This is an introduction and description of vipassana meditation [edit: actually, anapanasati, not vipassana as such] more than Buddhism. Nonetheless I hope it serves as some testament to the value of Buddhist thought outside of meditation.

One day I hope more people take up the mantle of the Buddhist Conspiracy, the Bayesanga, and preach the good word of Bayesian Buddhism for all to hear. Until then, though, I'd like to follow in the spirit of fellow Bayesian Buddhist Luke Grecki, and describe some of my personal experiences with anapanasati meditation in the hopes that they'll convince you to check it out.

Nearly everything I've learned about anapanasati/vipassana comes from this excellent guide. It's easy to read and it actually explains the reasoning behind all of the things you're asked to do in vipassana. I heavily encourage you to give it a look. Meditation without instruction didn't lead me anywhere: I spent hours letting my mind get tossed about while I tried in vain to think of nothing. Trying to think of nothing is not a good idea. Vipassana is the practice of mindfulness, and it is recommended that you focus on your breath (focusing on breath is sort of a form of vipassana, and sort of its own thing; I haven't quite figured it out yet). I chose that as my anchor for meditation as recommended. Since reading the above linked guide on meditation, I've meditated a mere 4 times, for a total of 100 minutes. I'm a total novice! So don't confuse my experiences for the wisdom of a venerable teacher. But I think that maybe since you, too, will be a novice, hearing a novice's experiences might be useful. A mere 100 minutes of practice, and I've had many insights that have helped me think more clearly about mindfulness, compassion, self-improvement, the nature of feedback cycles and cascades, relationships between the body and cognition, and other diverse subjects.

The first meditation session was for 10 minutes, the second for 40 minutes, the third for 10 minutes, and the fourth for 40 minutes again. Below are descriptions of the two 40 minutes sessions. In the first, I experienced a state of jhana (the second jhana, to be precise; I'm about 70% confident), which was profoundly moving and awe-inspiring. In the the second, my mind was a little too chatty to reach a jhana, but I did accidentally have a few insights that I think are important for me to have realized.

The below are very personal experiences, and I don't suspect that they're typical. But I hope that describing my experiences will inspire you to consider mindfulness meditation, or to continue with mindfulness meditation, even if your experiences end up being very different from mine. You might find that some of the 'physiological effects' I list are egregious, but I decided to leave them in, 'cuz they just might be relevant. For instance, I find that, quite surprisingly, my level of mindfulness seems to directly correlate with how numb various parts of my body are! Also, listing what parts of me were in pain at various points might alert future practitioners to what sorts of pain might be expected from sitting still for longer than thirty minutes. The most interesting observations will probably be in the 'insights' sections.

40 minutes, Evening/night, September 17, 2010.

Setting: First laying down on a bed with a pillow over my eyes, then sitting up on the bed on a pillow.

Physiological effects:

Before jhana:
I lay down on my bed with a pillow over my eyes. I think this is interesting, because many texts I've read emphasize the importance of sitting up straight. I don't think it is necessary. That said, they do seem to know what they're talking about, and I'm very new to this, so perhaps being able to enter a jhana from a position of lying down was something of a fluke.
I started concentrating on my breath.
My breath alternated between deep and slow and a more natural breath. As time went on and I became more comfortable, my breath became less slow and more normal.
I experienced numb facial muscles and random eye muscle flickers. I felt trong sense of peace, compassion, and wellbeing.
The numbness and joy gave way to a full-out jhana experience after about 5 to 10 minutes of meditation.
During jhana:
Incredibly intense feeling of bliss, compassion, and piece. I involuntarily laughed at loud about five times. I think there must have been some kind of feedback loop going on here. I felt clearheaded.
Incredibly intense body high. My whole body was quivering, including especially my eyelids. It was a numbness-like feeling, though perhaps different in that if felt like quivering. It could be that my perception of the feeling had changed.
I sat up on a pillow.
Watching the inside of my eyelids was entirely grey, where most of the time there are neon patterns on a black background. This was rather odd and the most obvious evidence that something really weird was going on with my perception.
I tried to sit in a half-lotus position. This was mildly painful, though the pain wasn't bad, if you take my meaning. I kept at it for about two to five minutes, after which I reverted to a normal cross-legged position.
I had a strong compulsion to sing out 108 'Om mane padme hum's, which I did, followed by 108 more, counting on my fingers.
I then got up and played a few blitz chess games online, still feeling the very strong effects of the meditation. Surprisingly, in the 3 games I played I was a tad subpar. I sorta expected to play amazingly well, though I wasn't sad when it turned out I was wrong. This might be a sign that my feelings of clearheadedness were not entirely justified, but the results aren't very indicative either way. By the third game the effects had mostly worn off, but I still felt very peaceful, compassionate, self-accepting, and joyful. The flittering quivering numbness and energy had mostly worn off.

Insights on breath:

I could feel the temperature difference of the air as it was inhaled and exhaled.
When I breathed heavily, inhalation was very slightly painful.
(A few others that I've forgotten.)
(I had the above insights before entering jhana. I think they helped achieve jhana.)

General insights:

Previously I'd heard that meditation could lead to feelings of profound bliss, compassion, and even a sort of very strong physical body high. I'd mostly discounted such reports on the grounds that 1) I've done some drugs and didn't expect the effects to be as strong as e.g. cannabis, and 2) it didn't seem clear how just focusing on your breath could cause significant physiological changes of the sort necessary to have such strong effects. After experiencing jhana, I can say I was wrong. However, I still do not understand the neurochemical mechanisms behind my experience, besides postulating the magical hypothesis of 'cascades'.
More generally, I realized more fully that the Buddhists really do have a lot of very good and very credible thoughts on mindfulness and rationality. I'd known this for awhile just by studying Buddhist texts and teachings, but feeling vipassana meditation working so strongly and obviously really made it sink in that Buddhism is very worth studying attentively.
Cascades and feedback loops in the mind are very, very strong. By becoming more mindful and more accepting, I allowed myself to become even more mindful and accepting, until the feedback loop led me to an incredible altered state. This led me to really believe that the mind is very messy and prone to accidentally allowing causation between two parameters when it'd probably be better to allow just one to push on the other, like happiness causing laughter and not the other way around. Nonetheless, I can use the messiness of my mind to my advantage by thinking the right kinds of thoughts. I got a better sense of this when I meditated again a two weeks later.
I am naturally rather severely self-critical. Previously I'd considered this, if not a virtue, then it least a necessary evil and a good habit that I should keep: it keeps me from being excessively narcissistic, it reminds me of areas where I can improve, it keeps me from feeling too justified in a dispute, and it allows me to better understand faults others see in me. However, becoming so accepting of both my faults and others' during meditation led me to think that perhaps the disgust I feel for myself and others is a needless emotion, and that simply acknowledging areas of improvement without associating them with negative affect is a much better way to make myself a more awesome person and understand the plights of others. The whole time I'd thought that getting angry at myself was a necessary part of being self-critical, but after meditating I realized that anger isn't a necessary part of realizing faults, just like self-love isn't a necessary part of realizing strengths. Both are affect-laden thoughts where simple awareness will do better. I have a feeling that this insight generalized to a lot of other problems.
If the Buddhist concept of Enlightenment is anything like a constant state of jhana (and this is somewhat implied by accounts of Gautama Buddha's path), then I can definitely see why people would want to aim for it, and I can see how it could be a very real, very effective, and very profound state of mind. It doesn't seem to me as if one has to postulate anything spiritual to think of Enlightenment as an amazing state of being that we should all aim for as rationalists. The magnaminity, compassion, competence, acceptance, and feeling of awesomeness created by the jhanas should be cultivated and drawn upon whenever possible.
Because of this, it is very worth researching ways to 'cheat' and induce jhana states without having to undergo careful meditation. Neurofeedback, isochronic beats, and transcranial magnetic stimulation all seem like potential paths towards easy Enlightenment. (The jhanas seem to allow strong clarity of mind where drugs do not; but it is possible that being on drugs as much as possible might also be an interesting path. I'd rather not go down it yet.) 'Course, we might still have to just do it the hard way.

40 minutes, Midnight, October 4, 2010.

Setting: Seated on a pillow on blanket on roof of my house in Tucson.

Physiological effects:

My left leg (quadricep) was mildly sore throughout from running/sprinting two days before. At times in went mildly numb, though not painfully so. My left foot also went slightly numb at various points throughout the sitting.
My shoulders and facial muscles would tense moderately at various times near the beginning of the sitting and slightly near the end. This normally followed losing track of my breath. My breathing also got heavier and faster during these times. When I focused on my breath again, my shoulders and facial muscles dropped and relaxed, and my breath returned to normal rapidity/intensity.
After 10 minutes and at various points after, for roughly 15 seconds each, I could feel certain facial muscles go slightly numb, though not painfully so.
Roughly 15 to 20 minutes in (not sure), my left hand went somewhat numb for one to three minutes.
Roughly 20 minutes in, my left arm went very numb for roughly two minutes, though I didn't feel any pain. My arm felt 'tight'. The numbness went away rather rapidly, followed immediately by what felt like increased blood flow and thus warmth in the rest of my body.
Roughly 25 minutes in I felt mild pain in my lower left back. It mostly went away within a minute or two.
After the meditation was over (40 minutes) I stood up and stretched. I felt very peaceful and happy. At first I felt a tad dizzy but soon felt fine.

Insights on breath:

Breathing was faster and more intense when I stopped focusing on it and thought of other things. (Sometimes it was slower and more intense. I think intensity was the real key change.) When I refocused on my breath, it naturally became smoother and at a more normal pace.
Previously, I'd always thought that air went 'up' my nose when I breathed in. I suddenly realized that air actually entered my nose diagonally, and this whole time I'd thought I'd been breating 'up' because of confirmation bias. All of a sudden it was obvious that I was breathing in diagonally. But moments later I realized I was actually mostly breathing 'up', and only a little diagonally: my new theory had also been subject to confirmation bias! So I settled on thinking that I did indeed breathe in 'up', but also a little diagonally.
I noticed that there are two types of breath. The first is very airy and goes through the top of your nose; it is the one that comes most naturally to me and I imagine most others. The second is throaty and maybe a little stuffy, and it seems as if less air is passing through. I tend to breath the second way a little more naturally when I try to tuck my chin in against my neck; but I can still breathe in the more airy way as well when I do this, so your mileage may very.

General insights:

Patterns of muscle contractions, patterns of thoughts, and patterns of breathing are all interrelated and can cause feedback loops. Being mindful of my thoughts helps me relax my muscles; relaxing my muscles helps my breathing be more natural; having a natural breath allows me to be more mindful; and so forth. This is good if I am diligent, but bad if I am not; I tend to gravitate towards whatever state I'm in. It takes effort to move between states of mind, but it seems that entropy and novel stimuli tend to push me toward patterns of thought that are irritant. I believe it is eminently possible that I could cultivate the disposition such that entropy and novel stimuli tend to push me towards mindfulness, compassion, and awesomeness.
Confirmation bias is there even at the very low instinctual level of breathing. As soon as you come up with a theory, even direct sensory experience doesn't always change it when it's wrong.
Psychic irritants, as they are sometimes called, are constantly mucking around in your brain, causing low level stress, anxiety, guilt, and general discomfort. It seems likely that this was the natural state of the brain for thousands upon thousands of years. I find it very odd that with an hour of focused mindfulness -- all you do is pay wordless attention to your breath! -- you can make a naturally fuzzy and pained human mind into a pure and blissful meditative engine. The difference is striking. It is hard for me to imagine why living in the moment has such a profound effect on cognition.

I'd love for others to share their meditative experiences, or offer feedback for this post. I'm not sure if it should become a top-level post or not. But hopefully LW starts moving in a more Buddhist and effectiveness-oriented direction.

Taken out of original essay for being egregious: I've talked previously of how there seems to be a libertarian/technophile/futurist set of rationalists and a liberal/Buddhist/scientist set of rationalists, and each eyes the other's origin with a cocked eyebrow. Well, I'm from the LBS origin group, and I still think it's the better of the two. We're better at cooperating and we're more okay with praise. But we also seem to lack an unfortunate meme that I've seen in the LTF crowd: uncharitable misinterpretation of what the best ideas of Buddhism really are, even if not every practitioner or teacher is at the standard of the best philosophers of that tradition. Hofstadter made Zen cool, but other easier and probably more useful forms of Buddhism have been left unplundered. I think it has more to do with an instinctual negative reaction towards anything that seems vaguely spiritual or religious. And don't get me wrong, there's a lot of religion and spirituality in Buddhist countries, especially of the Mahayana sort. But the best texts in the Theravada tradition have very good, very deep, and very insightful epistemology and rationality in them, of the kind that wasn't to be found anywhere else in the world for hundreds upon hundreds more years, if at all.