Articles of bayes theorem

Cox derivation of the laws of probability

I have read Jaynes’ Probability Theory: The Logic of Science a while ago, but mostly skimmed over parts of his derivations that I didn’t immediately understand. Now I’m trying to really understand it, but it appears he mostly skips over steps he sees as obvious or trivial. Now, I’ve been able to construct most of […]

How to calculate Pr(Diseased | 2 Positive Tests)?

Diagnosed with a rare disease, you know that there is only a 1% chance of getting it. Abbreviate D as the event “you have the disease” and T as “you test positive for the disease.” The test is imperfect: $\Pr(T | D) = 0.98$ and $\Pr(T^C | D^C) = 0.95.$ $\large{A.}$ Given that you test […]

How to solve probability with two conditions (with explanation)?

This is an extension over this question: Inter-causal reasoning: How to solve probability with two conditions? I’m a beginner in probability, and trying to deeply understand what is happening underneath. To sum up what the question is about: We’ve got a graph of (binary) events: $$ A \rightarrow C \leftarrow B $$ We’re given probabilities […]

Bayesian posterior with truncated normal prior

Suppose we observe one draw from the random variable $X$, which is distributed with normal distribution $\mathcal{N}(\mu,\sigma^2)$. The variance $\sigma^2$ is known, $\mu$ isn’t. We want to estimate $\mu$. Suppose further that the prior distribution is given by truncated normal distribution $\mathcal{N}(\mu_0,\sigma^2_0,t)$, i.e., density $f(\mu)=c/\sigma \phi((\mu-\mu_0)/\sigma_0)$ if $\mu<t$, and $f(\mu)=0$ otherwise, where $t>\mu$ and $c$ […]

Intuitively, why does Bayes' theorem work?

Why does Bayes’ theorem work? I’m not looking for a cryptic math demonstration. Rather, I’m interested in the intuition behind the theorem that reveals the a posteriori probability given the prior times the likelihood.

Why do knowers of Bayes's Theorem still commit the Base Rate Fallacy?

Source: 6 July 2014, “Do doctors understand test results?” by William Kremer, BBC World Service. A 50-year-old woman, with no prior symptoms of breast cancer, participates in routine mammography screening. She tests positive, is alarmed, and wants to know from the physician whether she has breast cancer or what the chances are. Apart from the […]

P(A|C)=P(A|B)*P(B|C)?

Is this formula P(A|C)=P(A|B)*P(B|C) correct according to Bayes’ theorem? I don’t think it correct but for a transition matrice system we can have something like thistransition matrice So I don’t know how to prove P(A|C)=P(A|B)*P(B|C) in transition matrice though it looks very intuitive

In Bayesian Statistic how do you usually find out what is the distribution of the unknown?

To estimate the posterior we have $$p(\theta|x) = \frac{p(\theta)*p(x|\theta)}{\sum p(\theta ‘)*p(x|\theta ‘)}$$ $x$ is usually the experimentally sampled data, and $\theta$ is the model, but both $p(x|\theta)$ and $p(\theta)$ is unknown, how do you usually measure those two quantities?

combining conditional probabilities

I’ve come across a physics paper in which pdf $$ p(a|b) $$ is desired, but only $$ p(a|c)\\ p(c|b) $$ are known. It is claimed that $$ p(a|b)=\int p(a|c)p(c|b) dc. $$ Is this correct wlog? I can’t prove it. I think it is in fact a conditional expectation for $p(a|c)$. Am I right? Is it […]

Understanding how the rules of probability apply to probability density functions

I’m trying to understand how the rules of probability apply with probability density functions. If I denote the two continuous random variables as $X$ and $Y$ and their corresponding joint distribution as $f_{X,Y}(x,y)$ are the following derivations legit / true / correctly understood: $$P(Y = y) = \lim_{\Delta y \rightarrow0} \int_{y}^{y+\Delta y}\int_{-\infty}^{\infty}f_{X,Y}(x,y)\,dx\,dy$$ $$P(X = x) […]