29 views

### Does maximum likelihood need to account for the probability of input covariates?

I am reading "Probabilistic Machine Learning" by K. Murphy. In it, he defines the likelihood of a dataset as However, this dataset $D$ is defined as: So if all $x_n, y_n$ are random ...
391 views

### Why is each observation in a sample considered a random variable in linear regression?

I have the following excerpt in my statistics textbook: I am confused by the sentence: "Another way statisticians treat this model is that, assume $X_1...X_n$ are random variables, we make ...
179 views

### Meaning of & intuition behind predictors being fixed in linear regression [duplicate]

My question is a bit naive. I'm trying to get the exact & clear meaning of the phrase "predictor variables are fixed and not random in linear regression". According to my understanding, ...
808 views

### Does the OLS estimator in simple linear regression converge a.s.?

Consider the following model. Assume $(x_i, u_i)$ is sequence of independent identically distributed random vectors in $\mathbf{R}^{d+1}:$ $x_i$ are $\mathbf{R}^d$-value random vectors, which will ...
62 views

### Notation of the Likelihood Term in Bayesian Neural Networks

I see that in Bayesian neural networks likelihood function is defined in two ways: $p(W|D) = Z^{-1} p(D|W)p(W)$ or $p(W|y,x)=Z^{-1}p(y|x,W)p(W)$ Are there a slight difference in interpreting $p(D|W)$ ...
49 views

### Do we assume that the regressors are uncorrelated with the unobserved error $\epsilon$ for least squares?

I recall seeing sources in the past state that the Gauss-Markov assumptions assume that the regressors are uncorrelated with $\epsilon$ in order to make $E[\hat{\beta}] = \beta$. But is this ...
241 views

### Clarification on the assumptions $E[u|x]=0$ and the $x_i$ being fixed in repeated samples in Wooldridge Introductory Econometrics

The author is writing on the assumption $E[u|x]=0$. The part of the text which is not clear to me is this (the red lines emphasize where the critical portions are located) : In the first piece I don'...
1k views

### Proof that Regression Sum of Squares and Residual Sum of Squares are independent random variables

Having consulted a number of sources, I still can't find a complete proof that Regression Sum of Squares ($SS_{regression}$) and ($SS_{residual}$) are independent random variables. I'll be doubly ...
29 views

162 views

### Confused with the fundamental assumptions of Frequentist and Bayesian Linear Regression

In Frequentist Linear Regression, I have seen 2 approaches which lead to basically similar models. We have $W,y,X,\epsilon$ related as $y=W^TX+\epsilon$, where $y$ is the dependent random variable, ...
2k views

1 vote
963 views

### Regression's population parameters

Suppose I've specified a linear regression model: $$Y = \beta_0 + \beta_1 X + \epsilon$$ where $\beta_0$, $\beta_1$ are the population parameters. My question is: why are these parameters ...
115 views

### Deriving the posterior distribution over the model parameters: are the model parameters and data independent?

We are told (in Section 9.2.3, Deisenroth et al.: Mathematics for Machine Learning) that we can compute the posterior over a model's parameters $\boldsymbol\theta$ (here in the context of linear ...