Tristan Mahr September 20, 2016
library("dplyr")
#>
#> Attaching package: 'dplyr'
#> The following objects are masked from 'package:stats':
#>
#> filter, lag
#> The following objects are masked from 'package:base':
#>
#> intersect, setdiff, setequal, union
library("tidyr")
library("ggplot2")
# We have some IQ scores
iqs <- car::Burt$IQbio
iqs
#> [1] 82 80 88 108 116 117 132 71 75 93 95 88 111 63 77 86 83
#> [18] 93 97 87 94 96 112 113 106 107 98
# Want to use Bayesian updating to estimate the mean of the data and the sd of
# the data.
# Use a uniform prior across a range of means and SDs
each_mean <- seq(from = 70, to = 130, by = 1)
each_sd <- 10:20
# equal probability
p_mean <- 1 / length(each_mean)
p_sd <- 1 / length(each_sd)
# We are using a grid approximation. Create every combination of mean and sd.
# Joint probability as product of each one's prior probability.
df <- data_frame(each_mean, p_mean) %>%
expand(nesting(each_mean, p_mean), nesting(each_sd, p_sd)) %>%
mutate(prior = p_sd * p_mean,
posterior = prior,
data_seen = 0)
df
#> # A tibble: 671 × 7
#> each_mean p_mean each_sd p_sd prior posterior
#> <dbl> <dbl> <int> <dbl> <dbl> <dbl>
#> 1 70 0.01639344 10 0.09090909 0.001490313 0.001490313
#> 2 70 0.01639344 11 0.09090909 0.001490313 0.001490313
#> 3 70 0.01639344 12 0.09090909 0.001490313 0.001490313
#> 4 70 0.01639344 13 0.09090909 0.001490313 0.001490313
#> 5 70 0.01639344 14 0.09090909 0.001490313 0.001490313
#> 6 70 0.01639344 15 0.09090909 0.001490313 0.001490313
#> 7 70 0.01639344 16 0.09090909 0.001490313 0.001490313
#> 8 70 0.01639344 17 0.09090909 0.001490313 0.001490313
#> 9 70 0.01639344 18 0.09090909 0.001490313 0.001490313
#> 10 70 0.01639344 19 0.09090909 0.001490313 0.001490313
#> # ... with 661 more rows, and 1 more variables: data_seen <dbl>
# Update the posterior given some data
update_distribution <- function(df, data) {
data_seen <- unique(df$data_seen) + length(data)
df %>%
# Last posterior is current prior
select(each_mean, each_sd, prior = posterior) %>%
# Attach data to each mean-sd pair
expand(nesting(each_mean, each_sd, prior), points = data) %>%
# Measure likelihood of data in each mean-sd pair
group_by(each_mean, each_sd, prior) %>%
summarise(likelihood = prod(dnorm(points, each_mean, each_sd))) %>%
ungroup %>%
# bayes theorem
mutate(posterior = likelihood * prior / (sum(likelihood * prior)),
data_seen = data_seen)
}
# Create each sequence of data samples
data_sequence <- Map(seq, 1, seq_along(iqs)) %>%
lapply(function(xs) iqs[xs])
iq_sequence <- data_sequence %>%
purrr::map(~ update_distribution(df, .x))
# include 0 data seen
iq_sequence <- c(list(df), iq_sequence)
iq_sequence %>%
purrr::map_dbl(~ sum(.x$posterior))
#> [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
df_fit <- iq_sequence %>%
purrr::accumulate(bind_rows) %>%
# Create animation steps
purrr::map(. %>% mutate(step = max(data_seen)) %>%
# Keep last five data_seen values.
filter(step - data_seen <= 5) %>%
# Create a column to control transparency using recency
mutate(alpha = 1 - ((step - data_seen) / 5))) %>%
bind_rows
df_fit
#> # A tibble: 102,663 × 10
#> each_mean p_mean each_sd p_sd prior posterior
#> <dbl> <dbl> <int> <dbl> <dbl> <dbl>
#> 1 70 0.01639344 10 0.09090909 0.001490313 0.001490313
#> 2 70 0.01639344 11 0.09090909 0.001490313 0.001490313
#> 3 70 0.01639344 12 0.09090909 0.001490313 0.001490313
#> 4 70 0.01639344 13 0.09090909 0.001490313 0.001490313
#> 5 70 0.01639344 14 0.09090909 0.001490313 0.001490313
#> 6 70 0.01639344 15 0.09090909 0.001490313 0.001490313
#> 7 70 0.01639344 16 0.09090909 0.001490313 0.001490313
#> 8 70 0.01639344 17 0.09090909 0.001490313 0.001490313
#> 9 70 0.01639344 18 0.09090909 0.001490313 0.001490313
#> 10 70 0.01639344 19 0.09090909 0.001490313 0.001490313
#> # ... with 102,653 more rows, and 4 more variables: data_seen <dbl>,
#> # step <dbl>, alpha <dbl>, likelihood <dbl>
# Also want a data-frame of the cumulative data-set as each new point is added.
# This lets us plot a rug of raw data.
data_sequence <- data_sequence %>%
purrr::map_df(~ data_frame(iq = .x, data_seen = length(iq))) %>%
group_by(data_seen) %>%
mutate(step = max(data_seen), sequence = seq_along(data_seen), current = sequence == max(sequence)) %>%
select(-sequence) %>%
ungroup
data_sequence
#> # A tibble: 378 × 4
#> iq data_seen step current
#> <int> <int> <int> <lgl>
#> 1 82 1 1 TRUE
#> 2 82 2 2 FALSE
#> 3 80 2 2 TRUE
#> 4 82 3 3 FALSE
#> 5 80 3 3 FALSE
#> 6 88 3 3 TRUE
#> 7 82 4 4 FALSE
#> 8 80 4 4 FALSE
#> 9 88 4 4 FALSE
#> 10 108 4 4 TRUE
#> # ... with 368 more rows
library("gganimate")
# plot posterior of means, one line per sd level
p <- ggplot(df_fit) +
aes(x = each_mean, y = posterior, frame = step) +
geom_line(aes(group = interaction(each_sd, data_seen), alpha = alpha)) +
ggtitle("Data observed:") +
xlab("possible mean") +
ylab("posterior probability") +
guides(alpha = FALSE) +
theme_grey(base_size = 14) +
geom_rug(aes(x = iq, y = 0), data_sequence, sides = "b") +
geom_rug(aes(x = iq, y = 0), filter(data_sequence, current), size = 1.5,
sides = "b")
animation::ani.options(interval = 1/10)
g <- gg_animate(p, ani.height = 400, ani.width = 640)
# g
gg_animate_save(g, filename = "./assets/updating.gif", saver = "gif",
ani.height = 400, ani.width = 640)
#> Executing:
#> ""convert" -loop 0 -delay 10 Rplot1.png Rplot2.png Rplot3.png
#> Rplot4.png Rplot5.png Rplot6.png Rplot7.png Rplot8.png
#> Rplot9.png Rplot10.png Rplot11.png Rplot12.png Rplot13.png
#> Rplot14.png Rplot15.png Rplot16.png Rplot17.png Rplot18.png
#> Rplot19.png Rplot20.png Rplot21.png Rplot22.png Rplot23.png
#> Rplot24.png Rplot25.png Rplot26.png Rplot27.png Rplot28.png
#> "updating.gif""
#> Output at: updating.gif
# plot posterior of sds, one line per mean level
p <- ggplot(df_fit) +
aes(x = each_sd, y = posterior, frame = step) +
geom_line(aes(group = interaction(each_mean, data_seen), alpha = alpha)) +
ggtitle("Data observed:") +
xlab("possible sd") +
ylab("posterior probability") +
guides(alpha = FALSE) +
theme_grey(base_size = 14)
g2 <- gg_animate(p, ani.height = 400, ani.width = 640)
gg_animate_save(g2, filename = "./assets/updating_sds.gif", saver = "gif",
ani.height = 400, ani.width = 640)
#> Executing:
#> ""convert" -loop 0 -delay 10 Rplot1.png Rplot2.png Rplot3.png
#> Rplot4.png Rplot5.png Rplot6.png Rplot7.png Rplot8.png
#> Rplot9.png Rplot10.png Rplot11.png Rplot12.png Rplot13.png
#> Rplot14.png Rplot15.png Rplot16.png Rplot17.png Rplot18.png
#> Rplot19.png Rplot20.png Rplot21.png Rplot22.png Rplot23.png
#> Rplot24.png Rplot25.png Rplot26.png Rplot27.png Rplot28.png
#> "updating_sds.gif""
#> Output at: updating_sds.gif
p <- iq_sequence %>%
bind_rows %>%
ggplot(.) +
aes(x = each_mean, y = each_sd, z = posterior, group = data_seen, frame = data_seen) +
geom_contour() +
ggtitle("Data observed:") +
xlab("possible mean") +
ylab("possible sd")
g3 <- gg_animate(p, ani.height = 400, ani.width = 640)
#> Warning in grDevices::contourLines(x = sort(unique(data$x)), y =
#> sort(unique(data$y)), : all z values are equal
#> Warning: Not possible to generate contour data
gg_animate_save(g3, filename = "./assets/updating_contour.gif", saver = "gif",
ani.height = 400, ani.width = 640)
#> Executing:
#> ""convert" -loop 0 -delay 10 Rplot1.png Rplot2.png Rplot3.png
#> Rplot4.png Rplot5.png Rplot6.png Rplot7.png Rplot8.png
#> Rplot9.png Rplot10.png Rplot11.png Rplot12.png Rplot13.png
#> Rplot14.png Rplot15.png Rplot16.png Rplot17.png Rplot18.png
#> Rplot19.png Rplot20.png Rplot21.png Rplot22.png Rplot23.png
#> Rplot24.png Rplot25.png Rplot26.png Rplot27.png
#> "updating_contour.gif""
#> Output at: updating_contour.gif
