Scene 2 · Priors as theories

A Bayes factor is a prediction contest

So far ADT has made one directional prediction: more enjoyment after the rewarded ending. Theory building usually means weighing rival explanations of the same result, so here are two rivals. To compare the three, each account now states how big δ should be. All three are stylized (simplified for this demo, not summaries of a literature):

Account	Mechanism	Prediction for δ
ADT	Viewers like the character who did good, so they enjoy seeing that character rewarded more than punished.	small and positive (around 0.3)
Craft	Enjoyment comes from how well the story is told, not from how it ends.	zero
Tragedy	Watching a good person suffer is moving, and moving stories are enjoyed more.	small and negative (around −0.3)

Predict first

Each account above makes a prediction about δ. Using the n = 50 data (observed d = 0.39), which account wins, and by roughly how much?

A theory that predicts “δ is around 0.3” can be written as a prior on δ. That prior implies which t statistics the study should produce. The Bayes factor is the ratio of how well two hypotheses predicted the t value we actually observed (Rouder et al., 2009; Gronau et al., 2020). Treating priors as part of the theory is the core of Vanpaemel’s (2010) defense of Bayes factors for theory testing.

Load the Study 1 data

s1_50  <- read.csv("data/story_n50.csv")
s1_150 <- read.csv("data/story_n150.csv")
s50  <- two_group(s1_50)
s150 <- two_group(s1_150)

hyps <- list(
  "Craft (δ = 0)" = "null",
  "ADT: Normal(0.3, 0.15)" = prior_adt,
  "Tragedy: Normal(−0.3, 0.15)" = prior_tragedy,
  "JASP default: Cauchy(0, 0.707)" = prior_default
)

Code

cols <- c("Craft (δ = 0)" = pal[["gray"]],
          "ADT: Normal(0.3, 0.15)" = pal[["blue"]],
          "Tragedy: Normal(−0.3, 0.15)" = pal[["orange"]],
          "JASP default: Cauchy(0, 0.707)" = pal[["aqua"]])
short <- c("Craft", "ADT", "Tragedy", "Default")
dl <- seq(-1.5, 1.5, by = 0.01)
pri <- do.call(rbind, lapply(names(hyps)[-1], function(h) data.frame(x = dl, y = hyps[[h]](dl), h = h)))
pri_lab <- data.frame(x = c(0.78, -0.78, 1.15), y = c(2.2, 2.2, 0.28), h = names(hyps)[-1],
                      lab = short[-1])
pA <- ggplot(pri, aes(x, y, color = h)) +
  geom_segment(aes(x = 0, xend = 0, y = 0, yend = 3.1), color = pal[["gray"]], linewidth = 1.3,
               arrow = arrow(length = unit(0.25, "cm")), inherit.aes = FALSE) +
  annotate("text", x = 0, y = 3.38, label = "Craft", hjust = 0.5, size = 4.3,
           color = pal[["ink2"]], fontface = "bold") +
  geom_line(linewidth = 1.1) +
  geom_text(data = pri_lab, aes(label = lab), size = 4.3, fontface = "bold") +
  scale_color_manual(values = cols) +
  coord_cartesian(ylim = c(0, 3.5), xlim = c(-1.4, 1.4)) +
  labs(x = "Standardized effect δ", y = "Prior density", title = "What each account predicts about δ") +
  theme_demo()

tg <- seq(-6, 8, by = 0.05)
pred <- do.call(rbind, lapply(list(list(s50, "n = 50 per group"), list(s150, "n = 150 per group")), function(z) {
  do.call(rbind, lapply(names(hyps), function(h) data.frame(t = tg, y = pred_t(tg, z[[1]], hyps[[h]]), h = h, n = z[[2]])))
}))
pred$n <- factor(pred$n, levels = c("n = 50 per group", "n = 150 per group"))
obs <- data.frame(n = factor(c("n = 50 per group", "n = 150 per group"), levels = levels(pred$n)),
                  t = c(s50$t, s150$t))
pk <- data.frame(
  n = factor(rep(levels(pred$n), each = 4), levels = levels(pred$n)),
  h = rep(names(hyps), 2), lab = rep(short, 2),
  t = c(-1.6, 4.7, -3.9, 5.6,  -1.6, 5.9, -4.4, 6.1),
  y = c(0.415, 0.25, 0.3, 0.1, 0.415, 0.2, 0.26, 0.085))
pB <- ggplot(pred, aes(t, y, color = h)) +
  geom_line(linewidth = 1.1) +
  geom_vline(data = obs, aes(xintercept = t), linewidth = 0.9, linetype = "dotted", color = pal[["ink2"]]) +
  geom_text(data = obs, aes(x = t, y = 0.49, label = sprintf("observed t = %.2f", t)),
            hjust = 1.05, size = 4.2, color = pal[["ink2"]], inherit.aes = FALSE) +
  geom_text(data = pk, aes(label = lab), size = 4, fontface = "bold", vjust = 0) +
  facet_wrap(~n) +
  scale_color_manual(values = cols) +
  coord_cartesian(ylim = c(0, 0.51)) +
  labs(x = "t statistic", y = "Predicted density", title = "What each account predicts about the t statistic") +
  theme_demo()
pA / pB

Figure 1: Top: each account’s prior on δ (the craft account is a spike at zero). Bottom: the t statistics each account predicts. The dotted vertical line is the observed t. The Bayes factor between two accounts is the ratio of their curve heights at that line.

How to read this figure

Top panel: each account’s prior on δ, its prediction about the true effect before seeing data. Normal(0.3, 0.15) means “centered at 0.3, with 95% of its weight between 0 and 0.6.” The craft account puts all its weight on exactly zero (the arrow).
Bottom panels: what each account predicts the study’s t statistic will be. Imagine running the study thousands of times with true effects drawn from the account’s prior, and recording each t. The curve is the distribution of those t values (the prior predictive distribution).
The dotted vertical line is the t we actually observed.
The Bayes factor between any two accounts is the ratio of their curves’ heights at that line. At n = 50, the blue (ADT) curve is about five times higher than the gray (craft) curve there.
Compare the two panels. The craft prediction does not change with sample size, but the other accounts’ predictions move away from zero as n grows, because the same true effect produces larger t values in larger samples. Bigger studies make theories easier to tell apart.

bf_table <- sapply(list(`n = 50` = s50, `n = 150` = s150), function(s)
  sapply(hyps[-1], function(pr) bf_vs_null(s, pr)))
rownames(bf_table) <- c("ADT vs. null", "Tragedy vs. null", "Default vs. null")
round(rbind(bf_table,
            "ADT vs. tragedy" = bf_table[1, ] / bf_table[2, ],
            "ADT vs. default" = bf_table[1, ] / bf_table[3, ]), 2)

                 n = 50 n = 150
ADT vs. null       4.90  307.31
Tragedy vs. null   0.12    0.30
Default vs. null   1.12   59.18
ADT vs. tragedy   40.74 1032.84
ADT vs. default    4.36    5.19

# Check our prediction-contest function against BayesFactor (the engine JASP uses)
c(ours = bf_vs_null(s50, prior_default),
  BayesFactor = extractBF(bf_default(s1_50))$bf)

       ours BayesFactor 
   1.123621    1.123621

Read the table as a contest against the craft account, which predicts δ = 0 (the null):

At n = 50, ADT beats the null by about 5 to 1, while the vague default alternative barely ties it (1.12). Same data, different verdicts, because the hypotheses made different predictions.
Tragedy loses badly (BF₁₀ ≈ 0.12): it predicted negative t values and the data came out positive. At n = 150 it looks slightly less bad against the null (0.30), only because the null is now failing too. Bayes factors are relative: they say which of two hypotheses predicted better, not whether either one is any good. At n = 150, tragedy loses to ADT by more than 1,000 to 1.
At n = 150, ADT is favored over the null by about 310 to 1, and over the default by about 5.2 to 1.

A precise theory sticks its neck out. When it is right, it earns more evidence than a vague one; when it is wrong, it loses more. Meehl (1967) argued that a theory earns corroboration only by making predictions precise enough to fail; the Bayes factor puts a number on that idea.

How much does the prior width matter?

This recreates JASP’s robustness check: the default hypothesis with Cauchy widths from 0.05 to 1.5.

Code

r <- seq(0.05, 1.5, by = 0.025)
rob <- rbind(data.frame(r, bf = sapply(r, function(w) bf_vs_null(s50,  function(x) dcauchy(x, 0, w))), n = "n = 50 per group"),
             data.frame(r, bf = sapply(r, function(w) bf_vs_null(s150, function(x) dcauchy(x, 0, w))), n = "n = 150 per group"))
rl <- do.call(rbind, lapply(split(rob, rob$n), function(z) z[z$r == 1.5, ]))
ggplot(rob, aes(r, bf, color = n)) +
  geom_hline(yintercept = c(1, 3, 10), color = pal[["gray"]], linetype = c("solid", "dashed", "dashed"), linewidth = 0.4) +
  geom_vline(xintercept = sqrt(2) / 2, color = pal[["gray"]], linetype = "dotted") +
  geom_line(linewidth = 1.2) +
  geom_text(data = rl, aes(label = n), hjust = 1, vjust = -0.8, size = 4.4, fontface = "bold") +
  annotate("text", x = 0.72, y = 150, label = "JASP default (0.707)", hjust = 0, size = 4, color = pal[["ink2"]]) +
  scale_y_log10(breaks = c(0.3, 1, 3, 10, 30, 100), limits = c(0.3, 200)) +
  scale_color_manual(values = c("n = 50 per group" = pal[["blue"]], "n = 150 per group" = pal[["aqua"]])) +
  labs(x = "Cauchy prior width r", y = "BF10 (log scale)") +
  theme_demo()

Figure 2: Bayes factor for a Cauchy alternative as its width changes. Wider priors predict larger effects and are penalized when the observed effect is small.

How to read this figure

Horizontal axis: the width (scale) of the default Cauchy prior. Small widths say “expect small effects”; large widths say “expect large ones.”
Vertical axis: BF₁₀ on a log scale. The solid line at 1 means no preference; the dashed lines at 3 and 10 mark JASP’s “moderate” and “strong” labels.
The dotted vertical line is JASP’s default width (0.707).
A flat curve means the conclusion does not depend on the prior’s width; a sloped curve means it does.

At n = 50 every width gives weak evidence at best, and wide priors tip toward the null. At n = 150 every reasonable width favors an effect. A prior sensitivity analysis is how you show readers which conclusions depend on your theory’s specifics and which do not.

The frequentist counterpart: test the theory’s value, not only zero

A p value can test any hypothesized value, not only zero (Greenland et al., 2016). Testing each account’s central prediction:

p_delta <- function(s, d0) { # exact test of H0: delta = d0 via the noncentral t
  lo <- pt(s$t, s$df, ncp = d0 * sqrt(s$neff)); 2 * min(lo, 1 - lo)
}
sapply(c(`Tragedy (delta = -0.3)` = -0.3, `Craft (delta = 0)` = 0, `ADT (delta = 0.3)` = 0.3),
       function(d0) c(`n = 50` = p_delta(s50, d0), `n = 150` = p_delta(s150, d0))) |> round(3)

        Tragedy (delta = -0.3) Craft (delta = 0) ADT (delta = 0.3)
n = 50                   0.001             0.054             0.660
n = 150                  0.000             0.000             0.316

At n = 50 the data reject the tragedy account’s value but cannot separate zero from 0.3. At n = 150 zero is rejected and 0.3 is not. The frequentist route falsifies values that are incompatible with the data; it does not quantify how much the data support one surviving value over another. For range predictions (“at least 0.2”), use minimum-effect and equivalence tests (Lakens et al., 2018; Scene 3).

What each framework lets you say

Frequentist: “The data are incompatible with the tragedy account’s effect of −0.3 (p < .001) and compatible with both zero and 0.3 (n = 50).”
Bayesian: “The data were about 5 times more likely under the ADT account than under no effect, and 8 times more likely under no effect than under the tragedy account.”

Theory-building takeaway. A Bayes factor is only as meaningful as the hypotheses it compares. Default priors test a default theory. To test your theory, write its prediction down as a prior, and report how conclusions change across defensible alternatives.

Your turn

Make the ADT prior more confident: Normal(0.3, 0.05). Does its Bayes factor go up or down at n = 50? Why?
Write a prior for a “large effect” account, Normal(0.8, 0.2). How does it fare at n = 150, where the observed d is 0.42?

--- title: "Scene 2 · Priors as theories" subtitle: "A Bayes factor is a prediction contest" --- ```{r} #| label: setup #| include: false source("R/helpers.R") suppressPackageStartupMessages({ library(BayesFactor) library(patchwork) }) knitr::opts_chunk$set(fig.align = "center") ``` So far ADT has made one directional prediction: more enjoyment after the rewarded ending. Theory building usually means weighing rival explanations of the same result, so here are two rivals. To compare the three, each account now states how big δ should be. All three are stylized (simplified for this demo, not summaries of a literature): | Account | Mechanism | Prediction for δ | |---|---|---| | **ADT** | Viewers like the character who did good, so they enjoy seeing that character rewarded more than punished. | small and positive (around 0.3) | | **Craft** | Enjoyment comes from how well the story is told, not from how it ends. | zero | | **Tragedy** | Watching a good person suffer is moving, and moving stories are enjoyed more. | small and negative (around −0.3) | ::: {.callout-important title="Predict first"} Each account above makes a prediction about δ. Using the n = 50 data (observed *d* = 0.39), which account wins, and by roughly how much? ::: A theory that predicts "δ is around 0.3" can be written as a **prior on δ**. That prior implies which *t* statistics the study should produce. The Bayes factor is the ratio of how well two hypotheses predicted the *t* value we actually observed (Rouder et al., 2009; Gronau et al., 2020). Treating priors as part of the theory is the core of Vanpaemel's (2010) defense of Bayes factors for theory testing. ```{r} #| label: load #| code-fold: true #| code-summary: "Load the Study 1 data" s1_50 <- read.csv("data/story_n50.csv") s1_150 <- read.csv("data/story_n150.csv") s50 <- two_group(s1_50) s150 <- two_group(s1_150) ``` ```{r} #| label: s2-priors hyps <- list( "Craft (δ = 0)" = "null", "ADT: Normal(0.3, 0.15)" = prior_adt, "Tragedy: Normal(−0.3, 0.15)" = prior_tragedy, "JASP default: Cauchy(0, 0.707)" = prior_default ) ``` ```{r} #| label: fig-predict #| code-fold: true #| fig-height: 7 #| fig-cap: "Top: each account's prior on δ (the craft account is a spike at zero). Bottom: the t statistics each account predicts. The dotted vertical line is the observed t. The Bayes factor between two accounts is the ratio of their curve heights at that line." cols <- c("Craft (δ = 0)" = pal[["gray"]], "ADT: Normal(0.3, 0.15)" = pal[["blue"]], "Tragedy: Normal(−0.3, 0.15)" = pal[["orange"]], "JASP default: Cauchy(0, 0.707)" = pal[["aqua"]]) short <- c("Craft", "ADT", "Tragedy", "Default") dl <- seq(-1.5, 1.5, by = 0.01) pri <- do.call(rbind, lapply(names(hyps)[-1], function(h) data.frame(x = dl, y = hyps[[h]](dl), h = h))) pri_lab <- data.frame(x = c(0.78, -0.78, 1.15), y = c(2.2, 2.2, 0.28), h = names(hyps)[-1], lab = short[-1]) pA <- ggplot(pri, aes(x, y, color = h)) + geom_segment(aes(x = 0, xend = 0, y = 0, yend = 3.1), color = pal[["gray"]], linewidth = 1.3, arrow = arrow(length = unit(0.25, "cm")), inherit.aes = FALSE) + annotate("text", x = 0, y = 3.38, label = "Craft", hjust = 0.5, size = 4.3, color = pal[["ink2"]], fontface = "bold") + geom_line(linewidth = 1.1) + geom_text(data = pri_lab, aes(label = lab), size = 4.3, fontface = "bold") + scale_color_manual(values = cols) + coord_cartesian(ylim = c(0, 3.5), xlim = c(-1.4, 1.4)) + labs(x = "Standardized effect δ", y = "Prior density", title = "What each account predicts about δ") + theme_demo() tg <- seq(-6, 8, by = 0.05) pred <- do.call(rbind, lapply(list(list(s50, "n = 50 per group"), list(s150, "n = 150 per group")), function(z) { do.call(rbind, lapply(names(hyps), function(h) data.frame(t = tg, y = pred_t(tg, z[[1]], hyps[[h]]), h = h, n = z[[2]]))) })) pred$n <- factor(pred$n, levels = c("n = 50 per group", "n = 150 per group")) obs <- data.frame(n = factor(c("n = 50 per group", "n = 150 per group"), levels = levels(pred$n)), t = c(s50$t, s150$t)) pk <- data.frame( n = factor(rep(levels(pred$n), each = 4), levels = levels(pred$n)), h = rep(names(hyps), 2), lab = rep(short, 2), t = c(-1.6, 4.7, -3.9, 5.6, -1.6, 5.9, -4.4, 6.1), y = c(0.415, 0.25, 0.3, 0.1, 0.415, 0.2, 0.26, 0.085)) pB <- ggplot(pred, aes(t, y, color = h)) + geom_line(linewidth = 1.1) + geom_vline(data = obs, aes(xintercept = t), linewidth = 0.9, linetype = "dotted", color = pal[["ink2"]]) + geom_text(data = obs, aes(x = t, y = 0.49, label = sprintf("observed t = %.2f", t)), hjust = 1.05, size = 4.2, color = pal[["ink2"]], inherit.aes = FALSE) + geom_text(data = pk, aes(label = lab), size = 4, fontface = "bold", vjust = 0) + facet_wrap(~n) + scale_color_manual(values = cols) + coord_cartesian(ylim = c(0, 0.51)) + labs(x = "t statistic", y = "Predicted density", title = "What each account predicts about the t statistic") + theme_demo() pA / pB ``` ::: {.callout-note appearance="simple" title="How to read this figure"} 1. **Top panel:** each account's prior on δ, its prediction about the true effect before seeing data. Normal(0.3, 0.15) means "centered at 0.3, with 95% of its weight between 0 and 0.6." The craft account puts all its weight on exactly zero (the arrow). 2. **Bottom panels:** what each account predicts the study's *t* statistic will be. Imagine running the study thousands of times with true effects drawn from the account's prior, and recording each *t*. The curve is the distribution of those *t* values (the *prior predictive distribution*). 3. **The dotted vertical line** is the *t* we actually observed. 4. **The Bayes factor between any two accounts is the ratio of their curves' heights at that line.** At n = 50, the blue (ADT) curve is about five times higher than the gray (craft) curve there. 5. **Compare the two panels.** The craft prediction does not change with sample size, but the other accounts' predictions move away from zero as *n* grows, because the same true effect produces larger *t* values in larger samples. Bigger studies make theories easier to tell apart. ::: ```{r} #| label: s2-bf bf_table <- sapply(list(`n = 50` = s50, `n = 150` = s150), function(s) sapply(hyps[-1], function(pr) bf_vs_null(s, pr))) rownames(bf_table) <- c("ADT vs. null", "Tragedy vs. null", "Default vs. null") round(rbind(bf_table, "ADT vs. tragedy" = bf_table[1, ] / bf_table[2, ], "ADT vs. default" = bf_table[1, ] / bf_table[3, ]), 2) ``` ```{r} #| label: s2-check # Check our prediction-contest function against BayesFactor (the engine JASP uses) c(ours = bf_vs_null(s50, prior_default), BayesFactor = extractBF(bf_default(s1_50))$bf) ``` Read the table as a contest against the craft account, which predicts δ = 0 (the null): - At *n* = 50, **ADT beats the null by about `r round(bf_table[1, 1])` to 1**, while the vague default alternative barely ties it (`r fmt_bf(bf_table[3, 1])`). Same data, different verdicts, because the hypotheses made different predictions. - **Tragedy loses badly** (BF₁₀ ≈ `r sprintf("%.2f", bf_table[2, 1])`): it predicted negative *t* values and the data came out positive. At *n* = 150 it looks slightly *less* bad against the null (`r sprintf("%.2f", bf_table[2, 2])`), only because the null is now failing too. Bayes factors are relative: they say which of two hypotheses predicted better, not whether either one is any good. At *n* = 150, tragedy loses to ADT by more than 1,000 to 1. - At *n* = 150, ADT is favored over the null by about `r round(bf_table[1, 2], -1)` to 1, and over the default by about `r round(bf_table[1, 2] / bf_table[3, 2], 1)` to 1. A precise theory sticks its neck out. When it is right, it earns more evidence than a vague one; when it is wrong, it loses more. Meehl (1967) argued that a theory earns corroboration only by making predictions precise enough to fail; the Bayes factor puts a number on that idea. ## How much does the prior width matter? This recreates JASP's robustness check: the default hypothesis with Cauchy widths from 0.05 to 1.5. ```{r} #| label: fig-robust #| code-fold: true #| fig-height: 4.2 #| fig-cap: "Bayes factor for a Cauchy alternative as its width changes. Wider priors predict larger effects and are penalized when the observed effect is small." r <- seq(0.05, 1.5, by = 0.025) rob <- rbind(data.frame(r, bf = sapply(r, function(w) bf_vs_null(s50, function(x) dcauchy(x, 0, w))), n = "n = 50 per group"), data.frame(r, bf = sapply(r, function(w) bf_vs_null(s150, function(x) dcauchy(x, 0, w))), n = "n = 150 per group")) rl <- do.call(rbind, lapply(split(rob, rob$n), function(z) z[z$r == 1.5, ])) ggplot(rob, aes(r, bf, color = n)) + geom_hline(yintercept = c(1, 3, 10), color = pal[["gray"]], linetype = c("solid", "dashed", "dashed"), linewidth = 0.4) + geom_vline(xintercept = sqrt(2) / 2, color = pal[["gray"]], linetype = "dotted") + geom_line(linewidth = 1.2) + geom_text(data = rl, aes(label = n), hjust = 1, vjust = -0.8, size = 4.4, fontface = "bold") + annotate("text", x = 0.72, y = 150, label = "JASP default (0.707)", hjust = 0, size = 4, color = pal[["ink2"]]) + scale_y_log10(breaks = c(0.3, 1, 3, 10, 30, 100), limits = c(0.3, 200)) + scale_color_manual(values = c("n = 50 per group" = pal[["blue"]], "n = 150 per group" = pal[["aqua"]])) + labs(x = "Cauchy prior width r", y = "BF10 (log scale)") + theme_demo() ``` ::: {.callout-note appearance="simple" title="How to read this figure"} 1. **Horizontal axis:** the width (scale) of the default Cauchy prior. Small widths say "expect small effects"; large widths say "expect large ones." 2. **Vertical axis:** BF₁₀ on a log scale. The solid line at 1 means no preference; the dashed lines at 3 and 10 mark JASP's "moderate" and "strong" labels. 3. **The dotted vertical line** is JASP's default width (0.707). 4. **A flat curve** means the conclusion does not depend on the prior's width; **a sloped curve** means it does. ::: At *n* = 50 every width gives weak evidence at best, and wide priors tip toward the null. At *n* = 150 every reasonable width favors an effect. **A prior sensitivity analysis is how you show readers which conclusions depend on your theory's specifics and which do not.** ## The frequentist counterpart: test the theory's value, not only zero A *p* value can test any hypothesized value, not only zero (Greenland et al., 2016). Testing each account's central prediction: ```{r} #| label: s2-freq p_delta <- function(s, d0) { # exact test of H0: delta = d0 via the noncentral t lo <- pt(s$t, s$df, ncp = d0 * sqrt(s$neff)); 2 * min(lo, 1 - lo) } sapply(c(`Tragedy (delta = -0.3)` = -0.3, `Craft (delta = 0)` = 0, `ADT (delta = 0.3)` = 0.3), function(d0) c(`n = 50` = p_delta(s50, d0), `n = 150` = p_delta(s150, d0))) |> round(3) ``` At *n* = 50 the data reject the tragedy account's value but cannot separate zero from 0.3. At *n* = 150 zero is rejected and 0.3 is not. The frequentist route **falsifies** values that are incompatible with the data; it does not quantify how much the data support one surviving value over another. For range predictions ("at least 0.2"), use minimum-effect and equivalence tests (Lakens et al., 2018; [Scene 3](scene-3.qmd)). ::: {.callout-tip title="What each framework lets you say"} - **Frequentist:** "The data are incompatible with the tragedy account's effect of −0.3 (*p* < .001) and compatible with both zero and 0.3 (*n* = 50)." - **Bayesian:** "The data were about `r round(bf_table[1, 1])` times more likely under the ADT account than under no effect, and `r round(1 / bf_table[2, 1])` times more likely under no effect than under the tragedy account." ::: **Theory-building takeaway.** A Bayes factor is only as meaningful as the hypotheses it compares. Default priors test a default theory. To test *your* theory, write its prediction down as a prior, and report how conclusions change across defensible alternatives. ::: {.callout-warning title="Your turn"} 1. Make the ADT prior more confident: Normal(0.3, 0.05). Does its Bayes factor go up or down at *n* = 50? Why? 2. Write a prior for a "large effect" account, Normal(0.8, 0.2). How does it fare at *n* = 150, where the observed *d* is 0.42? :::