I understand how MCMC can be useful for approximating the shape of a density function if you're able to generate random samples from that density function. However, I don't understand how this is applied towards Bayesian inference because any specific selection of model parameters using MCMC would yield a number, not a distribution.  For example, if I have a binomial likelihood function and a beta prior, I might use Metropolis-Hastings to select a random starting parameter, $\theta=.3$. When that starting parameter and some data are input into my likelihood function and prior, I get a deterministic number as the output. No randomness involved.  Framing this issue another way: Why does the MH algorithm generate the posterior distribution from the count of visits to a parameter value rather than using the posterior values generated directly?
My understanding is that when using a Bayesian approach to estimate parameter values:   The posterior distribution is the combination of the prior distribution and the likelihood distribution. We simulate this by generating a sample from the posterior distribution (e.g., using a Metropolis-Hasting algorithm to generate values, and accept them if they are above a certain threshold of probability to belong to the posterior distribution). Once we have generated this sample, we use it to approximate the posterior distribution, and things like its mean.   But, I feel like I must be misunderstanding something. It sounds like we have a posterior distribution and then sample from it, and then use that sample as an approximation of the posterior distribution. But if we have the posterior distribution to begin with why do we need to sample from it to approximate it?
Question:     Let $f$ be differentiable on $[0,+\infty)$, such as$$\lim_{x\to+\infty}\left([f'(x)]^2+f^3(x)\right)=0$$show that $$\lim_{x\to\infty}f(x)=\lim_{x\to\infty}f'(x)=0$$   I think this problem is similar to  and (the  background)  For my problem I guess that if  $$\lim_{x\to+\infty}\left([f'(x)]^{2m}+f^{2n+1}(x)\right)=0,m,n\in N^{+},n,m&gt;1$$ then we have $\lim_{x\to\infty}f(x)=\lim_{x\to\infty}f'(x)=0$. Suppose that $f:[0,\infty)\rightarrow\mathbb{R}$ is continuous. Is it true that   $$\int_{0}^\infty|f(t)|dt&lt;\infty\Rightarrow \lim_{t\rightarrow\infty}f(t)=0?$$  If so can you provide a proof, otherwise a counter example. Thank you. Creating a dictionary from a CSV file Berlin Public Transport Ticket Validity I have purchased an AB zone Berlin 7 day transport ticket and validated it on Sunday 17th April so hopefully it will be valid until Sunday 24th.  I validated the ticket at about 10:00am so I guess 1000 is the time I validated? But what are the numbers 15 and 6 and letters 'So' correspond to? How (if checked by a guard) can they tell if the ticket is valid? And how can I as a customer tell when it is validated to? Finding $\int_0^\infty \frac{\cos(ax)-\cos(bx)}{x}dx$ I am practicing calculus section of GRE Math Subject test, and I can't figure out a way to do the following integral: $$\int_0^\infty \frac{\cos (ax) - \cos (bx)}{x}dx.$$  I absolutely have no clue how to do this. Can someone show me the solution explicitly? Thank you. A group $G$ with $G'$ abelian and every abelian normal subgroup finite Let $G$ be a group such that  $G'$ abelian  and any abelian normal subgroup of $G$ is  finite. Show that $G$ is finite. Any way to list similar commands? When you write the command name wrong, bash often does this:  septi@norbert:~$ good No command 'good' found, did you mean:  Command 'gold' from package 'binutils' (main)  Command 'gmod' from package 'gmod' (universe)  Command 'goo' from package 'goo' (universe)  Command 'god' from package 'god' (universe)  Command 'geod' from package 'proj-bin' (universe)  Command 'gord' from package 'scotch' (universe) good: command not found   Or sometimes it does this:  septi@norbert:~$ nftp No command 'nftp' found, but there are 23 similar ones nftp: command not found   Is there any way to ask bash to show these 23 similar commands for me? And, is there a way to show similar commands, including those that aren't yet installed, instead of running the application, ftp for example? Change this behavior to allow for spelling corrections and the like: "Edits must be at least 6 characters" Is the energy density of vacuum uniform in the whole universe? Mac turns off randomly and has lines on screen For some reason my iMac keeps turning off around 5 to 10 minutes into boot.  Also just recently it turns on but has shaded lines going down the screen. I have not installed any weird programs and i have not used it in around 3 months.   Any ideas on what the issue is, i don't have any other information about the issue besides whats happening when i use it.  The image i'm showing you was found on google but this looks exactly like mine does.  The model of my imac is the late 2011 model. A few years ago I saw a scifi movie about a group of travelers stranded at what I believe might have been a bus station (I don't remember that detail with any certainty).  There were 2 children (a boy and a girl) who were running around and everyone thought they belonged to someone else there.  As it turned out, the children were aliens that reverted to their true form and began hunting down the travelers.  I've been searching all over the internet but can't come up with the name of the movie.  Does anyone know the name of it? Should I upgrade to the "mainline" kernels? Newer "mainline" versions of the kernel are available from the , e.g. 3.4 and 3.5, while Ubuntu 12.04 is on 3.2.   What are the mainline kernels? Are they better than my existing kernel? What are the pros and cons of upgrading to a mainline kernel? Will I easily be able to go back? In a moment of cheapness, I decided not to order a 5v to 3.3v level shifter from Sparkfun, but instead to put it together myself.  The  used a  MOSFET, but since I enjoy surface-mount soldering about as much as root canal treatment, I decided to use a  that is through-hole mounted and easily available from my preferred supplier.  The results of this were sub-optimal.  When pulling the 5v line to ground, all was fine - the 3.3v side went to 0.07v.  But when pulling the 3.3v to ground, the 5v line showed around 4.14v (it was otherwise a touch above 5.1v).  On a closer reading of the datasheets involved, as well as the original Philips application note on the subject, I began to come to the conclusion that the gate threshold voltage was the problem.  By altering the schematic and tying the MOSFET's gate to 5v instead of 3.3v, both sides seem to work fine.  Pulling either side low causes the other side to go low.  I'm not, however, at all confident that this is actually a sane thing to do.  My understanding of the original schematic isn't deep enough to form a sensible opinion.  Will this modified schematic work, or are the current good results I'm seeing mere fluke or a precursor to something releasing magic smoke? Restoring Windows XP PATH variable I installed an application that screwed my PATH variable. Now I cannot run anything from the command prompt. Is there any way I can restore PATH variable? How can I remove launcher drive icons? How to disable Import option in LibreOffice Calc while pasting text in a cell? I'm making a table in which I have to copy few lines from pdf to a table. I'm using LibreOffice calc for table making purpose. Problem is that even though I am copying simple text from the pdf, when I paste it to LibreOffice, Import Option keeps on poping up(randomly-sometimes it pop up sometime it does't). And due to this pasted text's font size also very small. Desired result is that when I paste the text in the cell, it just got pasted in the cell, without any time wasting popup which also disturbs the font size. Same question is asked here but no solution. Prove that inequality is true for $x>0$: $(e^x-1)\ln(1+x) > x^2$ I was given a task to prove that inequality is true for x>0: $(e^x-1)\ln(1+x) &gt; x^2$. I've tried to use derivatives to show that the $f(x) = (e^x-1)\ln(1+x)-x^2$ is greater than zero, but has never succeeded.  Any help will be appreciated. Why $∀x(P(x)→Q)$ is equivalent to $∃xP(x)→Q$ Given $x$ occurs free in $P$. If $x$ does not occur free in $Q$, then $∀x(P(x)→Q)$ is semantically equivalent with $∃xP(x)→Q$. How to understand this statement. And also, need an example to show that the formula will not necessarily be equivalent if $x$ does occur free in $Q$.  I guess what struggles me is I'm not quite understand what does "free" mean in this case. But it would be better to have answer for the above question. Bash : compare two strings with space I am trying to write a bash script which run a command and compare the result with another string.   #!/bin/bash -x  STATUS=`/root/setup_ha show --password-file=/root/password | grep "&gt;HA State" | awk '{print $3}' |  cut -c 2-`  TEST=`echo $STATUS` if [[ "$TEST" == "ON Master" ]]; then echo CLUSTER CRITICAL else   echo CLUSTER OK MASTER fi   As the Original string is on two lines, I echo it in a new variable TEST. The New variable have the output of the command on one line.  Here is the Bash debug output :  ++ /root/setup_ha show --password-file=/root/password ++ grep '&gt;HA State' ++ awk '{print $3}' ++ cut -c 2- + STATUS='ON Master' ++ echo 'ON' Master + TEST='ON Master' + [[ ON Master == \O\N\ \M\a\s\t\e\r ]] + echo CLUSTER OK MASTER CLUSTER OK MASTER   I also tried the following test :  if [[ "$TEST" =~ "ON Master" ]]   The thing is bash is not able to compare the strings it is always false.  Any idea ?   EDIT :  Here is the output with the first answer :  + STATUS='ON Master' ++ echo 'ON' Master + TEST='ON Master' + '[' 'ON Master' == 'ON Master' ']' + echo CLUSTER OK MASTER CLUSTER OK MASTER   Still  not working ON seems weird on line 3, plus in my bash it takes green color ! In January, some additional whitespace  below h1, h2 and h3 headings,  more whitespace below the headers than above them.  Could we have the whitespace above the headings be more than the whitespace below them?  I find some posts a bit hard to read, like the current rendering on the left:   &nbsp; &nbsp; &nbsp;