Have a look at the MCMC chapter in this; it discusses most of your required terms in a practical context

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Bayesian decision making using posterior probabilities and a variety of loss functions. We discussed how to minimize the expected loss for hypothesis testing. Moreover, we instroduced the concept of Bayes factors and gave some examples on how Bayes factors can be used in Bayesian hypothesis testing for comparison of two means.

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Bayesian inference methods to linear regression. We will first apply Bayesian statistics to simple linear regression models, then generalize the results to multiple linear regression models.

Bayesian inference in simple linear regressions. We will use the reference prior distribution on coefficients, which will provide a connection between the frequentist solutions and Bayesian answers.

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a baseline analysis for comparisons with more informative prior distributions. To illustrate the ideas, we will use an example of predicting body fat.

The plot and predictive intervals suggest that predictions for Case 39 are not well captured by the model. There is always the possibility that this case does not meet the assumptions of the simple linear regression model

Model diagnostics such as plots of residuals versus fitted values are useful in identifying potential outliers.

Now with the interpretation of Bayesian paradigm, we can go further to calculate the probability to demonstrate whether a case falls too far from the mean.

Chaloner and Brant (1988) suggested an approach for defining outliers and then calculating the probability that a case or multiple cases were outliers, based on the posterior information of all observations.

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The assumed model for our simple linear regression is yi=α+βxi+ϵiyi=α+βxi+ϵi, with ϵiϵi having independent, identical distributions that are normal with mean zero and constant variance σ2σ2, i.e., ϵiiid∼Normal(0,σ2)ϵi∼iidNormal(0,σ2). Chaloner & Brant considered outliers to be points where the error or the model

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The Bayesian vs Frequentist debate is one of those academic arguments that I find more interesting to watch than engage in. Rather than enthusiastically jump in on one side,

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I think it’s more productive to learn both methods of statistical inference and apply them where appropriate. In that line of thinking, recently, I have been working to learn and apply Bayesian inference methods to supplement the frequentist statistics covered in my grad classes.

One of my first areas of focus in applied Bayesian Inference was Bayesian Linear modeling.

The frequentist view of linear regression is probably the one you are familiar with from school: the model assumes that the response variable (y) is a linear combination of weights multiplied by a set of predictor variables (x).

y is the response variable (also called the dependent variable), β’s are the weights (known as the model parameters), x’s are the values of the predictor variables, and ε is an error term representing random sampling noise or the effect of variables not included in the model.

Linear Regression is a simple model which makes it easily interpretable: β_0 is the intercept term and the other weights, β’s, show the effect on the response of increasing a predictor variable. For example, if β_1 is 1.2, then for every unit increase in x_1,the response will increase by 1.2.

Since y|x,z or z|x,y are all easily simulated from. Probably you only need to sample from x|y,z using any appropriate MCMC sampler that you are familiar with. I recommend looking at Slice sampling, Neal 2003.

We can generalize the linear model to any number of predictors using matrix equations. Adding a constant term of 1 to the predictor matrix to account for the intercept

The goal of learning a linear model from training data is to find the coefficients, β, that best explain the data. In frequentist linear regression, the best explanation is taken to mean the coefficients, β, that minimize the residual sum of squares (RSS).

RSS is the total of the squared differences between the known values (y) and the predicted model outputs (ŷ, pronounced y-hat indicating an estimate).

The summation is taken over the N data points in the training set. We won’t go into the details here (check out this reference for the derivation), but this equation has a closed form solution for the model parameters, β, that minimize the error.

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Neter et al. (1996) describe a study of 54 patients undergoing a certain kind of liver operation in a surgical unit. The data set Surg contains survival time and certain covariates for each patient.

However, if you are interested in finding the probability that the coefficient is positive, Bayesian analysis offers a convenient alternative. You can use Bayesian analysis to directly estimate the conditional probability, , using the posterior distribution samples, which are produced as part of the output by PROC GENMOD.

Whereas, The example that follows shows how to use PROC GENMOD to carry out a Bayesian analysis of the linear model with a normal error term

The SEED= option is specified to maintain reproducibility; no other options are specified in the BAYES statement. By default, a uniform prior distribution is assumed on the regression coefficients. The uniform prior is a flat prior on the real line with a distribution that reflects ignorance of the location of the parameter, placing equal likelihood on all possible values the regression coefficient can take.

Using the uniform prior in the following example, you would expect the Bayesian estimates to resemble the classical results of maximizing the likelihood. If you can elicit an informative prior distribution for the regression coefficients, you should use the COEFFPRIOR= option to specify it. A default noninformative gamma prior is used for the scale parameter .

You should make sure that the posterior distribution samples have achieved convergence before using them for Bayesian inference. PROC GENMOD produces three convergence diagnostics by default. If ODS Graphics is enabled as specified in the following SAS statements, diagnostic plots are also displayed

Statistics of the posterior distribution samples are produced by default. However, these statistics might not be sufficient for carrying out your Bayesian inference, and further processing of the posterior samples might be necessary. The following SAS statements request the Bayesian analysis, and the OUTPOST= option saves the samples in the SAS data set PostSurg for further processing:

Suppose, for illustration, a question of scientific interest is whether blood clotting score has a positive effect on survival time. Since the model parameters are regarded as random quantities in a Bayesian analysis, you can answer this question by estimating the conditional probability of being positive, given the data, , from the posterior distribution samples. The following SAS statements compute the estimate of the probability of being positive:

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Establish the code's foundation on essential principles such as trust and integrity.

Confirmatory factor analysis (CFA) and exploratory factor analysis (EFA) are similar techniques, but in exploratory factor analysis (EFA), data is simply explored and provides information about the numbers of factors required to represent the data.

Confirmatory factor analysis (CFA) is the method for measuring latent variables (Hoyle 1995; 2011; Kline 2010; Byrne 2013).

https://www.researchgate.net/deref/https%3A%2F%2Fwww.cs.princeton.edu%2F~ken%2Fdynamiconline00.pdf

But in confirmatory factor analysis (CFA), researchers can specify the number of factors required in the data and which measured variable is related to which latent variable.  Confirmatory factor analysis (CFA) is a tool that is used to confirm or reject the measurement theory.

Confirmatory factor analysis estimates latent variables based on the correlated variations of the dataset (e.g., association, causal relationship) and can reduce the data dimensions, standardize the scale of multiple indicators, and account for the correlations inherent in the dataset (Byrne 2013)

Defining individual construct: First, we have to define the individual constructs. The first step involves the procedure that defines constructs theoretically. This involves a pretest to evaluate the construct items, and a confirmatory test of the measurement model that is conducted using confirmatory factor analysis (CFA), etc.

Constant imputation methods impute a constant value in the replacement of missing data in an observation.

The measurement model deals with the relationship between a latent variable and its indicators.

For example: In measuring the ATTITUDE(ATT) of employees towards the ADOPTION INTENTION of new technology at the workplace, the researcher has four (4) and three (3) items with good loadings ( > 0.70) measuring respectively the latent variables ATTITUDE i.e. ATT1 - ATT4; and ADOPTION INTENTION i.e. AI1-AI3. So, the link between each of these items/indicators with their respective latent variable is known as a measurement model.

In contrast, a structural model defines the relationship between the various constructs in a model.

The two measurement model becomes a structural model when they are linked together as shown below. Thus, specifying how latent variables directly or indirectly affect other latent variables in the model.

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Defining individual construct: First, we have to define the individual constructs. The first step involves the procedure that defines constructs theoretically. This involves a pretest to evaluate the construct items, and a confirmatory test of the measurement model that is conducted using confirmatory factor analysis (CFA), etc.

The main disadvantage of mean imputation is the fact that it tends to produce bias estimates for some parameters, particularly for the variance.

This method is preferred by the researcher because it estimates the multiple and interrelated dependence in a single analysis.  In this analysis, two types of variables are used endogenous variables and exogenous variables.  Endogenous variables are equivalent to dependent variables and are equal to the independent variable.

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