Best Gibbs Sampling Assignment Help

Gibbs sampling is an iterative process for generating a sequence of samples from a probability distribution. It works by randomly selecting one member of the population and then using it to generate another sample, repeating this process until all members have been used. Do you need Best Gibbs Sampling Assignment Help? Worry no more! We got experts to assist you!

Gibbs Sampling

It is a Markov Chain Monte Carlo algorithm that allows for the calculation of probability distributions over a set of variables. It is used to calculate a probability distribution for some data that is generated from a model of the system you’re studying based on some observed variables.  It works by randomly selecting one member from a sampled population and then using it to generate another sample, repeating this process until all members have been used.

Example 1:  calculate the distribution over an unknown parameter, x, using Gibbs Sampling and then use that distribution to conduct a randomization test with multiple hypotheses.

x is an unknown parameter that you will try to infer through the use of Gibbs Sampling. There are three sets: P, Q, and R. Each set has 5 values and each value is sampled 1000 times (yielding 5000 samples total). Each set is correlated with x.

The first step of this process is to sample from each set using Gibbs Sampling. To do that you’ll need to calculate the distribution for each set when given a certain value of x and when given another value of x. Once you have those probability distributions, then determine how many times in your 5000 samples each value x was sampled and calculate the p-value for your test.

Benefit of Gibbs Sampling

The main benefit is its ability to sample from a probability distribution without having to explicitly calculate the function for that distribution over and over again while the population is changing.  It is often used in Bayesian networks to infer posterior distributions, which allows it to be helpful in cases where parameters are interdependent.

Limitation of Gibbs Sampling

One of the big limitations of Gibbs Sampling is that it requires every variable to have a parent. This means that there cannot be any unknown variables or missing values in your data for Gibbs Sampling to work properly on a model.  if there are, then you’ll have to use another algorithm. You can also use it to do model checking where you find the maximum a posteriori estimate by sampling from the distribution of your prior and see if that sample is higher or lower than what you observed.

Randomized Test

Randomization tests are conducted with multiple hypotheses to determine whether the observed data are statistically significant while controlling the familywise error rate (FWER). Lack of control the FWER results to p-values not reflecting two aspects. These aspects include: how unlikely the observed results are given the null hypothesis and how unlikely they are just due to chance.

Failure to conduct a randomization test with multiple hypotheses and control for the FWER, it will result top values below alpha level (usually 0.05). It’s really likely that the observed results were generated under the null hypothesis.

The drawback of randomized test will be controlling familywise error rate. When multiple hypotheses are being tested against each other in the same experiment, the rate will increase according to the number of tests being conducted.  In conducting a single hypotheses test, the p value should not be what you report since the experiment has only a single hypothesis.

Basic steps on Gibbs Sampling Assignment.

 

1. Sample from Set 1 using a given value for x and then sample from it when given another value for x.
2. Do steps 1 and 2 for Sets Q and R, respectively.
3. Calculate the distribution of values in each set when given a certain value of x. You can do this by rolling up the distributions in steps 1 and 2.
4. Set the value of x to an unknown value x and conduct a randomization test with multiple hypotheses between all three sets (P, Q, and R).
5. Repeat steps 3-4 ,5 times (yielding 5000 samples total when you do this 10000 times).
6. Repeat steps 3-5 5 times for each value of x.
7. Calculate the p-value for your test for every x that you used.
8. Save all these p-values and chart them against their corresponding values of x. Then use a regression line to determine if there is a relationship between the two or not.
9. If there is a relationship between the two, use your regression line to estimate what value of x generated each p-value when they were sampled from the distribution in step 3.
10. Now that you have a way of estimating a given value for x given a specific p-value in step 5, repeat steps 1-9 but this time sample from the probability distributions of all three sets when given x.
11. Now that you have two different distributions for each set, repeat step 3 and calculate the p-value of your test (now using both P and Q).
12. Repeat steps 11-13 until there is no longer any significant difference between the resulting p-values.

This concludes the basic steps of Gibbs Sampling Assignment. Here are some tips to help you get started:

• Use the solver in Excel (while holding Ctrl) to estimate values for x which will give you similar p-values for step 3.
• Make sure your number of samples fits into memory before proceeding to the next step.
• If you take too long to finish this assignment, then Excel will probably stop making new samples (this can be indicated by P(x) = 1 for all values of x).
• Always save your work as you go along since this is a closed-book/closed-notes homework.

Steps of Gibbs Sampling In R

1. Set up the model you want to fit. The number of chains should be at least one more than the number of parameters, otherwise there are no degrees of freedom for an accurate estimate
2. Sample initial values from the prior distribution using sample(x) for each parameter x, or set all initial values using initialize ()
3. Run the posterior predictive samples using update (fit, new data=x) for each additional data set x. This step is repeated after every iteration of sampling from the prior distribution, so may be repeated multiple times per iteration.
4. Measure diagnostic measures of convergence and mixing (e.g. Gelman-Rubin statistic, autocorrelations of each parameter, variance inflation factors for all parameters), or fit more complex models with multiple data sets.
5. Save the model fit object so that you can run step 3 again later on additional data sets.

For example, if you are predicting new observations based on some training data, you can save the fit object containing posterior parameter estimates and use it to predict new data.

>5. Choose an appropriate departure point or model comparison strategy (e.g., calculating AIC for multiple models, BIC for hierarchical models)

>6. Draw random samples from the posterior distribution using sample(x) for each parameter x, or set all values using initialize(x)

>7. Inference for each parameter is drawn from the respective posterior distributions by default, whereas inference on other quantities (e.g., quantity of interest in an economic model), can be drawn directly if desired

8. Repeat steps 3-8 above many times to check convergence and mixing

>9. Compare posterior parameter estimates or other quantities of interest with the true values using plot(fit) or plot (fit, vary=name).

>10. Save the model fit object so that you can run step 3 again later on additional data sets and compare results. Note: if you use the update () function more than once during a single iteration of steps 3-7, you will overwrite the object containing the current model fit.

If you want to compare several models (e.g., alternative models of the same system) instead of just one model, you can use either compare() to get a summary table or run more than one set of chains in parallel (e.g., with different initial values) by running each chain using an index to select different data sets for prediction or model comparison.

Conclusion

Gibbs Sampling is a powerful technique that can be used to solve difficult problems. It is a MCMC technique that overcomes the problems of variance and boundaries.

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