n1post: Posterior Distribution for n=1 Data

In nisone: Inference for Samples of Size at Least One

Posterior Distribution for n=1 Data

Description

Density, probability, quantile, and random generation from the posterior distribution of the location parameter when one has n = 1 observation and uses a particular prior that results in valid confidence intervals (asymptotically in the confidence level).

Usage

nun1post(fam = c("normal", "cauchy"), ddist = NULL, ...)

dn1post(
  x,
  A,
  obs,
  nu = NULL,
  fam = c("normal", "cauchy"),
  ddist = NULL,
  log = FALSE,
  ...
)

pn1post(
  q,
  A,
  obs,
  nu = NULL,
  fam = c("normal", "cauchy"),
  pdist = NULL,
  lower.tail = TRUE,
  log.p = FALSE,
  ...
)

qn1post(
  p,
  A,
  obs,
  nu = NULL,
  fam = c("normal", "cauchy"),
  qdist = NULL,
  pdist = NULL,
  ...
)

rn1post(n, A, obs, nu = NULL, fam = c("normal", "cauchy"), rdist = NULL, ...)

Arguments

fam	One of "normal" or "cauchy". If ddist, pdist, qdist, or rdist are specified then this argument is ignored.
ddist	Density function of standard distribution \rho(). Should have a log argument.
...	Additional arguments for ddist, pdist, qdist, and rdist.
x, q	vector of quantiles
A	The prior value.
obs	The observed value. This is X in the description.
nu	The prior value of nu. If not known, use nun1post()
log, log.p	logical; if TRUE, probabilities p are given as log(p).
pdist	Cumulative distribution function of standard distribution.
lower.tail	logical; if TRUE (default), probabilities are P(X<=x) otherwise, P(X>x).
p	vector of probabilities
qdist	Quantile function of standard distribution.
n	sample size
rdist	Random generation of standard distribution.

Details

Let X have PDF \frac{1}{\sigma}\rho((x - \mu)/\sigma) for a known and symmetric \rho(\cdot). Let \nu = |\mu - A| / \sigma for a pre-specified A. We place prior \pi(\mu) = |\mu - A|^{-1} and \nu = \text{argmax}_{a > 0}a\rho(a) with probability 1.

These functions will calculate the posterior distribution and allow you to interact with it through the distribution, density, quantile, and random generation functions.

Value

density, distribution, quantile, or random values.

Functions

• nun1post(): Obtains prior value of \nu.

• dn1post(): Density function.

• pn1post(): Distribution function.

• qn1post(): Quantile function.

• rn1post(): Random generation.

Author(s)

David Gerard

Examples

set.seed(1)
# Observe x = 2, assume t with 2 df, prior value is A = 1
nun1post(ddist = dt, df = 2) ## nu should be 1
x <- seq(-10, 10, length.out = 500)
y <- dn1post(x = x, A = 1, obs = 2, nu = 1, ddist = dt, df = 2)
z <- rn1post(n = 10000, A = 1, obs = 2, nu = 1, rdist = rt, df = 2)
z <- z[z >= -10 & z <= 10]
graphics::hist(z, freq = FALSE, breaks = 200, border = "grey", col = "grey")
graphics::lines(x, y, type = "l", col = "#E69F00")
graphics::abline(v = c(1, 2), col = c("#56B4E9", "#009E73"), lty = c(2, 3))
pn1post(q = 2, A = 1, obs = 2, nu = 1, pdist = pt, df = 2)
qn1post(p = 0.7113, A = 1, obs = 2, nu = 1, qdist = qt, pdist = pt, df = 2)

nisone documentation built on Sept. 8, 2026, 5:08 p.m.