Bayesian estimation with uSDT

This article fits the tutorial’s model and data with brms and Stan, then summarises the posterior and computes Bayes factors.

Bayesian fitting needs a C++ toolchain and either rstan (with BH and RcppEigen) or cmdstanr with CmdStan. The brms.backend option selects the backend. The default is rstan and this article uses cmdstanr. Compiled models are cached for later sessions.

The data

We use the paired data exactly as the tutorial prepares them:

data_usdt <- usdt_data_tasks(
  awareness = vadillo_awareness,
  influence = vadillo_cuing,
  subject_col      = "subj",
  condition_col    = "condition",
  condition_levels = c(signal = "old", noise = "new"),
  response_col     = list(awareness = "judged.old", influence = "rt"),
  response_levels  = list(awareness = c(signal = 1, noise = 0),
                          influence = c(signal = "faster", noise = "slower")),
  dichotomize      = list(awareness = FALSE, influence = TRUE)
)

Choosing the priors

usdt_priors() specifies the priors. Each prior is a string in Stan notation, and the defaults are weakly informative on the probit scale:

priors <- usdt_priors()
priors

── Priors ────────────────────────────────────────────────────────────────────── 

  Parameter    Task       Prior
  mean d'      awareness  normal(0, 1)
  mean d'      influence  normal(0, 1)
  sd(d')       awareness  student_t(4, 0, 0.5)
  sd(d')       influence  student_t(4, 0, 0.5)
  cor(d')      both       lkj(1)
  mean c       awareness  normal(0, 0.5)
  mean c       influence  normal(0, 0.5)
  sd(c)        awareness  student_t(4, 0, 0.5)
  sd(c)        influence  student_t(4, 0, 0.5)
  cor(c)       both       lkj(1)
  sd(signal)   awareness  lognormal(0, 0.5)
  sd(signal)   influence  lognormal(0, 0.5)
  intercept    both       normal(0, 1)

  Standard deviation priors are truncated at zero. cor(c) applies when
  both criteria are estimated, and sd(signal) under unequal variances.
  Under the latent regression, intercept replaces the influence mean d'.

Means of $d'$ and of the criterion: normal priors centred at zero, normal(0, 1) for $d'$ and normal(0, 0.5) for $c$.
Between-subject standard deviations: Student-$t$ priors truncated at zero, student_t(4, 0, 0.5).
Correlations: lkj(1), a uniform prior on $(-1, 1)$. Larger values of its argument concentrate the correlation around zero.
Intercept: used only by the latent regression parameterization, at the end of this article, in place of the prior of the influence mean.

Supported families are listed in usdt_priors(). Means need full real-line support. SDs need positive support or truncation at zero.

The model only uses the priors of the parameters it estimates. Here the median split fixes the influence criterion at zero and both tasks have equal variances, so the influence criterion, the criterion correlation and the signal standard deviations play no role.

Checking what the priors imply

Before fitting anything, plot() simulates the priors to show what they say about the data and the hypotheses:

plot(priors, data = data_usdt)

plot of chunk prior-check

Hit and false-alarm rates (top left): the rates of a new subject spread over the whole $(0, 1)$ range without piling up at either end. The influence rates are more concentrated because the median split fixes their criterion.
H1 and H2 (top right, bottom left): the priors of $\Delta d'$, a normal centred at zero, and of $\rho$, uniform.
H3 (bottom right): the intercept of the latent regression has no prior of its own. Its prior follows from those of the means, standard deviations and correlation, and is symmetric around zero with heavy tails.

To change a prior, pass its new value to usdt_priors(), either for both tasks or per task, and the result to hsdt():

priors_custom <- usdt_priors(
  dprime    = list(awareness = "normal(0, 1)", influence = "normal(0, 0.5)"),
  sd_dprime = "gamma(2, 4)"
)
hsdt(data_usdt, estimation = "bayesian", priors = priors_custom)

Fitting the model

Setting estimation = "bayesian" in hsdt() fits the model with brms:

fit_bayes <- hsdt(data_usdt, estimation = "bayesian", seed = 2026)

Defaults are four chains, each with 1,000 warmup iterations and 5,000 kept draws. Inspect divergences, tree-depth warnings, R-hat and effective sample size before interpreting the fit. Sampling options include chains, iter (including warmup), warmup, cores, seed and control. backend selects the Stan interface.

The summary has the same structure as the frequentist one:

summary(fit_bayes)

── Model summary ─────────────────────────────────────────────────────────────── 

  Subjects:       104
  Observations:   416 aggregated rows (46,592 trials)
  Family:         binomial (probit), equal variances
  Criteria:       Awareness estimated, Influence fixed to 0 (Meyen split, mean |c| = 0.0000)
  Estimation:     brms via cmdstanr, 4 chains x 5,000 draws (warmup 1,000)
  Diagnostics:    max R-hat 1.002, min ESS 1,880, 0 divergent

Loading required namespace: rstan


── Fixed effects ───────────────────────────────────────────────────────────────

  Parameter      Task           Mean       SD  95% CrI             R-hat     ESS
  criterion      Awareness   -0.0356   0.0289  [ -0.092,  0.021]   1.000   9,191
  d'             Awareness    0.2350   0.0330  [  0.170,  0.300]   1.000  32,301
  d'             Influence    0.1283   0.0153  [  0.098,  0.158]   1.000  26,275

── Random effects ──────────────────────────────────────────────────────────────

  Parameter      Task           Mean       SD  95% CrI             R-hat     ESS
  sd(criterion)  Awareness    0.2513   0.0249  [  0.205,  0.303]   1.000   9,337
  sd(d')         Awareness    0.1073   0.0597  [  0.006,  0.224]   1.000   4,354
  sd(d')         Influence    0.0864   0.0219  [  0.039,  0.127]   1.000   5,186
  cor(d')        both         0.2902   0.4448  [ -0.740,  0.954]   1.002   1,880

── Hypotheses ──────────────────────────────────────────────────────────────────

H1: Group-level sensitivity difference (Δd' = Influence d' - Awareness d')
  Parameter           Mean       SD  95% CrI            p-value
  Δd' (I - A)     -0.1067   0.0358  [ -0.177, -0.036]     .002

H2: Correlation between sensitivities across tasks
  Parameter           Mean       SD  95% CrI            p-value
  rho               0.2902   0.4448  [ -0.740,  0.954]     .484

H3: Latent regression of Influence d' on Awareness d'
  Parameter           Mean       SD  95% CrI            p-value
  Intercept         0.0026   7.9765  [ -0.780,  0.761]     .512
  Slope             0.5406  34.1561  [ -2.712,  3.862]     .484

── Priors ──────────────────────────────────────────────────────────────────────

  Parameter    Task       Prior
  mean d'      Awareness  normal(0, 1)
  mean d'      Influence  normal(0, 1)
  sd(d')       Awareness  student_t(4, 0, 0.5)
  sd(d')       Influence  student_t(4, 0, 0.5)
  cor(d')      both       lkj(1)
  mean c       Awareness  normal(0, 0.5)
  sd(c)        Awareness  student_t(4, 0, 0.5)

── Notes ───────────────────────────────────────────────────────────────────────

  Mean, SD and CrI are the posterior mean, posterior SD and central
  credible interval. The p-value is the two-sided posterior p-value.

Estimates are posterior means, with posterior SDs and central 95% credible intervals (CrIs). Mean sensitivities are $d' = 0.24$ in the awareness task and $d' = 0.13$ in the influence task.

The brms fit

fit_bayes$fit is a brmsfit. Its non-linear parameters include dA and dI (sensitivities) and cA (the classical awareness criterion). Inspect their priors with:

prior_table <- brms::prior_summary(fit_bayes$fit)
prior_table[prior_table$source == "user", ]

                prior class coef  group resp dpar nlpar lb ub tag source
 lkj_corr_cholesky(1)     L        subj                             user
       normal(0, 0.5)     b                          cA             user
         normal(0, 1)     b                          dA             user
         normal(0, 1)     b                          dI             user
 student_t(4, 0, 0.5)    sd        subj              dA             user
 student_t(4, 0, 0.5)    sd        subj              dI             user
 student_t(4, 0, 0.5)    sd      subj_c              cA             user

Criteria use a separate subject block, subj_c. Check model adequacy with posterior predictive comparisons as well as sampler diagnostics:

brms::pp_check(fit_bayes$fit)

The three hypotheses

The p-value of each hypothesis is now the two-sided posterior p-value, $2 \min\{P(\theta > 0 \mid y), P(\theta < 0 \mid y)\}$. It falls below .05 when the central 95% credible interval excludes zero (up to finite-draw rounding). This is posterior sign uncertainty, not a frequentist p-value or a Bayes factor.

H1 ($\Delta d'$): awareness of the repeated patterns is again stronger than their influence on search ($\Delta d' = -0.11$, 95% CrI $[-0.18, -0.04]$, $p = .002$), the same conclusion as in the tutorial.
H2 ($\rho$): the posterior of the latent correlation spans almost the whole range (95% CrI $[-0.74, 0.95]$). The data say little about it.
H3 (latent regression): the 95% credible interval of the intercept covers zero ($[-0.78, 0.76]$). Its posterior mean ($0.00$) and standard deviation ($7.98$) are dominated by heavy tails: the awareness sensitivity barely varies across subjects ($\text{sd}(d') = 0.11$), so the slope $\rho \, \sigma_I / \sigma_A$ can take very large values. With posteriors like this one, emphasise the credible interval.

Bayes factors

A credible interval that covers zero does not show that an effect is absent. usdt_bf() weighs the evidence for and against each hypothesis. By default, it tests the three point nulls with the Savage-Dickey density ratio and draws the prior and posterior of each tested quantity:

set.seed(2026)
bf <- usdt_bf(fit_bayes)

plot of chunk bf

bf

 uSDT hypothesis ─ Savage-Dickey density ratio test
════════════════════════════════════════════════════════════════════════════════
  Hypothesis (H1)             : Δd' != 0
  Hypothesis (H0)             : Δd' = 0
  Posterior mean              : -0.107
  Posterior SD                : 0.036
  95% CrI                     : [-0.177, -0.036]
  log BF10                    : 0.809
  log BF01                    : -0.809
  Evidence                    : anecdotal for H1
────────────────────────────────────────────────────────────────────────────────

 uSDT hypothesis ─ Savage-Dickey density ratio test
════════════════════════════════════════════════════════════════════════════════
  Hypothesis (H1)             : rho != 0
  Hypothesis (H0)             : rho = 0
  Posterior mean              : 0.290
  Posterior SD                : 0.445
  95% CrI                     : [-0.740, 0.954]
  log BF10                    : -0.151
  log BF01                    : 0.151
  Evidence                    : anecdotal for H0
────────────────────────────────────────────────────────────────────────────────

 uSDT hypothesis ─ Savage-Dickey density ratio test
════════════════════════════════════════════════════════════════════════════════
  Hypothesis (H1)             : intercept != 0
  Hypothesis (H0)             : intercept = 0
  Posterior mean              : 0.003
  Posterior SD                : 7.976
  95% CrI                     : [-0.780, 0.761]
  log BF10                    : -1.909
  log BF01                    : 1.909
  Evidence                    : moderate for H0
────────────────────────────────────────────────────────────────────────────────

A point statement names H0. An inequality or region names H1. Reports give both log Bayes factors. Each plot panel reports evidence for the hypothesis in its title: BF01 for a point null, BF10 for a direction or region.

$\Delta d' = 0$: the evidence for a difference is anecdotal ($\log \text{BF}_{01} = -0.81$, $\text{BF}_{10} = 2.2$), even though the credible interval excludes zero. The Bayes factor compares the posterior with the whole prior, and a $\text{normal}(0, 1)$ prior on each mean expects much larger differences than $-0.11$.
$\rho = 0$: the evidence is anecdotal ($\log \text{BF}_{01} = 0.15$).
Intercept $\beta_0 = 0$: there is moderate evidence for the null ($\log \text{BF}_{01} = 1.91$, $\text{BF}_{01} = 6.7$) under these priors. This compares the zero latent intercept with its specified alternative. It does not identify individual participants without awareness.

Solid curves show posteriors and dashed curves priors. For a point null, BF01 is posterior density divided by prior density at the dotted line. Use usdt_bf(..., plot = FALSE) to compute without drawing, then plot(bf) to draw later. caption = FALSE hides the caption.

You can also test directional hypotheses and regions, such as a region of practical equivalence around zero:

set.seed(2026)
bf_other <- usdt_bf(fit_bayes, c("intercept > 0", "diff in [-0.1, 0.1]"),
                    plot = FALSE)
bf_other

 uSDT hypothesis ─ Directional test
════════════════════════════════════════════════════════════════════════════════
  Hypothesis (H1)             : intercept > 0
  Hypothesis (H0)             : intercept <= 0
  Posterior mean              : 0.003
  Posterior SD                : 7.976
  95% CrI                     : [-0.780, 0.761]
  Posterior probability (H1)  : 0.744
  log BF10                    : 1.067
  log BF01                    : -1.067
  Evidence                    : anecdotal for H1
────────────────────────────────────────────────────────────────────────────────

 uSDT hypothesis ─ Interval test
════════════════════════════════════════════════════════════════════════════════
  Hypothesis (H1)             : Δd' in [-0.1, 0.1]
  Hypothesis (H0)             : Δd' out [-0.1, 0.1]
  Posterior mean              : -0.107
  Posterior SD                : 0.036
  95% CrI                     : [-0.177, -0.036]
  Posterior probability (H1)  : 0.421
  log BF10                    : 2.503
  log BF01                    : -2.503
  Evidence                    : strong for H1
────────────────────────────────────────────────────────────────────────────────

These Bayes factors compare the posterior odds of H1, the share of posterior draws inside it, with its prior odds. A positive intercept has a posterior probability of .744 against .500 a priori, which is anecdotal evidence. The region $[-0.1, 0.1]$ holds 42% of the posterior but only about 6% of the prior, so the Bayes factor is strong ($\log \text{BF}_{10} = 2.50$) even though the posterior mean lies just outside it. Read interval Bayes factors together with the posterior probability.

Bayes factors depend on the priors. Report the priors and check sensitivity to other defensible choices by refitting. Changing prior width need not have the same effect for every hypothesis.

Visualizing the posterior

All the plots of the tutorial work with a Bayesian fit, and show posterior means with credible intervals.

The latent regression

plot(fit_bayes)

plot of chunk plot-regression

The left panel is unchanged. In the right panel, the latent line uses the posterior means, and its 95% credible band fans out away from the group means because the slope is so uncertain. The vertical segment at $d'_{\text{Awareness}} = 0$ is the credible interval of the intercept.

Subject intervals

plot(fit_bayes, type = "caterpillar")

plot of chunk plot-caterpillar

The right panels show each participant’s posterior mean $d'$ with its 95% credible interval. Partial pooling pulls the estimates towards the group mean and narrows their intervals. type = "shrinkage" and type = "roc" work as in the tutorial.

The latent regression parameterization

The intercept of H3 has no prior of its own above: the model places its priors on the means, standard deviations and correlation of the two sensitivities, and the intercept prior they induce has very heavy tails. Its Bayes factor depends on all of them. With latent = "regression", the prior of the influence mean moves to the intercept $\beta_0$ of the latent regression, the last row of usdt_priors(). The influence mean follows from it as $\mu_I = \beta_0 + \rho \, (\sigma_I / \sigma_A) \, \mu_A$, and the priors of the awareness mean, the standard deviations and the correlation do not change.

When $\sigma_A$ approaches zero, the ratio $\sigma_I/\sigma_A$ can produce difficult posterior geometry. A prior that gives less weight near zero can help, if substantively justified. Here we use gamma(2, 4) for $\sigma_A$ and retain the default for $\sigma_I$:

priors_reg <- usdt_priors(sd_dprime = list(awareness = "gamma(2, 4)",
                                           influence = "student_t(4, 0, 0.5)"))
plot(priors_reg, data = data_usdt, latent = "regression")

plot of chunk prior-check-regression

H3 now has the specified intercept prior (default normal(0, 1)), while H1 has the induced prior $\Delta d' = \beta_0 + (\rho\,\sigma_I/\sigma_A-1)\mu_A$. Inspect both the hypothesis and predictive implications: this choice changes the prior model, even though it reports the same quantities.

fit_reg <- hsdt(data_usdt, estimation = "bayesian", priors = priors_reg,
                latent = "regression", seed = 2026)

The model samples the same parameters as before, so the fit reports the same tables and hypotheses, and its header names the parameterization:

summary(fit_reg)

── Model summary ─────────────────────────────────────────────────────────────── 

  Subjects:       104
  Observations:   416 aggregated rows (46,592 trials)
  Family:         binomial (probit), equal variances
  Model:          latent regression of Influence d' on Awareness d'
  Criteria:       Awareness estimated, Influence fixed to 0 (Meyen split, mean |c| = 0.0000)
  Estimation:     brms via cmdstanr, 4 chains x 5,000 draws (warmup 1,000)
  Diagnostics:    max R-hat 1.002, min ESS 2,578, 0 divergent

── Fixed effects ───────────────────────────────────────────────────────────────

  Parameter      Task           Mean       SD  95% CrI             R-hat     ESS
  criterion      Awareness   -0.0357   0.0294  [ -0.094,  0.021]   1.001   9,838
  d'             Awareness    0.2345   0.0343  [  0.167,  0.301]   1.000  34,365
  d'             Influence    0.1282   0.0154  [  0.098,  0.158]   1.001  25,396

── Random effects ──────────────────────────────────────────────────────────────

  Parameter      Task           Mean       SD  95% CrI             R-hat     ESS
  sd(criterion)  Awareness    0.2515   0.0254  [  0.205,  0.305]   1.000   8,970
  sd(d')         Awareness    0.1330   0.0529  [  0.034,  0.238]   1.001   6,784
  sd(d')         Influence    0.0866   0.0213  [  0.041,  0.126]   1.000   6,732
  cor(d')        both         0.3342   0.4010  [ -0.560,  0.954]   1.002   2,578

── Hypotheses ──────────────────────────────────────────────────────────────────

H1: Group-level sensitivity difference (Δd' = Influence d' - Awareness d')
  Parameter           Mean       SD  95% CrI            p-value
  Δd' (I - A)     -0.1062   0.0368  [ -0.178, -0.034]     .004

H2: Correlation between sensitivities across tasks
  Parameter           Mean       SD  95% CrI            p-value
  rho               0.3342   0.4010  [ -0.560,  0.954]     .402

H3: Latent regression of Influence d' on Awareness d'
  Parameter           Mean       SD  95% CrI            p-value
  Intercept         0.0653   0.1236  [ -0.204,  0.266]     .365
  Slope             0.2695   0.5224  [ -0.568,  1.421]     .402

── Priors ──────────────────────────────────────────────────────────────────────

  Parameter    Task       Prior
  mean d'      Awareness  normal(0, 1)
  sd(d')       Awareness  gamma(2, 4)
  sd(d')       Influence  student_t(4, 0, 0.5)
  cor(d')      both       lkj(1)
  mean c       Awareness  normal(0, 0.5)
  sd(c)        Awareness  student_t(4, 0, 0.5)
  intercept    both       normal(0, 1)

── Notes ───────────────────────────────────────────────────────────────────────

  Mean, SD and CrI are the posterior mean, posterior SD and central
  credible interval. The p-value is the two-sided posterior p-value.

$\Delta d' = -0.11$ (95% CrI $[-0.18, -0.03]$) as before. The intercept, with a prior of its own and $\sigma_A$ kept away from zero, has the 95% credible interval $[-0.20, 0.27]$, against $[-0.78, 0.76]$ when its prior was induced by the others.

The interval of $\rho$ is now $[-0.56, 0.95]$, against $[-0.74, 0.95]$ before. Both the intercept prior and the SD prior changed. This comparison does not isolate either one’s effect. The intercept prior also affects the correlation posterior and Bayes factor:

set.seed(2026)
usdt_bf(fit_reg, c("intercept = 0", "rho = 0"), plot = FALSE)

 uSDT hypothesis ─ Savage-Dickey density ratio test
════════════════════════════════════════════════════════════════════════════════
  Hypothesis (H1)             : intercept != 0
  Hypothesis (H0)             : intercept = 0
  Posterior mean              : 0.065
  Posterior SD                : 0.124
  95% CrI                     : [-0.204, 0.266]
  log BF10                    : -1.752
  log BF01                    : 1.752
  Evidence                    : moderate for H0
────────────────────────────────────────────────────────────────────────────────

 uSDT hypothesis ─ Savage-Dickey density ratio test
════════════════════════════════════════════════════════════════════════════════
  Hypothesis (H1)             : rho != 0
  Hypothesis (H0)             : rho = 0
  Posterior mean              : 0.334
  Posterior SD                : 0.401
  95% CrI                     : [-0.560, 0.954]
  log BF10                    : -0.185
  log BF01                    : 0.185
  Evidence                    : anecdotal for H0
────────────────────────────────────────────────────────────────────────────────

Where to go from here

Reliability

usdt_reliability() returns the posterior of each task’s reliability:

reliability <- usdt_reliability(fit_bayes)
reliability

── Reliability summary ───────────────────────────────────────────────────────── 

  Task         Subjects Group mean       SD  95% CrI              By-subject mean median [min, max]
  Awareness         104       .118     .096  [  0.000,  0.331]    .119 [.110, .120]
  Influence         104       .311     .102  [  0.086,  0.495]    .311 [.310, .312]

  Means, SDs and credible intervals summarise the posterior draws.

The posterior means (.118 and .311) are close to the frequentist estimates, and the credible intervals show how uncertain they are: the reliability of the awareness task could be anywhere between 0.00 and .331.

Unequal signal and noise variances

unequal_variances = TRUE is available for Bayesian fits:

fit_uv <- hsdt(data_usdt, estimation = "bayesian",
               unequal_variances = TRUE, seed = 2026)

With one binary operating point per task, signal SDs are weakly identified, so check prior sensitivity. H1–H3 then use the separation $d$ in noise-SD units. To compare discriminability across different signal SDs, transform each draw to $d_a=d\sqrt{2/(1+\sigma_s^2)}$. See hsdt(). The equal-variance tutorial does not fit this extension.

Custom models fitted with brms

When you need covariates, you can fit your own model with brms and test it with the same functions. usdt_aggregate() counts the trials first, which makes the fit much faster. usdt_bf() reads the priors from the model, so the two sensitivities need proper priors from the families of usdt_priors():

trials <- rbind(
  data.frame(vadillo_awareness[c("subj", "condition", "set.size")],
             task = "A", resp = vadillo_awareness$judged.old),
  data.frame(vadillo_cuing[c("subj", "condition", "set.size")],
             task = "I", resp = meyen_split(vadillo_cuing$rt,
                                            by = vadillo_cuing$subj))
)
trials <- within(trials, {
  cond <- ifelse(condition == "old", 0.5, -0.5)
  size <- ifelse(set.size == "set size 16", 0.5, -0.5)
  awareness <- as.integer(task == "A")
})
counts <- usdt_aggregate(trials, response = "resp",
                         by = c("subj", "task", "cond", "size", "awareness"))

fit_brms <- brms::brm(
  y | trials(n) ~ 0 + awareness + task:size + task:cond +
    (0 + awareness | subj) + (0 + task:cond | subj),
  data = counts, family = binomial("probit"),
  prior = brms::prior(normal(0, 1), class = b)
)
usdt_tests(fit_brms, awareness = "taskA:cond", influence = "taskI:cond")
usdt_bf(fit_brms, awareness = "taskA:cond", influence = "taskI:cond")

The examples of usdt_tests() build counts from the Vadillo data with set size as a covariate.