I have trouble discerning conceptually between these two notions. I am aware of their formal relations, proprieties and what not, but I just can't wrap my head around what they "mean", if that even makes sense. I have checked some answers, such as , but I think none of these addresses my problem.  So, given some data $\mathcal{D}$ and parameters $\theta$,  $$ p(\theta|\mathcal{D})=\frac{p(\mathcal{D}|\theta)p(\theta)}{p(\mathcal{D})} $$  I have learned that the posterior is "the probability of $\theta$ being the statistical parameter underlying $\mathcal{D}$". And the likelihood is "the likelihood of $\theta$ having generated $\mathcal{D}$". In my head, these two notions are exactly the same. So how can I distinguish them?
We can write Bayes' theorem as  $$p(\theta|x) = \frac{f(X|\theta)p(\theta)}{\int_{\theta} f(X|\theta)p(\theta)d\theta}$$  where $p(\theta|x)$ is the posterior, $f(X|\theta)$ is the conditional distribution, and $p(\theta)$ is the prior.  or  $$p(\theta|x) = \frac{L(\theta|x)p(\theta)}{\int_{\theta} L(\theta|x)p(\theta)d\theta}$$  where $p(\theta|x)$ is the posterior, $L(\theta|x)$ is the likelihood function, and $p(\theta)$ is the prior.  My question is   Why is Bayesian analysis done using the likelihood function and not the conditional distribution? Can you say in words what the difference between the likelihood and conditional distribution is?  I know the likelihood is not a probability distribution and $L(\theta|x) \propto f(X|\theta)$.
Lie bracket for Lie algebra of $SO(n,m)$ How does one show that the bracket of elements in the Lie algebra of $SO(n,m)$ is given by  $$[J_{ab},J_{cd}]  ~=~ i(\eta_{ad} J_{bc} + \eta_{bc} J_{ad} - \eta_{ac} J_{bd} - \eta_{bd}J_{ac}),$$  where $\eta$ has is the definite symmetric form with signature $(n,m)$? Align equation left Here is my split equation:  \documentclass{article} \usepackage{amsmath} \begin{document} \begin{equation} \begin{split} a_{ijk} =&amp; \frac {Pr(M_{I} =2  \&amp;  M_J=1 \&amp; M_K =1 | I=i , J=j , K=k)}{Pr (M_I =1 \&amp; M_J =1 \&amp; M_K=1 | I=i , J=j)} \\             =&amp; \frac {\mu_{ijk211}}{\mu_{ijk111}}\\ \end{split} \end{equation} \end{document}   I want to align this equation to the left. Is this possible in an equation environment? What is the use/meaning of 'swap area' while linux installation? How to get the IPV4 address for an interface from /proc Is there a clean, simple way to get an IP address for a network interface from /proc, similar to the way I can get the MAC address for a network interface? Ideally I would just type cat /proc/&lt;foo&gt;/{interface_name} and get the IPv4 address.  I'd rather not run anything other than cat. Generated $\sigma$-algebras with cylinder set doesn't contain the space of continuous functions Word for obscuring bad or immoral acts with verbiage I have been in many scenarios where people try to obscure/reduce the intensity of a seemingly bad or immoral act using grammar. E.g. Assuming Thomas killed a dog, you could hear something like:     There's essentially nothing wrong about palliating the miserable life   of a despondent being. If there was any issue here, I'd say it's not   consulting a veterinary doctor first but I've come to understand that some people are just straightforward.   How does one react to the above mumbo jumbo? I think there are similar situations also found in law courts. What can this be referred to as? Grammatification, obscure tactics? Finding the Public IP address in a shell script I can find my IP address using ifconfig or hostname -i command.  But how do I find my Public IP?  (I have a static public IP but I want to find it out using unix command) Why is inductance (L) proportional with turns-square (N²)? We start from the   $$ \mathbf{\nabla} \times \mathbf{B} = \mu \mathbf{J} + \overbrace{\mu \epsilon \dfrac{\partial \mathbf{E}}{\partial t}}^0. $$  We take surface integration of both sides, for the surface (\$s\$) inside the mean path (\$c\$) of the core.  $$ \int_s \left( \mathbf{\nabla} \times \mathbf{B} \right) \cdot d\mathbf{s} = \mu \int_s \mathbf{J} \cdot d\mathbf{s} $$  We use the  to rewrite the left hand side; where \$c\$ is in the same direction with the magnetic flux \$\Phi\$.  $$ \oint_c \mathbf{B} \cdot d \mathbf{\ell} = \mu N I $$  (The integral on the left hand side results \$NI\$, because there are \$N\$ different wires on the winding.)  The magnetic field density inside these kind of cores are considered to be uniform. So, we can write  $$ B \ell_c \overset\sim= \mu NI \implies B = \dfrac{\mu NI}{\ell_c}; $$  where \$\ell_c\$ is the mean path lenght of the core.  We can find magnetic flux from the magnetic flux density we have found by using the cross sectional area of the core \$A_c\$.  $$ \Phi = BA_c = \dfrac{\mu NIA_c}{\ell_c} $$  By definition, inductance is the amount of magnetic flux generated per applied current, that is  $$ L \overset\triangle= \dfrac{\Phi}{I}. $$  So, we find inductance of the system as  $$ \boxed{ L = \dfrac{\Phi}{I} = \dfrac{\dfrac{\mu NIA_c}{\ell_c}}{I} = \dfrac{\mu NA_c}{\ell_c} }. $$  But, all other sources () give inductance of an inductor like this as  $$ L = \dfrac{\mu N^2A_c}{\ell_c}. $$  What is the mistake I did in my derivation? Please explain in detail. How can I figure out what's slowly eating my HD space? I accidentally opened Orca Screen Reader from the Dash when I was trying to open Screenshot. Now everything I type or click on is spoken out loud.  How do I make it stop?  For reference, here is a screenshot of Orca Screen Reader running: How to play blu ray .mkv files on ubuntu 20.04 LTS? How to play blu ray .mkv files on Linux? e.g:1080p.BluRay.DD5.1.x264-NTb.mkv If $m,n\in N$ Prove that there is such a positive integer k, such that $(\sqrt{m}+\sqrt{m+1})^n=\sqrt{k}+\sqrt{k+1}$ If $m,n\in N$ Prove that there is such a positive integer k, such that $(\sqrt{m}+\sqrt{m+1})^n=\sqrt{k}+\sqrt{k+1}$ I attempted to solve this question using binomial coefficients, saying: $(\sqrt{m}+\sqrt{m+1})^n=\sqrt{k}+\sqrt{k+1}=\sum\limits_{a=0}^n {n\choose a}*\sqrt{m}^{n-a}*\sqrt{m+1}^a$ and from here I was thinking that I had to do something with $\sqrt{m}^{n-a}$ with $\sqrt{k}$ and do something else with $\sqrt{k+1}$ and $\sqrt{m+1}^a$. Unfortunately I couldn't think of what to do with this circumstance. I then thought of maybe taking $(\sqrt{k}+\sqrt{k+1})^2=(\sum\limits_{a=0}^n {n\choose a}*\sqrt{m}^{n-a}*\sqrt{m+1}^a)^2$, but this immediately over-complicated the question. Could you please explain to me how to solve this question and how to solve similar questions in the future? Why is 8 so special? I have been reading about multi-dimensional numbers, and found out that it's been proven that the Octonions are the composition algebra of the largest dimension. I was wondering why, despite having infinitely many different dimensions of numbers, the only composition algebras are of 1, 2, 4, and 8 dimensions. What's so special about 8? Axiom of Choice: Where does my argument for proving the axiom of choice fail? Help me understand why this is an axiom, and not a theorem. Coin sequence paradox from Martin Gardner's book Is "89.9%" singular or plural? I’m writing a dissertation on network security. I’m unsure which of the following to use:   ...89.9%, which is 57 million users.  ...89.9%, which are 57 million users.   Technically it should be are because it refers to a plural amount of people, but it doesn’t sound right to me considering that 57 million is in a sense one group. What causes the 'Connection Refused' message? This is a  about Connection Refused    We see a lot of questions to the effect      When I try to connect to a system I get a message      Connection refused      Why is this ? Let $\lambda(A)$ be the Lebesgue measure of $A$. There exists an interval $I$ such that $\lambda(E \cap I) / \lambda(I) < 1-\epsilon$ (Not mentioned in title but $\epsilon$ is a number greater than $0$ and $E$ a Lebesgue measurable subset of $\mathbb{R}^n$.) I know a question equivalent to this one has been asked (). But it was not answered. Also my thinking is a bit different. So I would love to get some help on this problem. So far I've got:  $$\frac{\lambda(E \cap I)}{\lambda(I)} &gt; 1- \epsilon \iff \frac{\lambda(I)-\lambda(E\cap I)}{\lambda(I)} &lt; \epsilon$$  And also, a theorem states that, for all $\epsilon$ there exists a closed set $F\subset E$ such that $\lambda(E\setminus F) &lt; \epsilon$ so it would sufice to prove  that there exists an interval $I$ such that $$\frac{\lambda(I)-\lambda(E\cap I)}{\lambda(I)}&lt;\lambda(E\setminus F).$$ I'm not sure if this helps me much but any help is appreciated. What would you do? When I look in sys.sql_logins, I see a column called is_policy_checked. Can I trust that my password policy has been checked for all of the logins where this column value is 1? How to make a file (e.g. a .sh script) executable, so it can be run from a terminal I have a script.sh file and type of this file is shellscript file. I want to make this file as application/x-executable file. How can I make it?