## Metropolis Hastings sampling from a Bivariate Normal distribution

### Description

This function uses the MetropolisHastings algorithm to draw a sample from a correlated bivariate normal target density using a random walk candidate and an independent candidate density respectively where we are drawing both parameters in a single draw. It can also use the blockwise Metropolis Hastings algorithm and Gibbs sampling respectively to draw a sample from the correlated bivariate normal target.

### Usage

```bivnormMH(rho, rho1 = 0.9, sigma = c(1.2, 1.2),
steps = 1000, type = 'ind')
```

### Arguments

 `rho` the correlation coefficient for the bivariate normal `rho1` the correlation of the candidate distribution. Only used when type = 'ind' `sigma` the standard deviations of the marginal distributions of the independent candidate density. Only used when type = 'ind' `steps` the number of Metropolis Hastings steps `type` the type of candidate generation to use. Can be one of 'rw' = random walk, 'ind' = independent normals, 'gibbs' = Gibbs sampling or 'block' = blockwise. It is sufficient to use 'r','i','g', or 'b'

### Value

returns a list which contains a data frame called targetSample with members x and y. These are the samples from the target density.

### Examples

```## independent chain
chain1.df<-bivnormMH(0.9)\$targetSample

## random walk chain
chain2.df<-bivnormMH(0.9, type = 'r')\$targetSample

## blockwise MH chain
chain3.df<-bivnormMH(0.9, type = 'b')\$targetSample

## Gibbs sampling chain
chain4.df<-bivnormMH(0.9, type = 'g')\$targetSample

oldPar <- par(mfrow=c(2,2))
plot(y ~ x, type = 'l', chain1.df, main = 'Independent')
plot(y ~ x, type = 'l', chain2.df, main = 'Random Walk')
plot(y ~ x, type = 'l', chain3.df, main = 'Blockwise')
plot(y ~ x, type = 'l', chain4.df, main = 'Gibbs')
par(oldPar)
```