I've been studying Bayesian Statistics lately, and just came across the Metropolis-Hastings Algorithm. I understand that the goal is to sample from an intractable posterior - but I'm not really able to understand how the algorithm achieves what it sets out to achieve.  Why and how does it work? What's the intuition behind the algorithm?    To clarify the parts I've problems with, in particular, I've attached the algorithm above.   How is the $q$ distribution (the proposal) related to the intractable posterior? I don't see how $q$ popped out of nowhere. Why is the acceptance ratio calculated the way it is? It doesn't make intuitive sense to me - it'd be great if someone could explain that better. In Step 3, we accept the $X$ we sampled from the $q$ distribution with some probability - why is that? How does that get me something closer to the intractable posterior, which is our goal? (right?)   Please help me out here. Thanks!
Maybe the concept, why it's used, and an example.
How can I know what speed card to get for my camera? Cameras rarely specify exactly what speed cards you should be using in them - I just checked the manual for my DSLR, and all it says is to use an SDHD card, and a "high speed" card should be used for recording video.  On the other hand, there are a number of different rating systems for memory card speed (33x! 150x! Class 2! Class 10!) and price seems to increase with speed.  Is there any way to test the maximum write speed the camera is capable of?  Are the read/write speeds documented anywhere? To explain the expansion of space I have often heard people saying that space is continuously created. This picture is usually applied to cosmological scales but I`m nevertheless curious if some microscopic description of this process could be found (or probably is already available?).  The first thing which comes into my mind by thinking about this is that in theories like loop quantum gravity, there is an idea that space is quantized. Would space expansion then be explained as the creation of new quanta in loop quantum cosmology? Or are there other different ideas, such as the quanta themselves changing properties? Do people actually working on this already have some interesting results? How does extra damage work for critical hits? I'm a bit confused on how extra damage works for critical hits. Let's take the following example: A thief with sneak attack rolls a crit. I'm under the impression he rolls double his normal attack damage dice AND double his Sneak Attack damage dice. Is this accurate? Another example, I finished DM-ing my first campaign recently. The Giant Spider enemy monster has an melee attack that causes a constitution saving throw. If the player fails, the enemy rolls an additional 2D8 of damage. In this situation, if there was a critical hit, and the player failed their saving throw, would I also roll double dice on the additional damage? Or is that considered a separate attack? The usual Schrodinger Lagrangian is $$ \tag 1 i(\psi^{*}\partial_{t}\psi ) + \frac{1}{2m} \psi^{*}(\nabla^2)\psi, $$ which gives the correct equations of motion, with conjugate momentum for $\psi^{*}$ vanishing. This Lagrangian density is not real but differs from a real Lagrangian density  $$ \tag 2 \frac{i}{2}(\psi^{*}\partial_{t}\psi -\psi \partial_{t}\psi^{*} ) + \frac{1}{2m} \psi^{*}(\nabla^2)\psi $$ by a total derivative.  My trouble is that these two lagrangian densities lead to different conjugate momenta and hence when setting equal time commutation relations, I am getting different results (a factor of 2 is causing the problem). I can rescale the fields but then my Hamiltonian also changes. Then applying Heisenberg equation of motion, I don't get the operator Schrodinger equation.   Is it possible to work with the real Lagrangian density and somehow get the correct commutation relations? I would have expected two Lagrangians differing by total derivative terms to give identical commutation relations (since canonical transformations preserve them). But perhaps I am making some very simple error. Unless all conjugate momenta are equivalent for two Lagrangians differing by total derivatives, how does one choose the correct one?   I guess the same thing happens for other first order systems like Dirac Lagrangian also. Grub does not show a Windows 8 option after dual boot So, I've successfully dual-booted my Windows 8 machine with Ubuntu 12.04 . However, I still don't have a convenient method of choosing what OS to load at boot time.  After installing Ubuntu, my computer still loads Windows 8 directly. I then added grubx64.efi to the white list of my boot loader. But after that, my machine loads Ubuntu directly without even a shadow of GRUB showing up!  I used boot-repair and I got this paste.ubuntu URL: . After running boot-repair (and re-white listing the grubx64.efi file), GRUB now shows up but without any Windows 8 option!  Lastly, I ran sudo fdisk -l and it gave me this:  WARNING: GPT (GUID Partition Table) detected on '/dev/sda'! The util fdisk doesn't support GPT. Use GNU Parted.   Disk /dev/sda: 750.2 GB, 750156374016 bytes 255 heads, 63 sectors/track, 91201 cylinders, total 1465149168 sectors Units = sectors of 1 * 512 = 512 bytes Sector size (logical/physical): 512 bytes / 4096 bytes I/O size (minimum/optimal): 4096 bytes / 4096 bytes Disk identifier: 0x6396389f     Device Boot      Start         End      Blocks   Id  System /dev/sda1               1  1465149167   732574583+  ee  GPT Partition 1 does not start on physical sector boundary.   I'm guessing my problem has something to do with the warning from fdisk above but I don't know what to do with it. How do I proceed now?  Edit  My etc/default/grub is as follows:  # If you change this file, run 'update-grub' afterwards to update # /boot/grub/grub.cfg. # For full documentation of the options in this file, see: #   info -f grub -n 'Simple configuration'  GRUB_DEFAULT=0 #GRUB_HIDDEN_TIMEOUT=0 GRUB_HIDDEN_TIMEOUT_QUIET=true GRUB_TIMEOUT=10 GRUB_DISTRIBUTOR=`lsb_release -i -s 2&gt; /dev/null || echo Debian` GRUB_CMDLINE_LINUX_DEFAULT="quiet splash" GRUB_CMDLINE_LINUX=""  # Uncomment to enable BadRAM filtering, modify to suit your needs # This works with Linux (no patch required) and with any kernel that obtains # the memory map information from GRUB (GNU Mach, kernel of FreeBSD ...) #GRUB_BADRAM="0x01234567,0xfefefefe,0x89abcdef,0xefefefef"  # Uncomment to disable graphical terminal (grub-pc only) #GRUB_TERMINAL=console  # The resolution used on graphical terminal # note that you can use only modes which your graphic card supports via VBE # you can see them in real GRUB with the command `vbeinfo' #GRUB_GFXMODE=640x480  # Uncomment if you don't want GRUB to pass "root=UUID=xxx" parameter to Linux #GRUB_DISABLE_LINUX_UUID=true  # Uncomment to disable generation of recovery mode menu entries #GRUB_DISABLE_RECOVERY="true"  # Uncomment to get a beep at grub start #GRUB_INIT_TUNE="480 440 1"   Also, just for the record,  in installing my dual-boot. The tutorial (which uses, 11.10, I must note), doesn't mention anything about this problem. Is there anything there I shouldn't have done? What is the center of $D_{2n}/Z(D_{2n})$ What is the center of $D_{2n}/Z(D_{2n})$. I see that when $n=2^k$ then I have a $p$-group so the center is not trivial. But when $n$ is not power of $2$ how can i know what is the center of this group?. Surely when $n$ is odd then the center is trivial. Proof: For all integers $x$ and $y$, if $x^3+x = y^3+y$ then $x = y$ I entered chmod -R 777 /usr/bin and now sudo is not working.  It says sudo must be setuid root.  Some advice online said to run chown root:root /usr/bin/sudo chmod 4755 /usr/bin/sudo.  On entering chown root:root /usr/bin/sudo it shows an opened in readonly mode error. Poking around the people's profiles to see how the  I keep seeing things like this:  It looks strange to have a 25/5 progress on a badge, but the dialog explains that what this user is missing is a positive record of asking. The  is: (total questions - negative questions - closed - deleted)/total questions &gt;= 0.5  If you look at , it's hard to see how his asking record could be anything but exceptional. According to  his asking record is 0.875. The problem is that he asked many of his questions in the early days of the site that were well-received, but later deleted. Some were deleted after the question was . My suggestion is that we change the asking record formula: (total questions - negative questions - closed - early deleted)/total questions &gt;= 0.5  The early deleted variable counts only questions deleted within 60 days of being asked. I picked 60 days because it's the dividing line between being able to . After looking through the list of people who would get the Curious badge on Stack Overflow with this change, I noticed a few things:  Many of them have self-deleted questions on their record. Often these are questions that were asked and basically ignored. After a few months, the original asker went back and deleted their question. It's hard to know what happened in each individual case, but my guess is that they never found a solution to their specific question and worked around the issue instead. (I have a few of those, but I'm not courteous enough to delete them.)  These users don't tend to ask the same question over and over again. I ran a similar test that excluded self-deleted questions. That group included a lot of cases of people asking a question, not getting an answer after a day or two, deleting the question, and asking the same question again. This strategy doesn't work if you wait two months to delete your first question.  Two mods (George Stocker and Gordon), Tim Post, and myself are on the list of users who would get a Curious badge under this system. On Meta Stack Exchange 5 current employees would be newly eligible for the badge. In my (admittedly biased) opinion, these are curious users.  I did some spot check on the lower end of the reputation scale. Every user I looked at fit my model of the ideal recipient of the Curious badge; they asked mostly upvoted questions with few closed questions. This change would add about 3286 new Curious badges on Stack Overflow, so there will certainly be a few exceptions I didn't find. But I think they are more than balanced by the clearly deserving users who will be awarded new badges.    Proposal: For the Curious, Inquisitive, and Socratic badges, don't count questions deleted 60 or more days after asking as deleted for the purposes of calculating asking record. These questions may still count as negatively scored or closed, if applicable. Days that include deleted questions, no matter when they are deleted, do not count toward the days asked criteria.   Modified Proposal: For the Curious, Inquisitive, and Socratic badges, don't count questions deleted or closed 60 or more days after asking as deleted or closed for the purposes of calculating asking record. These questions may still count as negatively scored, if applicable. Days that include deleted or closed questions, no matter when they are deleted or closed, do not count toward the days asked criteria. On Stack Overflow, that would add about 665 Curious badges, including two more for moderators (casperOne and 0x7fffffff) and one for an employee (Oded). On the opposite end of the reputation scale my spot checks are a bit muddier. Instead of a small number of deleted (and usually closed) questions, this list turns up users whose latest questions still aren't all that interesting. Even so, on balance a few extraneous bronze badges seem worthwhile in exchange for acknowledging truly curious users. I want to attach a code file to my question, but I am not able to find a link to do it. Is attaching files possible at all? I have already dual systems : Windows 7 and Ubuntu installed on my laptop. The Windows 7 had been installed when I bought this laptop, and I don't have a CD.  I just installed VitrualBox under Ubuntu, hoping that I could install another Windows in this virtual machine, so that I could run some applications of Windows even under Ubuntu.  But it seems that I need a CD or ISO of Windows to boot inside this virtual machine. Otherwise, there is an error: FATAL: No bootable medium found! System halted. My question is where I could find a CD or ISO?  At the moment, I am downloading ISO from this , but do not know if it will work. By the way, I am not sure if I should pick the 32-bit or 64-bit? As my CPU is 64-bit but my Ubuntu is i386.  Could anyone help? Thank you very much. Explicit vs implicit SQL joins Is there any efficiency difference in an explicit vs implicit inner join? For example:  SELECT * FROM table a INNER JOIN table b ON a.id = b.id;   vs.  SELECT a.*, b.* FROM table a, table b WHERE a.id = b.id; gedit: search and replace with regular expressions I want to replace this with gedit:   1299465 |  2003415 | 2015-09-06 05:35:34.59662+02  1299449 |  2009400 | 2015-09-06 05:35:32.301683+02  1299450 |  2008465 | 2015-09-06 05:35:32.451393+02  1299457 |  2015211 | 2015-09-06 05:35:33.451049+02   to that:  2003415 2009400 2008465 2015211   How can I do this with gedit?  Background: I want to do it with gedit and not sed/python/.... since the replace needs to be done interactive. Checking the way to stop duplicate names - validation rules or triggers Which has the heavier tail, lognormal or gamma? (This is based on a question that just came to me via email; I've added some context from a previous brief conversation with the same person.)  Last year I was told that the gamma distribution is heavier tailed than the lognormal, and I've since been told that's not the case.   Which is heavier tailed? What are some resources I can use to explore the relationship? Can regular expressions be used to match nested patterns? Is it possible to write a regular expression that matches a nested pattern that occurs an unknown number of times? For example, can a regular expression match an opening and closing brace when there are an unknown number of open/close braces nested within the outer braces?  For example:  public MyMethod() {   if (test)   {     // More { }   }    // More { } } // End   Should match:  {   if (test)   {     // More { }   }    // More { } } Removing las2dem streaks from raster DEM? I've created a series of DEM files using the  tool from LAStools; however since I don't have a licensed version they have black streaks running across them. Does anyone know a good method for removing these streaks? Command to open a URL Is there a command I can use in Bash that will open a specific URL in the default browser? I downloaded the iso file for XUbuntu. I created a live usb using Pen Drive Linux. I turned on my computer, booted to the XUbuntu installation screen, and selected "check disk for defects".  A minute later, it says that it found one defect. How do I figure out what this defect is? Laravel - Populating Select List with Derived Values