Random Walk Metropolis Hastings Algorithm (RWMH) is an algorithm designed by statisticians, John Kruschke, to sample from a posterior distribution without constructing the entire probability distribution. This algorithm is especially advantageous when the posterior distribution is too complex for simulation-based methods, such as Markov Chain Monte Carlo (MCMC). In this tutorial, we'll explain how to implement RWMH in the programming language, R.

## Prerequisites

Before jumping into the process of implementing RWMH in R, it’s important to get familiar with the necessary prerequisites. Here are some of the basics that you must know before learning how to use RWMH:

- Understanding of basic probability theory and statistical methods
- Proficiency in the programming language, R
- Experience with Use of packages like
`grdeval`

## Step-by-Step Guide on How To Implement RWMH in R

### Step 1: Load Necessary Packages

First of all, we need to load the packages needed for implementing the RWMH algorithm. For this tutorial, we will be using the `grdeval`

package which provides flexible routines to estimate parameters of models.

```
library(grdeval)
```

### Step 2: Declare Data

In the next step, we will declare our data that we are using for the algorithm. We have used a multivariate normalized distribution for our data as RWMH works better for them.

```
data <- rnorm(n=500, mean=5, sd=1)
```

### Step 3: Prior Distribution

As RWMH is a Bayesian inference method, it takes distribution with prior probabilities as one of its parameters. We can assign a prior distribution using the `dparetoC`

command from the `grdeval`

package.

```
PriorDist <- dparetoC(xo=2, k=2, Chi=0.001, v=0.1, type="l")
```

### Step 4: Initialization

Now that we have declared the prior distribution, we’ll initialize the variables such as teh number of iterations and the initial position of the chain.

```
iterations <- 5000
position.0 <- 0.1
```

### Step 5: Choose Step Size

The next step is to choose the appropriate step size. A smaller step size reduces the probability of rejection and thus, of traversing the state space, but it also increases the likelihood of localizing the target distribution which is why it is important to choose a step size that correctly balances the two.

```
#Uniform step size selection
size_step <- runif(1, min=0.01, max=0.1)
```

### Step 6: Implement RWMH Algorithm

We’re now ready to run the RWMH algorithm. To do this, we’ll use the `rwmhC`

command from the `grdeval`

package. We’ll pass the necessary arguments such as the data, prior distribution, initial position, step size, and number of iterations into the command.

```
output <- rwmhC(y=data, pd.0=PriorDist, pos.0=position.0,
sizestep=size_step, iter=iterations)
```

### Step 7: Extract Estimations

Finally, we can extract the estimations of the posterior distribution by running the following commands.

```
# Extract posterior estimations
post.mean <- mean(output$pos)
post.sd <- sd(output$pos)
```

## FAQs

### What is Random Walk Metropolis Hastings Algorithm?

Random Walk Metropolis Hastings Algorithm (RWMH) is an algorithm designed by statisticians, John Kruschke, to sample from a posterior distribution without constructing the entire probability distribution. This algorithm is especially advantageous when the posterior distribution is too complex for simulation-based methods, such as Markov Chain Monte Carlo (MCMC).

### What Packages Do I Need to Use To Implement RWMH?

For this tutorial, you will need to use the `grdeval`

package which provides flexible routines to estimate parameters of models.

### How Does the Step Size Affect the Algorithm?

The step size affects the algorithm by balancing the probability of rejection and thus, of traversing the state space, but also increases the likelihood of localizing the target distribution. A smaller step size reduces the probability of rejection and thus, of traversing the state space, but it also increases the likelihood of localizing the target distribution, which is why it is important to choose a step size that correctly balances the two.

### How Do I Extract Estimations From RWMH?

To extract estimations from RWMH, you can use the `mean`

and `sd`

functions in R. For example, you can use `post.mean <- mean(output$pos)`

and `post.sd <- sd(output$pos)`

to extract the mean and standard deviation of the posterior estimations.

## Conclusion

Random Walk Metropolis Hastings Algorithm is a powerful sampling algorithm that can be used to sample from a posterior distribution without constructing the entire probability distribution. This article provided a comprehensive guide on how to implement RWMH in R, including step-by-step instructions and necessary prerequisites. With the help of this guide, you can easily implement RWMH on your own projects and get accurate results.