21	0.03	0.09	0.10
0.60	1	0.22	0.35	0.26	0.22	0.35	0.26	0.04	0.10	0.12
0.65	1	0.27	0.38	0.30	0.27	0.38	0.30	0.05	0.11	0.13
0.70	1	0.34	0.41	0.32	0.34	0.41	0.32	0.07	0.12	0.14
0.75	1	0.42	0.42	0.31	0.42	0.42	0.31	0.08	0.13	0.14
0.80	1	0.51	0.41	0.28	0.51	0.41	0.28	0.10	0.12	0.12
0.85	1	0.61	0.37	0.21	0.61	0.37	0.21	0.12	0.11	0.09
0.90	1	0.73	0.29	0.12	0.73	0.29	0.12	0.14	0.09	0.06
0.95	1	0.86	0.17	0.04	0.86	0.17	0.04	0.16	0.05	0.02
1.00	1	1.00	0.00	0.00	1.00	0.00	0.00	0.19	0.00	0.00

Jetzt können wir das Diagramm zeichnen.

Mit ggpubr:

library(ggpubr)
ggline(dist, 
       x = "p_grid", 
       y = "std_post_1")

Oder mit ggplot2:

ggplot(dist) +
  aes(x = p_grid, y= std_post_1) +
  geom_line()+
  geom_point() +
  labs(x = "p(W)",
       y = "Posteriori-Wahrscheinlichkeit",
       title = "Daten: WWW")

ggplot(dist) +
  aes(x = p_grid, y= std_post_2) +
  geom_line()+
  geom_point() +
  labs(x = "p(W)",
       y = "Posteriori-Wahrscheinlichkeit",
       title = "Daten: WWWL")

ggplot(dist) +
  aes(x = p_grid, y= std_post_3) +
  geom_line()+
  geom_point() +
  labs(x = "p(W)",
       y = "Posteriori-Wahrscheinlichkeit",
       title = "Daten: LWWLWWW")

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Categories:

• probability
• bayesbox
• bayes
• rethink-chap2
• string