### Fit Bayesian Linear Models

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

Examples demonstrating how to fit a Bayesian linear model using bayes.lm with custom priors and comparing results with standard lm.

```R
data(bears)
bears = subset(bears, Obs.No==1)
bears = bears[,-c(1,2,3,11,12)]
bears = bears[ ,c(7, 1:6)]
bears$Sex = bears$Sex - 1
log.bears = data.frame(log.Weight = log(bears$Weight), bears[,2:7])

b0 = rep(0, 7)
V0 = diag(rep(1e6,7))

fit = bayes.lm(log(Weight)~Sex+Head.L+Head.W+Neck.G+Length+Chest.G, data = bears,
               prior = list(b0 = b0, V0 = V0))
summary(fit)
print(fit)


## Dobson (1990) Page 9: Plant Weight Data:
ctl <- c(4.17,5.58,5.18,6.11,4.50,4.61,5.17,4.53,5.33,5.14)
trt <- c(4.81,4.17,4.41,3.59,5.87,3.83,6.03,4.89,4.32,4.69)
group <- gl(2, 10, 20, labels = c("Ctl","Trt"))
weight <- c(ctl, trt)

lm.D9 <- lm(weight ~ group)
bayes.D9 <- bayes.lm(weight ~ group)

summary(lm.D9)
summary(bayes.D9)
```

--------------------------------

### Custom Prior Function Example

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

Illustrates how to define a custom prior function for a Bayesian analysis, specifically replicating Dianna's prior, and then using it with the `poisgcp` function.

```APIDOC
## Example 10.1 Dianna's prior

### Description
This example demonstrates how to define and use a custom prior function for a Bayesian analysis.

### Usage
```R
f = function(mu){
   result = rep(0, length(mu))
   result[mu >=0 & mu <=2] = mu[mu >=0 & mu <=2]
   result[mu >=2 & mu <=4] = 2
   result[mu >=4 & mu <=8] = 4 - 0.5 * mu[mu >=4 & mu <=8]
   
   A = 2 + 4 + 4
   result = result / A
   
   return(result)
 }
 
results = poisgcp(y, mu = c(0, 10), mu.prior = f)
```

### Details
- The function `f` defines a specific prior distribution.
- `poisgcp` is used with the custom prior `f`.
```

--------------------------------

### Perform Bayesian inference with normdp

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

Examples demonstrating how to use normdp with uniform and non-uniform discrete priors.

```R
## generate a sample of 20 observations from a N(-0.5,1) population
x = rnorm(20,-0.5,1)

## find the posterior density with a uniform prior on mu
normdp(x,1)

## find the posterior density with a non-uniform prior on mu
mu = seq(-3,3,by=0.1)
mu.prior = runif(length(mu))
mu.prior = sort(mu.prior/sum(mu.prior))
normdp(x,1,mu,mu.prior)

## Let mu have the discrete distribution with 5 possible
## values, 2, 2.5, 3, 3.5 and 4, and associated prior probability of
## 0.1, 0.2, 0.4, 0.2, 0.1 respectively. Find the posterior
## distribution after a drawing random sample of n = 5 observations
## from a N(mu,1) distribution y = [1.52, 0.02, 3.35, 3.49, 1.82]
mu = seq(2,4,by=0.5)
mu.prior = c(0.1,0.2,0.4,0.2,0.1)
y = c(1.52,0.02,3.35,3.49,1.82)
normdp(y,1,mu,mu.prior)
```

--------------------------------

### xdesign

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

Simulates completely randomized design and randomized block designs.

```APIDOC
## Function: xdesign

### Description
Monte Carlo study of randomized and blocked designs. Simulates completely randomized design and randomized block designs from a population of experimental units with underlying response values `yyy` and underlying other variable values `xxx` (possibly lurking).

### Usage
```R
xdesign(
  x = NULL,
  y = NULL,
  corr = 0.8,
  size = 20,
  n.treatments = 4,
  n.rep = 500,
  ...
)
```

### Arguments
* `x` (numeric vector): A set of lurking values which are correlated with the response.
* `y` (numeric vector): A set of response values.
* `corr` (numeric): The correlation between the response and lurking variable. Defaults to 0.8.
* `size` (integer): The size of the treatment groups. Defaults to 20.
* `n.treatments` (integer): The number of treatments. Defaults to 4.
* `n.rep` (integer): The number of Monte Carlo replicates. Defaults to 500.
* `...`: Additional parameters which are passed to `Bolstad.control`.

### Value
If the output of `xdesign` is assigned to a variable, then a list is returned with the following components:
* `block.means` (vector): A vector of the means of the lurking variable from each replicate of the simulation stored by treatment number within replicate number.
* `treat.means` (vector): A vector of the means of the response variable from each replicate of the simulation stored by treatment number within replicate number.
* `ind` (vector): A vector containing the treatment group numbers. Note that there will be twice as many group numbers as there are treatments corresponding to the simulations done using a completely randomized design and the simulations done using a randomized block design.

### Examples
```R
# Carry out simulations using the default parameters
xdesign()

# Carry out simulations using a simulated response with 5 treatments,
# groups of size 25, and a correlation of -0.6 between the response
# and lurking variable
xdesign(corr = -0.6, size = 25, n.treatments = 5)
```
```

--------------------------------

### Generic Print Method for sintegral Objects

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

S3 print method for objects of class 'sintegral', designed to display the approximate value of the integral.

```R
print(x, ...)
```

--------------------------------

### Create prior default method

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

Default method for creating a prior distribution with specified x values and weights.

```APIDOC
## Create prior default method

### Description
Create prior default method 

### Usage
```R
## Default S3 method:
createPrior(x, wt, ...)
```

### Arguments
`x` | a vector of x values at which the prior is to be specified (the support of the prior). This should contain unique values in ascending order. The function will sort values if x is unsorted with a warning, and will halt if x contains any duplicates or negative lag 1 differences.
`wt` | a vector of weights corresponding to the weight of the prior at the given x values.
`...` | optional exta arguments. Not currently used.
```

--------------------------------

### Analyze Bear Dataset

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

Load the bears dataset and visualize weight distribution by sex.

```R
data(bears)
boxplot(Weight~Sex, data = bears)
```

--------------------------------

### Simplest Binomial Mixture Prior Call

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

Demonstrates the simplest usage of binomixp with 6 successes in 8 trials and a 50:50 mix of two uniform beta(1,1) priors.

```R
binomixp(6,8)
```

--------------------------------

### Print Method for Bolstad Objects

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

Provides a print summary method for objects of class 'Bolstad', which are typically outputs from the `bayes.lm` function. It handles different class combinations for printing.

```APIDOC
## Print method for objects of class `Bolstad`

### Description
This function provides a print summary method for the output of `bayes.lm`.

### Usage
```R
## S3 method for class 'Bolstad'
print(x, digits = max(3L, getOption("digits") - 3L), ...)
```

### Arguments
- `x` (Bolstad): An object of class `Bolstad`.
- `digits` (numeric): Number of digits to print.
- `...`: Any other arguments to be passed to `print.default`.

### Details
If `x` has both class `Bolstad` and `lm`, a print method similar to `print.lm` is called; otherwise, `print.default` is called.

### Author(s)
James Curran
```

--------------------------------

### Create Prior Default Method

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

The default S3 method for createPrior. It takes x values and corresponding weights to define the prior, ensuring the resulting function integrates to 1.

```R
createPrior(x, wt, ...)
```

--------------------------------

### Simulate Experimental Designs

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

Simulates completely randomized and randomized block designs. Use default parameters for a quick simulation or customize correlation, group size, number of treatments, and replicates.

```R
xdesign()
```

```R
xdesign(corr = -0.6, size = 25, n.treatments = 5)
```

--------------------------------

### Print Method for sintegral Objects

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

A generic print method for objects of class 'sintegral', designed to display the approximate value of the integral.

```APIDOC
## Generic print method for sintegral objects

### Description
Print the value of an `sintegral` object, specifically the (approximate) value of the integral.

### Usage
```R
## S3 method for class 'sintegral'
print(x, ...)
```

### Arguments
- `x` (sintegral): An object of type `sintegral`.
- `...`: Other parameters passed to `paste0`.
```

--------------------------------

### Perform Bayesian inference with general conjugate priors using normgcp

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

Demonstrates generating samples, calculating posterior densities with uniform and user-defined priors, and computing credible intervals using CDF and summary functions.

```R
## generate a sample of 20 observations from a N(-0.5,1) population
x = rnorm(20,-0.5,1)

## find the posterior density with a uniform U[-3,3] prior on mu
normgcp(x, 1, params = c(-3, 3))

## find the posterior density with a non-uniform prior on mu
mu = seq(-3, 3, by = 0.1)
mu.prior = rep(0, length(mu))
mu.prior[mu <= 0] = 1 / 3 + mu[mu <= 0] /9
mu.prior[mu > 0] = 1 / 3 - mu[mu > 0] / 9
normgcp(x, 1, density = "user", mu = mu, mu.prior = mu.prior)

## find the CDF for the previous example and plot it
## Note the syntax for sintegral has changed
results = normgcp(x,1,density="user",mu=mu,mu.prior=mu.prior)
cdf = sintegral(mu,results$posterior,n.pts=length(mu))$cdf
plot(cdf,type="l",xlab=expression(mu[0])
             ,ylab=expression(Pr(mu<=mu[0])))

## use the CDF for the previous example to find a 95%
## credible interval for mu. Thanks to John Wilkinson for this simplified code

lcb = cdf$x[with(cdf,which.max(x[y<=0.025]))]
ucb = cdf$x[with(cdf,which.max(x[y<=0.975]))]
cat(paste("Approximate 95% credible interval : ["
           ,round(lcb,4)," ",round(ucb,4),"]\n",sep=""))

## use the CDF from the previous example to find the posterior mean
## and std. deviation
dens = mu*results$posterior
post.mean = sintegral(mu,dens)$value

dens = (mu-post.mean)^2*results$posterior
post.var = sintegral(mu,dens)$value
post.sd = sqrt(post.var)

## use the mean and std. deviation from the previous example to find
## an approximate 95% credible interval
lb = post.mean-qnorm(0.975)*post.sd
ub = post.mean+qnorm(0.975)*post.sd


cat(paste("Approximate 95% credible interval : ["
   ,round(lb,4)," ",round(ub,4),"]\n",sep=""))

## repeat the last example but use the new summary functions for the posterior
results = normgcp(x, 1, density = "user", mu = mu, mu.prior = mu.prior)

## use the cdf function to get the cdf and plot it
postCDF = cdf(results) ## note this is a function
plot(results$mu, postCDF(results$mu), type="l", xlab = expression(mu[0]),
     ylab = expression(Pr(mu <= mu[0])))

## use the quantile function to get a 95% credible interval
ci = quantile(results, c(0.025, 0.975))
ci

## use the mean and sd functions to get the posterior mean and standard deviation
postMean = mean(results)
postSD = sd(results)
postMean
postSD

## use the mean and std. deviation from the previous example to find
## an approximate 95% credible interval
ciApprox = postMean + c(-1,1) * qnorm(0.975) * postSD
ciApprox
```

--------------------------------

### Poisson Sampling with User-Specified Continuous Prior

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

Performs Poisson sampling using a user-defined continuous prior distribution for the mean rate (mu). This allows for flexible prior specifications beyond standard parametric forms. It also demonstrates how to obtain a 99% credible interval using the alpha parameter.

```R
y = c(3,4,3,0,1)
mu = seq(0,8,by=0.001)
mu.prior = c(seq(0,2,by=0.001),rep(2,1999),seq(2,0,by=-0.0005))/10
poisgcp(y,"user",mu=mu,mu.prior=mu.prior,print.sum.stat=TRUE,alpha=0.01)
```

--------------------------------

### Load and Plot Sample Data

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

Loads the sscsample.data dataset and creates a scatter plot of income versus ethnicity. Ensure the data is loaded before plotting.

```R
data(sscsample.data)
plot(income~ethnicity, data = sscsample.data)
```

--------------------------------

### Print Method for Bolstad Objects

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

S3 print method for objects of class 'Bolstad', providing a summary similar to 'print.lm' if the object also has class 'lm', otherwise using 'print.default'.

```R
print(x, digits = max(3L, getOption("digits") - 3L), ...)
```

--------------------------------

### Binomial Mixture Prior with Multiple Non-Uniform Priors and Plotting

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

Performs binomial sampling with 4 successes in 12 trials, using a 90:10 mix of non-uniform beta(3,3) and beta(4,12) priors. It also plots the prior, likelihood, and posterior densities.

```R
results = binomixp(4, 12, c(3, 3), c(4, 12), 0.9)$mix

par(mfrow = c(3,1))
y.lims = c(0, 1.1 * max(results$posterior, results$prior))

plot(results$pi,results$prior,ylim=y.lims,type='l'
,xlab=expression(pi),ylab='Density',main='Prior')
polygon(results$pi,results$prior,col='red')

plot(results$pi,results$likelihood,type='l',
     xlab = expression(pi), ylab = 'Density', main = 'Likelihood')
polygon(results$pi,results$likelihood,col='green')

plot(results$pi,results$posterior,ylim=y.lims,type='l'
,xlab=expression(pi),ylab='Density',main='Posterior')
polygon(results$pi,results$posterior,col='blue')
```

--------------------------------

### Perform Bayesian inference with mixture of normal priors using normmixp

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

Demonstrates generating samples and calculating posterior densities using different normal prior mixtures.

```R
## generate a sample of 20 observations from a N(-0.5, 1) population
x = rnorm(20, -0.5, 1)

## find the posterior density with a N(0, 1) prior on mu - a 50:50 mix of
## two N(0, 1) densities
normmixp(x, 1, c(0, 1), c(0, 1))

## find the posterior density with 50:50 mix of a N(0.5, 3) prior and a
## N(0, 1) prior on mu
normmixp(x, 1, c(0.5, 3), c(0, 1))

## Find the posterior density for mu, given a random sample of 4
## observations from N(mu, 1), y = [2.99, 5.56, 2.83, 3.47],
```

--------------------------------

### Posterior Quantiles Method for Bolstad Objects

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

S3 quantile method for objects of class 'Bolstad'. It calculates posterior quantiles using numerical integration and linear interpolation.

```R
quantile(x, probs = seq(0, 1, 0.25), ...)
```

--------------------------------

### Binomial Sampling with Continuous Prior

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

Evaluates posterior density for binomial success probability with a general continuous prior. Supports various prior distributions like beta, exponential, normal, or user-defined.

```R
binogcp(6, 8)
```

```R
binogcp(6, 8,density = "beta", params = c(2, 2))
```

```R
binogcp(5, 10, density = "normal", params = c(0.5, 0.25))
```

```R
pi = seq(0, 1,by = 0.001)
pi.prior = rep(0, length(pi))
priorFun = createPrior(x = c(0, 0.5, 1), wt = c(0, 2, 0))
pi.prior = priorFun(pi)
results = binogcp(4, 12, "user", pi = pi, pi.prior = pi.prior)
```

```R
myCdf = cdf(results)
plot(myCdf, type = "l", xlab = expression(pi[0]),
	   ylab = expression(Pr(pi <= pi[0])))
```

```R
qtls = quantile(results, probs = c(0.025, 0.975))
cat(paste("Approximate 95% credible interval : ["
	, round(qtls[1], 4), " ", round(qtls, 4), "]\n", sep = ""))
```

```R
post.mean = mean(results)
post.var = var(results)
post.sd = sd(results)
```

```R
# calculate an approximate 95% credible region using the posterior mean and
```

--------------------------------

### Posterior CDF and Credible Region Calculation

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

Demonstrates how to calculate the posterior Cumulative Distribution Function (CDF) and find credible intervals using the results from a Bayesian analysis. It also shows how to calculate the posterior mean, variance, and standard deviation.

```APIDOC
## Posterior CDF and Credible Region Calculation

This section shows how to compute the posterior CDF and credible regions from a Bayesian analysis.

### Usage
```R
results = poisgcp(y,"user",mu=mu,mu.prior=mu.prior)
cdf = cdf(results)
curve(cdf,type="l",xlab=expression(mu[0]), ylab=expression(Pr(mu<=mu[0])))

ci = quantile(results, c(0.025, 0.975))
cat(paste0("Approximate 95% credible interval : [",
	round(ci[1],4)," ",round(ci[2],4),"]\n"))

post.mean = mean(results)
post.var = var(results)
post.sd = sd(results)

ci = post.mean + c(-1, 1) * qnorm(0.975) * post.sd
cat(paste("Approximate 95% credible interval : [" ,
	round(ci[1],4)," ",round(ci[2],4),"]\n",sep=""))
```

### Details
- `poisgcp`: Function to perform Poisson generalized conjugate prior analysis.
- `cdf`: Extracts the cumulative distribution function from the results.
- `curve`: Plots the CDF.
- `quantile`: Calculates quantiles of the posterior distribution.
- `mean`, `var`, `sd`: Calculate posterior mean, variance, and standard deviation.
```

--------------------------------

### Perform Bayesian Linear Regression with bayes.lin.reg

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

Demonstrates fitting linear models with various prior configurations and generating predictions.

```R
## generate some data from a known model, where the true value of the
## intercept alpha is 2, the true value of the slope beta is 3, and the
## errors come from a normal(0,1) distribution
set.seed(123)
x = rnorm(50)
y = 2 + 3*x + rnorm(50)

## use the function with a flat prior for the slope beta and a
## flat prior for the intercept, alpha_xbar.

bayes.lin.reg(y,x)

## use the function with a normal(0,3) prior for the slope beta and a
## normal(30,10) prior for the intercept, alpha_xbar.

bayes.lin.reg(y,x,"n","n",0,3,30,10)

## use the same data but plot it and the credible interval

bayes.lin.reg(y,x,"n","n",0,3,30,10, plot.data = TRUE)

## The heart rate vs. O2 uptake example 14.1
O2 = c(0.47,0.75,0.83,0.98,1.18,1.29,1.40,1.60,1.75,1.90,2.23)
HR = c(94,96,94,95,104,106,108,113,115,121,131)
plot(HR,O2,xlab="Heart Rate",ylab="Oxygen uptake (Percent)")

bayes.lin.reg(O2,HR,"n","f",0,1,sigma=0.13)

## Repeat the example but obtain predictions for HR = 100 and 110

bayes.lin.reg(O2,HR,"n","f",0,1,sigma=0.13,pred.x=c(100,110))
```

--------------------------------

### Poisson sampling with a discrete prior

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

Evaluates and plots the posterior density for the mean rate of a Poisson process using a discrete prior.

```R
## simplest call with an observation of 4 and a uniform prior on the
## values mu = 1,2,3
poisdp(4,1:3,c(1,1,1)/3)

##  Same as the previous example but a non-uniform discrete prior
mu = 1:3
mu.prior = c(0.3,0.4,0.3)
poisdp(4,mu=mu,mu.prior=mu.prior)

##  Same as the previous example but a non-uniform discrete prior
mu = seq(0.5,9.5,by=0.05)
mu.prior = runif(length(mu))
mu.prior = sort(mu.prior/sum(mu.prior))
poisdp(4,mu=mu,mu.prior=mu.prior)

## A random sample of 50 observations from a Poisson distribution with
## parameter mu = 3 and  non-uniform prior
y.obs = rpois(50,3)
mu = c(1:5)
mu.prior = c(0.1,0.1,0.05,0.25,0.5)
results = poisdp(y.obs, mu, mu.prior)

##  Same as the previous example but a non-uniform discrete prior
mu = seq(0.5,5.5,by=0.05)
mu.prior = runif(length(mu))
mu.prior = sort(mu.prior/sum(mu.prior))
y.obs = rpois(50,3)
poisdp(y.obs,mu=mu,mu.prior=mu.prior)
```

--------------------------------

### Plot Prior, Likelihood, and Posterior

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

Plots the prior, likelihood, and posterior distributions on a single graph to visualize their influence. Ensure the input is an object of class 'Bolstad'.

```R
decomp(x, ...)
```

```R
# an example with a binomial sampling situation
results = binobp(4, 12, 3, 3, plot = FALSE)
decomp(results)
```

```R
# an example with normal data
y = c(2.99,5.56,2.83,3.47)
results = normnp(y, 3, 2, 1, plot = FALSE)
decomp(results)
```

--------------------------------

### Calculate Posterior CDF and Quantiles

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

Finds the posterior CDF and calculates the 95% credible interval using the quantile function on the results from a poisgcp model.

```R
results = poisgcp(y,"user",mu=mu,mu.prior=mu.prior)
cdf = cdf(results)
curve(cdf,type="l",xlab=expression(mu[0])
	,ylab=expression(Pr(mu<=mu[0])))

ci = quantile(results, c(0.025, 0.975))
cat(paste0("Approximate 95% credible interval : ["
	,round(ci[1],4)," ",round(ci[2],4),"]\n"))
```

--------------------------------

### Binomial Sampling with Discrete Prior

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

Evaluate posterior density for binomial sampling using a discrete prior.

```R
binodp(x, n, pi = NULL, pi.prior = NULL, n.pi = 10, ...)
```

--------------------------------

### Binomial Mixture Prior with Non-Uniform Priors

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

Shows binomixp with 6 successes in 8 trials, using a 20:80 mix of a non-uniform beta(0.5,6) prior and a uniform beta(1,1) prior.

```R
binomixp(6,8,alpha0=c(0.5,6),alpha1=c(1,1),p=0.2)
```

--------------------------------

### Calculate Credible Interval for Poisson Mean with Gamma Prior (Quantile Method)

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

Calculates a 95% credible interval for the mean (mu) of a Poisson distribution using a gamma prior. This method uses the quantile function on the results of poisgamp to determine the interval bounds.

```R
y = c(3,4,4,3,3,4,2,3,1,7)
results = poisgamp(y, 6, 3)
ci = quantile(results, c(0.025, 0.975))
cat(paste("95% credible interval for mu: [",round(ci[1],3), "," ,round(ci[2],3)),"]
)
```

--------------------------------

### Binomial sampling with a beta mixture prior

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

Evaluates and plots the posterior density for π, the probability of a success in a Bernoulli trial, with binomial sampling when the prior density for π is a mixture of two beta distributions.

```APIDOC
## Binomial sampling with a beta mixture prior

### Description
Evaluates and plots the posterior density for `πππ`, the probability of a success in a Bernoulli trial, with binomial sampling when the prior density for `πππ` is a mixture of two beta distributions, `beta(a0,b0)beta(a_0,b_0)beta(a0​,b0​)` and `beta(a1,b1)beta(a_1,b_1)beta(a1​,b1​)`.

### Usage
```R
binomixp(x, n, alpha0 = c(1, 1), alpha1 = c(1, 1), p = 0.5, ...)
```

### Arguments
`x` | the number of observed successes in the binomial experiment.
`n` | the number of trials in the binomial experiment.
`alpha0` | a vector of length two containing the parameters, `a0a_0a0​` and `b0b_0b0​`, for the first component beta prior - must be greater than zero. By default the elements of alpha0 are set to 1.
`alpha1` | a vector of length two containing the parameters, `a1a_1a1​` and `b1b_1b1​`, for the second component beta prior - must be greater than zero. By default the elements of alpha1 are set to 1.
`p` | The prior mixing proportion for the two component beta priors. That is the prior is `pimesbeta(a0,b0)+(1−p)imesbeta(a1,b1)pimes beta(a_0,b_0)+(1-p)imes beta(a_1,b_1)pimesbeta(a0​,b0​)+(1−p)imesbeta(a1​,b1​)`.
`...` | additional arguments that are passed to `Bolstad.control`

### Value
A list will be returned with the following components: 
`pi` | the values of `πππ` for which the posterior density was evaluated
`posterior` | the posterior density of `πππ` given `nnn` and `xxx`
`likelihood` | the likelihood function for `πππ` given `xxx` and `nnn`, i.e. the `binomial(n,π)binomial(n,π)binomial(n,π)` density
`prior` | the prior density of `πππ` density

### See Also
`binodp` `binogcp` `normmixp`

### Examples
```R
# simplest call with 6 successes observed in 8 trials and a 50:50 mix
# of two beta(1,1) uniform priors
binomixp(6,8)

# 6 successes observed in 8 trials and a 20:80 mix of a non-uniform
# beta(0.5,6) prior and a uniform beta(1,1) prior
binomixp(6,8,alpha0=c(0.5,6),alpha1=c(1,1),p=0.2)

# 4 successes observed in 12 trials with a 90:10 non uniform beta(3,3) prior
# and a non uniform beta(4,12).
# Plot the stored prior, likelihood and posterior
results = binomixp(4, 12, c(3, 3), c(4, 12), 0.9)$mix

par(mfrow = c(3,1))
y.lims = c(0, 1.1 * max(results$posterior, results$prior))

plot(results$pi,results$prior,ylim=y.lims,type='l'
,xlab=expression(pi),ylab='Density',main='Prior')
polygon(results$pi,results$prior,col='red')

plot(results$pi,results$likelihood,type='l',
     xlab = expression(pi), ylab = 'Density', main = 'Likelihood')
polygon(results$pi,results$likelihood,col='green')

plot(results$pi,results$posterior,ylim=y.lims,type='l'
,xlab=expression(pi),ylab='Density',main='Posterior')
polygon(results$pi,results$posterior,col='blue')
```
```

--------------------------------

### Binomial Sampling with Beta Prior

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

Evaluate posterior density for binomial sampling using a beta prior.

```R
binobp(x, n, a = 1, b = 1, pi = seq(0, 1, by = 0.001), ...)
```

```R
binobp(6,8)
```

```R
binobp(6,8,0.5,6)
```

```R
results = binobp(4, 12, 3, 3)
decomp(results)
```

--------------------------------

### Poisson sampling with a gamma prior

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

Evaluates and plots the posterior density for the mean rate of a Poisson process using a gamma prior.

```R
## simplest call with an observation of 4 and a gamma(1, 1), i.e. an exponential prior on the
## mu
poisgamp(4, 1, 1)

##  Same as the previous example but a gamma(10, ) prior
poisgamp(4, 10, 1)

##  Same as the previous example but an improper gamma(1, ) prior
poisgamp(4, 1, 0)

## A random sample of 50 observations from a Poisson distribution with
## parameter mu = 3 and  gamma(6,3) prior
set.seed(123)
y = rpois(50,3)
poisgamp(y,6,3)

## In this example we have a random sample from a Poisson distribution
```

--------------------------------

### Plotting Bolstad objects

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

Visualizes prior, likelihood, and posterior distributions from a Bolstad object.

```R
x = rnorm(20,-0.5,1)
## find the posterior density with a N(0,1) prior on mu
b = normnp(x,sigma=1)
plot(b)
plot(b, which = 1:3)
plot(b, overlay = FALSE, which = 1:3)
```

--------------------------------

### binogcp

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

Evaluates binomial sampling with a general continuous prior.

```APIDOC
## binogcp

### Description
Evaluates and plots the posterior density for the probability of success in a Bernoulli trial with a continuous prior.

### Parameters
#### Arguments
- **x** (numeric) - Required - The number of observed successes.
- **n** (numeric) - Required - The number of trials.
- **density** (string) - Required - One of 'beta', 'exp', 'normal', 'student', 'uniform', or 'user'.
- **params** (vector) - Optional - Parameters for the chosen density.
- **n.pi** (numeric) - Optional - Number of possible pi values.
- **pi** (vector) - Optional - Vector of possibilities for pi (required if density='user').
- **pi.prior** (vector) - Optional - Associated prior probability mass (required if density='user').

### Response
#### Success Response
- **likelihood** (vector) - The scaled likelihood function.
- **posterior** (vector) - The posterior probability.
- **pi** (vector) - The vector of possible pi values.
- **pi.prior** (vector) - The associated probability mass.
```

--------------------------------

### Print Method for sscsample Objects

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

Provides a print summary method for objects of class 'sscsamp', generated by the `sscsample` function. It displays sample means and stratum counts.

```APIDOC
## Print method for objects of class `sscsample`

### Description
This function provides a print summary method for the output of `sscsample`. The `sscsample` produces a large number of samples from a fixed population using either simple random, stratified, or cluster sampling. This function provides the means of each sample plus the number of observations from each ethnicity stratum in the sample.

### Usage
```R
## S3 method for class 'sscsamp'
print(x, ...)
```

### Arguments
- `x` (sscsamp): An object of class `sscsamp` produced by `sscsample`.
- `...`: Any other arguments that are to be passed to `cat`.

### Author(s)
James Curran
```

--------------------------------

### POST /poisgamp

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

Evaluates and plots the posterior density for the mean rate of occurrence in a Poisson process using a gamma prior.

```APIDOC
## POST /poisgamp

### Description
Evaluates and plots the posterior density for μ, the mean rate of occurrence in a Poisson process and a gamma prior on μ.

### Method
POST

### Endpoint
/poisgamp

### Parameters
#### Request Body
- **y** (vector) - Required - A random sample from a Poisson distribution.
- **shape** (numeric) - Required - The shape parameter of the gamma prior.
- **rate** (numeric) - Optional - The rate parameter of the gamma prior.
- **scale** (numeric) - Optional - The scale parameter of the gamma prior.
- **alpha** (numeric) - Optional - The width of the credible interval.

### Response
#### Success Response (200)
- **prior** (vector) - The prior density assigned to μ.
- **likelihood** (vector) - The scaled likelihood function for μ given y.
- **posterior** (vector) - The posterior probability of μ given y.
- **shape** (numeric) - The shape parameter for the gamma posterior.
- **rate** (numeric) - The rate parameter for the gamma posterior.
```

--------------------------------

### Posterior Quantiles for Bolstad Objects

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

Calculates posterior quantiles for objects of class 'Bolstad'. This method uses numerical integration and linear interpolation if necessary.

```APIDOC
## Posterior quantiles for Bolstad objects

### Description
Calculates posterior quantiles for objects of class 'Bolstad'.

### Usage
```R
## S3 method for class 'Bolstad'
quantile(x, probs = seq(0, 1, 0.25), ...)
```

### Arguments
- `x` (Bolstad): An object of class `Bolstad`.
- `probs` (numeric): Vector of probabilities in the range [0, 1].
- `...`: Any extra arguments needed.

### Details
If `x` is of class `Bolstad`, this function finds the quantiles of the posterior distribution using numerical integration and linear interpolation.
```

--------------------------------

### Summary Method for bayes.lm

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

Provides a summary of a 'Bolstad' object, which is the output of the bayes.lm function. This method is used to inspect the results of a Bayesian linear model fit.

```R
summary(object, ...)
```

--------------------------------

### Create prior generic

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

Creates a prior distribution function through linear interpolation.

```APIDOC
## Create prior generic

### Description
Create prior generic 

### Usage
```R
createPrior(x, ...)
```

### Arguments
`x` | a vector of x values at which the prior is to be specified (the support of the prior).
`...` | optional exta arguments. Not currently used.

### Value
a linear interpolation function where the weights have been scaled so the function (numerically) integrates to 1.
```

--------------------------------

### summary.Bolstad

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

Provides a summary method for objects created by the bayes.lm function.

```APIDOC
## S3 Method: summary.Bolstad

### Description
`summary` method for output of `bayes.lm`.

### Usage
```R
## S3 method for class 'Bolstad'
summary(object, ...)
```

### Arguments
* `object` (Bolstad): An object of class 'Bolstad' that is the result of a call to `bayes.lm`.
* `...`: Any further arguments to be passed to `print`.

### See Also
* `bayes.lm`: The function to fit the model.
* `coef`: Function to extract the matrix of posterior means along with standard errors and t-statistics.
```

--------------------------------

### Binomial Probability with Discrete Prior

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

Calculates binomial probabilities with a discrete prior. Use for scenarios with a fixed number of trials and successes, and a defined prior distribution for the success probability.

```R
binodp(6,8)
```

```R
binodp(6, 8, n.pi = 100)
```

```R
pi = seq(0, 1, by = 0.01)
pi.prior = runif(101)
pi.prior = sort(pi.prior / sum(pi.prior))
binodp(6, 8, pi, pi.prior)
```

```R
pi = c(0.3, 0.4, 0.5)
pi.prior = c(0.2, 0.3, 0.5)
results = binodp(5, 6, pi, pi.prior)
```

```R
results.matrix = rbind(results$pi.prior,results$posterior)
colnames(results.matrix) = pi
barplot(results.matrix, col = c("red", "blue"), beside = TRUE,
	      xlab = expression(pi), ylab=expression(Probability(pi)))
box()
legend("topleft", bty = "n", cex = 0.7, 
       legend = c("Prior", "Posterior"), fill = c("red", "blue"))
```

--------------------------------

### Plot Moisture Data

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

Plots the final moisture level against the in-process moisture level using the 'moisture.df' dataset.

```R
data(moisture.df)
plot(final.level~proc.level, data = moisture.df)
```

--------------------------------

### POST /poisdp

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

Evaluates and plots the posterior density for the mean rate of occurrence in a Poisson process using a discrete prior.

```APIDOC
## POST /poisdp

### Description
Evaluates and plots the posterior density for μ, the mean rate of occurrence in a Poisson process and a discrete prior on μ.

### Method
POST

### Endpoint
/poisdp

### Parameters
#### Request Body
- **y.obs** (vector) - Required - A random sample from a Poisson distribution.
- **mu** (vector) - Required - A vector of possibilities for the mean rate of occurrence.
- **mu.prior** (vector) - Required - The associated prior probability mass.

### Response
#### Success Response (200)
- **likelihood** (vector) - The scaled likelihood function for μ given y.obs.
- **posterior** (vector) - The posterior probability of μ given y.obs.
- **mu** (vector) - The vector of possible μ values used in the prior.
- **mu.prior** (vector) - The associated probability mass for the values in μ.
```

--------------------------------

### median(x, na.rm = FALSE, ...)

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

Computes the posterior median of the posterior distribution.

```APIDOC
## median(x, na.rm = FALSE, ...)

### Description
Compute the posterior median of the posterior distribution for a Bolstad object.

### Parameters
#### Arguments
- **x** (Bolstad object) - Required - An object of class Bolstad.
- **na.rm** (boolean) - Optional - Ideally removes missing values (not currently used).
- **...** (dots) - Optional - Not currently used.
```

--------------------------------

### Create Prior Function

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

A generic function to create a prior distribution. It takes a vector of x values (support) and returns a linear interpolation function scaled to integrate to 1.

```R
createPrior(x, ...)
```

--------------------------------

### Poisson Sampling with Gamma Prior

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

Performs Poisson sampling with a gamma prior distribution for the mean rate (mu). This is a common choice for a conjugate prior in Poisson models.

```R
y = c(3,4,3,0,1)
poisgcp(y, density = "gamma", params = c(6,8))
```

--------------------------------

### sscsample

Source: https://cran.r-project.org/web/packages/Bolstad/refman/Bolstad.html

Samples from a fixed population using either simple random sampling, stratified sampling or cluster sampling.

```APIDOC
## sscsample

### Description
Samples from a fixed population using either simple random sampling, stratitified sampling or cluster sampling. 

### Usage
```R
sscsample(
  size,
  n.samples,
  sample.type = c("simple", "cluster", "stratified"),
  x = NULL,
  strata = NULL,
  cluster = NULL
)
```

### Arguments
`size` |  the desired size of the sample  
`n.samples` |  the number of repeat samples to take  
`sample.type` |  the sampling method. Can be one of "simple", "stratified", "cluser" or 1, 2, 3 where 1 corresponds to "simple", 2 to "stratified" and 3 to "cluster"  
`x` |  a vector of measurements for each unit in the population. By default x is not used, and the builtin data set sscsample.data is used  
`strata` |  a corresponding vector for each unit in the population indicating membership to a stratum  
`cluster` |  a corresponding vector for each unit in the population indicating membership to a cluster  

### Value
A list will be returned with the following components: 
`samples` |  a matrix with the number of rows equal to size and the number of columns equal to n.samples. Each column corresponds to a sample drawn from the population  
`s.strata` |  a matrix showing how many units from each stratum were included in the sample  
`means` |  a vector containing the mean of each sample drawn  

### Author(s)
James M. Curran, Dept. of Statistics, University of Auckland. Janko Dietzsch, Proteomics Algorithm and Simulation,Zentrum f. Bioinformatik Tuebingen Fakultaet f. Informations- und Kognitionswissenschaften, Universitaet Tuebingen 
### Examples
```R

## Draw 200 samples of size 20 using simple random sampling
sscsample(20,200)

## Draw 200 samples of size 20 using simple random sampling and store the
## results. Extract the means of all 200 samples, and the 50th sample
res = sscsample(20,200)
res$means
res$samples[,50]
```
```