Frequentist perspective of regression coefficients and significance (coming from Bayesian background)? I come from a primarily Bayesian background when using performing statistical analysis. In the context of linear regression, I would look at the posterior distributions for each regression coefficient. If a given coefficient was distributed ~N(1,1), I might not be terribly convinced of the associated variable's influence on the dependent variable.
From a frequentist perspective, regression coefficients often have p-values assigned. And I'm not quite sure what to make of this coming from a bayesian, full posterior interpretation. Is the idea that there's only an x% chance that we would have observed this specific regression coefficient inferred by accident?

A p value is the probability of observing a test statistic as or more extreme than the researcher's own test statistic, assuming the null hypothesis, and an assumed distribution model are both true.
So when you see frequentist statistical software provide regression output that includes something like:




name
estimate
S.E.
t (or z)
p value




cons
3.0
0.015
200
<0.001


beta
0.5
0.18
2.78
0.003




The default null hypothesis for regression coefficients in most stats software I am familiar with is $\text{H}_{0}\text{: }\beta = 0$, with $\text{H}_{\text{A}}\text{: }\beta \ne 0$. You should read the $p = 0.003$ as $p = P(|T_{\text{df}}|\ge |t|)$ given $\text{H}_{0}$, or in plain language: the probability of observing a value of $\widehat{\beta}$ as or more extreme than 0.5 is 0.003, assuming the null hypothesis is true (and assuming Student's t distribution truly describes the distribution of your test statistic). If a priori you had an $\alpha =0.01$, then you would interpret that p value as evidence against the null hypothesis, and for the alternative hypothesis (i.e. you found evidence that $\beta \ne 0$).
