# TeachBayes Package

The TeachBayes package was written to assist in the teaching of a “Beginning Bayes” course. There are a number of functions dealing with spinners, proportion, and mean inference. The purpose of this introduction to give a simple example of all of the functions. The material is divided into sections (1) Spinner Functions, (2) Inference for a Proportion, (3) Inference for a Mean, (4) Inference for Two Proportions and (5) Utilities

library(TeachBayes)

### Spinner Functions

(spinner_data, spinner_likelihoods, spinner_plot, spinner_probs, dspinner, many_spinner_plots)

A useful randomization device is a spinner that is defined by a vector of areas.

sp_regions <- c(2, 1, 1, 2)
1. spinner_plot will construct a plot of the spinner.
spinner_plot(sp_regions)

1. spinner_probs displays the spinner probabilities:
spinner_probs(sp_regions)
##   Region      Prob
## 1      1 0.3333333
## 2      2 0.1666667
## 3      3 0.1666667
## 4      4 0.3333333
1. spinner_data simulates spins from the spinner:
spinner_data(sp_regions, nsim=20)
##  [1] 2 1 4 4 1 4 3 4 1 4 3 4 2 4 4 1 1 4 2 4

Suppose we define multiple spinners:

sp1 <- c(1, 1, 1)
sp2 <- c(1, 2, 2, 1)
sp3 <- c(1, 1, 1, 1)
sp4 <- c(2, 2, 3, 3, 4)
1. We can use many_spinner_plots to display all of the spinners.
many_spinner_plots(list(sp1, sp2, sp3, sp4))

1. Function spinner_likelihoods will compute a probability matrix for a group of spinners. This is used in a Bayes’ rule calculation.
(LIKE <- spinner_likelihoods(list(sp1, sp2, sp3, sp4)))
##                   1         2         3         4         5
## Spinner A 0.3333333 0.3333333 0.3333333 0.0000000 0.0000000
## Spinner B 0.1666667 0.3333333 0.3333333 0.1666667 0.0000000
## Spinner C 0.2500000 0.2500000 0.2500000 0.2500000 0.0000000
## Spinner D 0.1428571 0.1428571 0.2142857 0.2142857 0.2857143
1. Function dspinner computes the likelhood function of spinners given a vector of spinner outcomes.
dspinner(c(1, 3, 4, 1, 2), LIKE)
##              [,1]
## [1,] 0.0000000000
## [2,] 0.0005144033
## [3,] 0.0009765625
## [4,] 0.0001338728

### Bayes’ Rule

(bayesian_crank, prior_post_plot)

Given a data frame with variables Model, Prior, and Likelihood, function bayesian_crank computes the posterior probabilities.

bayes_df <- data.frame(Model=paste("Spinner", 1:4),
Prior=rep(1/4, 4),
Likelihood=dspinner(c(1, 2, 1), LIKE))
(bayes_df <- bayesian_crank(bayes_df))
##       Model Prior  Likelihood     Product  Posterior
## 1 Spinner 1  0.25 0.037037037 0.009259259 0.57123527
## 2 Spinner 2  0.25 0.009259259 0.002314815 0.14280882
## 3 Spinner 3  0.25 0.015625000 0.003906250 0.24098988
## 4 Spinner 4  0.25 0.002915452 0.000728863 0.04496604

Function prior_post_plot will display the prior and posterior probabilities.

prior_post_plot(bayes_df)

### Inference for a Proportion

(beta_area, beta_data, beta_draw, beta_interval, beta_prior_post, beta_quantile, ChooseBeta)

1. beta_draw displays a single beta curve
beta_draw(c(10, 5))

1. beta_area displays a beta area (probability)
beta_area(.4, .6, c(10, 5))

1. beta_quantile displays a beta quantile
beta_quantile(.7, c(10, 5))

1. beta_interval displays an equal-tails interval that contains a specific probability content
beta_interval(.8, c(10, 5))

1. beta_data simulates from a beta density
beta_data(c(10, 5), nsim=20)
##  [1] 0.7426011 0.5611435 0.5968928 0.6804467 0.4029381 0.7818348 0.7938118
##  [8] 0.6169155 0.6079032 0.7382036 0.8758459 0.8533475 0.6888096 0.8409967
## [15] 0.8141908 0.5539010 0.6670882 0.6144303 0.9105439 0.6363769
1. beta_prior_post will graphically compare two beta curves such as a prior and posterior for a proportion
beta_prior_post(c(4, 4), c(20, 10))

1. Function ChooseBeta runs a Shiny app for selecting a beta curve based on knowledge of the prior median and prior 90th percentile.

### Inference for a Mean

(normal_area, normal_draw, normal_interval, normal_quantile, many_normal_plots, normal_update)

1. Function normal_area displays and computes a normal probability.
normal_area(90, 105, c(100, 15))

1. Function normal_draw will draw a single normal curve.
normal_draw(c(100, 10))

1. Function normal_interval will compute an equal tails probability interval.
normal_interval(.8, c(100, 10))

1. Function normal_quantile will compute and show graphically a normal quantile.
normal_quantile(.3, c(100, 10))

1. Function many_normal_plots will display many normal curves on the same graph.
many_normal_plots(list(c(100, 10), c(110, 10), c(120, 10)))

1. Function normal_update will give the posterior mean and standard deviation given the normal prior and the sample info.
prior <- c(100, 10)
ybar <- 120
se <- 15
normal_update(prior, c(ybar, se))
## [1] 106.153846   8.320503
normal_update(prior, c(ybar, se), teach=TRUE)
##        Type     Mean   Precision Stand_Dev
## 1     Prior 100.0000 0.010000000 10.000000
## 2      Data 120.0000 0.004444444 15.000000
## 3 Posterior 106.1538 0.014444444  8.320503

### Inference for Two Proportions

(testing_prior, draw_two_p, two_p_update, twoproplike, two_p_summarize)

1. Function testing_prior will construct a discrete prior for two proportions.
testing_prior(.1, .5, 5)
##       0.1   0.2   0.3   0.4   0.5
## 0.1 0.100 0.025 0.025 0.025 0.025
## 0.2 0.025 0.100 0.025 0.025 0.025
## 0.3 0.025 0.025 0.100 0.025 0.025
## 0.4 0.025 0.025 0.025 0.100 0.025
## 0.5 0.025 0.025 0.025 0.025 0.100
1. Function draw_two_p constructs a graph of a discrete distribution of two proportions.
draw_two_p(testing_prior(.1, .5, 5))

1. Function two_p_update will update a 2-proportion discrete prior with data.
prior <- testing_prior(.1, .5, 5)
(post <- two_p_update(prior, c(2, 10), c(4, 10)))
##              0.1         0.2         0.3         0.4         0.5
## 0.1 0.0394392508 0.048580661 0.064700702 0.043771965 0.017259318
## 0.2 0.0121451654 0.239363641 0.079697327 0.053917632 0.021259762
## 0.3 0.0071889668 0.035421034 0.188697782 0.031914927 0.012584079
## 0.4 0.0027357478 0.013479408 0.017952146 0.048580661 0.004788848
## 0.5 0.0006903727 0.003401562 0.004530269 0.003064863 0.004833910
1. Function twopropllike computes a likelihood for two binomial proportions (currently this calculation is already in the two_p_update function).

2. Function two_p_summarize will find the probability distribution for the difference in two proportions.

prior <- testing_prior(.1, .5, 5)
post <- two_p_update(prior, c(2, 10), c(4, 10))
two_p_summarize(post)
## # A tibble: 9 × 2
##   diff21         Prob
##    <dbl>        <dbl>
## 1   -0.4 0.0006903727
## 2   -0.3 0.0061373097
## 3   -0.2 0.0251986433
## 4   -0.1 0.0685832086
## 5    0.0 0.5209152453
## 6    0.1 0.1649817632
## 7    0.2 0.1312024124
## 8    0.3 0.0650317267
## 9    0.4 0.0172593180

### Utilities

1. bar_plot constructs a frequency bar plot of numeric data
bar_plot(spinner_data(c(1, 2, 3)))

1. prob_plot constructs a plot of a discrete probability distribution
prob_plot(data.frame(x=1:5, prob=c(.2, .3, .4, .1, .1)))