Exposure. The word contains a certain notion of risk and a certain amount of irremediable commitment. Once the exposure has been set on a camera and the subsequent photograph taken there is no way to physically change this exposure. You are done, set and have to live with the consequences. Hopefully the exposure is "right on" and the photograph holds details everywhere.
But what if the photograph does not hold detail everywhere? What if your calculated exposure isn't absolutely, 100% correct? What if your calculations were off, what if you made a mistake! Then what? Is there a way out, a way back to a great image despite the fact you could have done better in the field exposure-wise?
More importantly, in order to avoid post-photographic-exposure stress, how can one calculate, with absolute accuracy, the perfect exposure, each and every time, in the field?
So many questions and, I am afraid, so few answers. Exposure isn't the most widely discussed subject in photography. Articles on it are rare and many photographers skirt the issue. Not to be surprised. Exposure isn't rocket science. There is a certain amount of leeway, adjustment and artistic interpretation involved.
Let's take a closer look at an aspect of photography that is rarely closely looked at. And let's try to relax and have fun in the process. After all, poorly exposed photographs are not deadly. They are just annoying and disappointing. The good news is, we can expose other photographs, and hopefully do better after reading this article.
Exposing each photograph properly is of utmost importance in order to get the best print possible from this one image. This is because, whether you use film or digital capture, your original film or digital image contains all the information you will ever get of the scene you photographed.
Ansel Adams was originally trained as a professional pianist. He often made comparisons between photography and music. In regard to exposure, he compared the negative to the score and the print to the performance. Today, we can extend this score to transparencies and Raw files. The performance, the print, has not changed. Neither has the overall idea behind Adams' comparison. The better the score, the better the performance will be.
In order to get the best score possible, you want as much information from the scene you photographed recorded on film, or on the Raw file, as possible. Ideally, you want details everywhere in the scene. Since the two most important areas of this scene are the shadows and the highlights areas, you want to make sure that you have details in both.
The problem is that getting details in both shadow and highlight areas is difficult. This difficulty stems from the contrast range that can be captured by the specific film, or digital sensor, you are using. This contrast range, which is measured in f-stops, varies from film to film and from sensor to sensor. From this statement we can already deduce one of the most important tenets of exposure: in order to expose your film, or digital file, properly, you must know its contrast range.
Before proceeding any further it is necessary to explain some of the terms commonly used when discussing exposure.
A grey card is a card painted Medium grey. These cards are available from Kodak. Grey cards have a precisely 18% reflectivity meaning they reflect 18% of the light that falls on them.
This is important to know since light meters all over the world, irregardless of the manufacturer, are calibrated to give an 18% grey reading. This means that reading a uniform surface and photographing it without adjusting the exposure given by the meter will turn any surface into an 18% reflectivity grey surface. Whether you photograph a black or a white surface it will become 18% grey in the photograph.
In other words, a light meter is simply a machine designed to render the world 18% grey and calculate each exposure accordingly. To get a lighter or darker photograph based on the reading from a light meter you have to adjust the exposure accordingly and either overexpose or underexpose. If you trust what your meter tells you, you will get an 18% grey photograph of every surface you photograph!
If you photograph in black and white you will actually get a medium, 18% shade of grey. If you photograph in color you will get a color equivalent to an 18% reflectivity. It will not be grey, but it will have the same reflectivity as an 18% grey card.
Numerous cameras today calculate the exposure and then compare the scene in front of them to a database of thousands of photographs pre-recorded in the camera's digital memory. Such cameras do not blindly follow what their light meter tells them. Instead, they compare the exposure provided by the meter to correct exposures for similar scenes pre-calculated and recorded in the camera's build-in database. They then adjust the final exposure accordingly. These cameras will deliver a much higher percentage of correct exposures. However, in extreme exposure situations, you may still have to make exposure corrections yourself.
Overexposure gives you a brighter image.
Underexposure is the process of exposing a photograph less than what your light meter is telling you. Again, you are not following what your meter is telling you instead. This time you are making a call to expose less than what the meter is saying.
Underexposure gives you a darker image.
This photograph could have easily been overexposed had I used an average meter reading of the scene. To hold details in the moon and clouds I had to considerably underexpose the arch. There is actually some details on the arch but I decided to print it pure black, except for the triangle of directly-lit rock at the top left corner, to emphasize the stark quality of this scene.
– Find a uniformly lit and colored textured surface. You can use a wall that has a texture to it and is in open shade. Choose a wall that isn't painted in a dark shade. Ideally, you want this wall medium grey or as close to this as possible.
– Get as close to the chosen surface as possible so that you see only this surface in your viewfinder. You don't want anything else in the photograph.
– Take a photograph at the reading given by your camera. That is set the exposure that your camera, or your light meter if you use a hand-held meter, says is "correct".
-Take a photograph underexposed by one stop, two stops, three stops, four stops and five stops. If you are not familiar with how to do this look in your camera manual .
– Take a photograph overexposed by one stop, two stops, three stops, four stops and five stops. Again, if you are not familiar about how to do this consult your camera manual.
– If you use film get your film developed normally by your lab. Do not ask for any compensation of any kind during film development. Do not scan the film.
– If you use digital capture import your raw files in your raw converter and do not adjust the exposure compensation whatsoever. Make no adjustments in the raw converter. Simply convert the images and open them in your image editing software.
– Look at your film (transparencies or negatives) on a light table and look at your raw-converted files on a closely-calibrated monitor. You should have 11 images in front of you. One exposed "correctly", 5 overexposed and 5 underexposed.
– What you are looking for are images that are either so light or so dark that they do not show any detail – any texture – either on the film or on the monitor.
– Once you have determined the images that have detail in them count them. Count only those that are underexposed or overexposed. Do not count the one which shows the exposure given by your meter.
– If you have 6 images showing details (not counting the "correctly exposed" image) it means your film or digital sensor is able to record a 6-stops range. If you have 5 images with details you can record a 5 stops range and so on. Your results will vary from film to film and sensor to sensor as I said previously.
For example, I use almost exclusively Fuji Provia 100F. I know, from conducting the test I just described, that I can safely record a range of 5 stops: 3 under and 2 over. Your own tests will show the exact contrast range your film or digital camera can record.
– First, measure, either with your camera or a hand-held meter, the darkest and lightest area of the scene in front of you. Using a spot meter will make this process much easier.
– Second calculate how many stops are between the darkest and lightest area of the scene. For example, if the shadows measure at f/2.8 at 1/60th of a second, and the highlights at f/16 at 1/60th of a second, you have a 5 stops contrast range in the scene in front of you. Keeping either the f-stop or the film speed constant will make this calculation a lot easier. In this example I kept the speed constant, using 1/60th of a second, and changed the f-stops.
Each of the settings on either scale are one stop apart. Going up or down from one setting to the next means either overexposing or underexposing by one stop.
On the f-stop scale changing to a higher number means underexposing and changing to a smaller number means overexposing (provided you do not change the shutter speed).
On the shutter speed scale a smaller number means overexposing and a larger number means underexposing (provided you do not change the f-stop).
This photograph shows an extreme range of contrast from the shaded part of the ruin to the Mountains and the sky. To get details in the ruin I had to almost overexpose the mountains. I could not have calculated the exposure correctly had I not known precisely the contrast range of the film I was using.
In this respect the histogram is best thing that happened to photographers since the invention of the light meter. A histogram does not replace a light meter. We still need a meter to calculate the exposure. But the histogram gives us what we previously could only see in the darkroom, or after scanning, and that is a map of where each area falls.
Histograms come in two flavors: monochrome and tri-color. Monochrome histograms show where the brightness values are located on the photograph. Tri-color histograms show how the brightness values of the three primary colors – red, green and blue – are recorded. Tri-color histograms allow the photographer one additional level of control since we now can tell what the blue, red and green channels of a digital image will look like and thus monitor each of the three channels prior to taking the photograph. Since the light level of red, green and blue objects is often different this control is very important.
For years photographers have used a simple remedy to the difficulty of determining the perfect exposure: bracketing. Bracketing refers to the widely-used practice of taking several photographs of the same scene, each with a slightly different exposure, in the near-certain hope that one of these exposures will turn out to be right on.
Bracketing shows first that empiricism is still a valid approach. It also shows that despite our constant improvements in equipment, technique and methodology a certain amount of guesswork remains.
Bracketing also offers a universal remedy to exposure difficulties. All you have to do is "get in the ballpark" to use one of the many baseball metaphors found in the American language.
Such a scene can be a real challenge to meter properly. I actually took only one exposure and got it right because I knew that I wanted the shaded snow area medium grey. I therefore took a spot meter reading of the shaded snow area at the top left of the photograph. As a result the cliff in the top right of the image is darker and the snow covered trees in direct sunlight nearly white, just the way I wanted it. This scene is a perfect candidate for bracketing if you are not 100% sure what the proper exposure may be. It is better to be safe than sorry.
Personally I bracket with both film and digital capture. With film the cost of the extra film is worth the peace of mind I get. With digital cost is no longer an issue and bracketing should become second nature in situations where finding the correct exposure is either challenging or overly time-consuming.
In practice I normally bracket one stop over and one stop under my calculated exposure. I may bracket in full stops or in half stops. For example, if my calculated exposure is f/22 at 1/4th of a second, I will make an exposure at this setting, one exposure at f/22 and 1/2th and one exposure at f/22 and 1/8h. This way I have a two stops bracketing range which, in nearly all situations, assures me that I will have at least one properly exposed image.
With digital capture it is recommended that you "expose to the right" as Michael Reichmann says. This approach stems from digital cameras being able to record more information in the high values than in the low values. Exposing to the right means overexposing your image. How much? As much as it takes to place the highest value all the way to the right of the histogram without going out of this histogram. It also means bringing the exposure level down in the raw conversion software to bring the image back to a normal exposure. The gain is better image quality, especially in the midtones and shadows, where additional information was recorded through overexposure.
When exposing to the right it is important to keep in mind that each image is composed of three channels (red, green and blue) and that brightness levels are different in each individual channel. If your camera only has a single channel histogram you need to be very careful when exposing to the right since you may accidentally overexpose one or two of the individual channels. Ideally you want to use a camera which has a three-channels histogram, since only a three-channels histogram will show you what is happening in each of the red, green and blue channels. Unfortunately, as of April 2004, only a few cameras offer three channel histograms.
When I expose to the right with a single channel histogram camera I also take a photograph underexposed by one stop and sometimes one underexposed by two stops. Why? First, because this insures me that I will not accidentally clip one of the individual channels and that I will have an image in which all the information is recorded. Second, because there is no extra cost, third because it takes hardly any additional time to do so and fourth because this allows me to see a properly exposed image on the camera's LCD screen instead of an overexposed image. Seeing this image is important to me in terms of estimating how each color and brightness value will look on the final print.
Another challenging scene exposure-wise. First, the light level was changing very fast as the sun was rising requiring constant monitoring with my light meter. Second, the contrast was very high due to the presence of sunlit snow and relatively deep shadows. Third, I had to make a decision about how dark I wanted to shaded snow. I did not have time to bracket since the clouds disappeared shortly after I made this exposure. The scene holds details throughout fortunately. Shaded snow turns blue on film because shadows are lit mostly by blue light from the open sky.
Very bright objects will nearly always "throw your meter off" and result in underexposed photographs. The reason being that your meter wants to turn all objects in the scene into an 18%, medium grey value. If the object is white, and you want it white on film, you will have to overexpose in order to counter balance the meter's desire to make all objects grey.
A typical situation in landscape photography is snow. Snow is by definition white, highly reflective and the cause of many underexposed photographs. With most light meters, either hand held or built in, it is necessary to overexpose snow by one to two stops. Of course, you have to take into account whether the snow is in the sunlight or in the shade. Snow in the shade does not need to be overexposed very much unless you want to show it brighter than it really is.
Shiny and reflective surfaces are another potential subject which may result in underexposed photographs. The sun reflecting off a pool of water, or a metallic object lit by direct sunlight, or even a very bright canyon wall for example also need to be overexposed from what the light meter is saying by one stop or more. Again, the meter is reading the brightness of these objects correctly but wants to render them 18% grey while they are actually much closer to pure white or actually pure white.
Very dark objects will also confuse your meter, this time causing it to overexpose in order to turn into medium grey what is actually a black or nearly black object. A typical situation is a backlit scene at sunrise with the outline of cliffs, mountains, monuments or trees. It is best to let these elements be pure black on the final photograph because getting details in both the monuments and the sky is very difficult if at all feasible. However, your meter will want to average the scene to 18% grey resulting in greyish blacks which are less than pleasing in my estimate. In this situation you need to underexpose by one to two stops in order to get the proper exposure.
Similarly, other dark objects, such as a very dark tree, coal, black sand, etc. will end up being underexposed if you do not adjust -ie underexpose- the reading your meter gives you.
I cannot stress enough the importance of constantly monitoring the light level of a scene at critical times such as sunrise, sunset and active weather. Constantly increasing light levels, at sunrise for example, and constantly diminishing light levels, at sunset for example, are very difficult to monitor with the naked eye. The presence of clouds, alternatively hiding and revealing the sun, offer the same challenge. When the light is changing fast our eyes adapt accordingly and we are not always able to see the change in brightness levels precisely.
In such situations it is necessary to constantly monitor the light level. This is particularly true if you use a manual camera and a hand-held light meter, my situation when I work with 4×5. However, this also applies to automatic cameras. What was a dark object at dawn may have become a very bright object after sunrise and this bright object may be confusing your camera's built in meter. Same thing at dusk where the camera may be overexposing what it, after all, a dark landscape.
What happens if you do not succeed in exposing a photograph properly in the field or if the range of values is to high to record details both in the highlights and in the shadows?
Well, first, rest assured you are not the only one in this situation. This is a problem that has plagued photographers since the very beginning of photography. Fortunately, there are a number of solutions available. Let's take a look at them.
A split neutral density filter is a filter which is divided in two parts: a clear area at the bottom and a neutral grey area at the top. The clear area of the filter is larger than the neutral density part, the ratio being usually 2 to 1 or 3 to 1. The transition between the clear and neutral grey area is gradual.
Split neutral density filters are available in either one stop, two stops, three stops or more. I personally recommend either a one or two stops split neutral density filter. More than that and the effect becomes obvious and unnatural. I also recommend a soft transition between the graduated and the clear areas of the filter so that this transition is not visible on film.
Using a split neutral density filter is easy. Just place it in a holder in front of the lens and move it up and down until the transition is where you want it. A typical use for this filter is to compensate for the brightness of the sky at sunrise or sunset. In this case you place the neutral density area over the sky and the transition along the horizon. Other uses are possible, the limit being your own imagination.
These 3 photographs show a perfect example of digital sandwiching. I took a single photograph of this scene in Paris. When I converted the image from Raw I realized I could not get details in both highlights and shadows on a single converted image. I therefore decided to do two raw conversions, one for highlights and one for shadows. I then combined these two files in Photoshop. This technique takes only a few minutes and works extremely well. Furthermore, it is virtually invisible since you can adjust the brush size and the areas removed at will. The third photograph, on the far right, shows the final result.
There isn't much more you can do in the field besides optimizing the exposure and compensating for a large contrast range with split neutral density filters. You can certainly look for scenes which contrast range does not exceed that of your film or digital sensor, but doing so will severely limit what you can photograph.
If you capture a scene which contrast range exceed that of your film or digital camera the next step is to compensate for this excessive contrast range in the darkroom.
One very effective technique for digital images is to make a separate raw conversion for highlights and for shadows. This technique is perfect when you have all the information you want in the raw file but cannot successfully display all this information in a single raw conversion.
To apply this technique start first with the highlights and adjust the exposure control in the raw conversion software so that highlight areas look just the way you want it on your monitor. Typically you will have to lower the exposure of your photograph. Convert this image. Then, going back to the original raw file, make adjustments to the shadows so they too look just the way you want them. Typically you will have to increase exposure in the shadows. Make a second conversion for the same image.
Open both images in your image processing software and drag one of the images on top of the other in a separate channel making sure they are in perfect registration. Add a layer mask to this channel and, using the eraser brush and a soft, wide brush, remove the parts of the top image which you do not want to see. For example, if you placed the image with the proper highlights on top erase the shadow part of this image to reveal the properly converted shadows on the layer underneath it.
A similar technique can be used with film this time by scanning the same transparency or negative two times, one time with the scan settings optimized for the highlights and the second time with the scan settings optimized for the shadows. Import both images in your image processing software and proceed as described above. Make sure you do not move your film in the scanner between the two scans otherwise your two image will not register perfectly.
You may ask "why not scan or convert from raw only once and then use image adjustments — such as curves, dodging and burning or the new highlight-shadow filter in Photoshop CS?" You certainly can use this approach for many images which require only a minor amount of contrast correction. But for images which require significant contrast correction you will get better quality results using the technique I just described. This is because you will work with images that have the information you need from the start instead of images that have little or no data either in the highlight or shadow areas.
In some extreme situations the above technique will not work because nothing was recorded in the shadow or highlight areas on a single photograph, either film or digital. However, you did take several photographs and you have one exposed for the highlights and one exposed for the shadows. Shadow details are present on the first photograph and highlight details are present in the second photograph.
In this instance your only resource is to sandwich the two images and use the areas with details from each of them. Doing this successfully takes more time than the previous technique. Furthermore, sandwiching works better with digital since digital images are always in perfect registration if the camera was placed on a tripod, if the camera wasn't moved between exposures and if the same lens or zoom setting were used for both images. With film things are more problematic since films are almost never exactly in the same place in the camera in successive frames, no matter what format you use.
At any rate open both images and bring them together in one file on two different layers. Then use the same approach I described previously to erase the areas you do not want to keep and show the areas you want to keep. Use a very soft brush and a relatively wide transition zone. You may also want to play with the opacity setting of the top layer and choose a setting lower than 100% to show part of both images in areas where the transition from one to the other may be noticeable.
As you practice these techniques or ponder if you should use them keep in mind that because you can combine images doesn't mean you should. Nobody says you must have details in both highlight and shadows in each and every image you create. In fact, with some photographs having details in both highlights and shadows does not result in the strongest image possible. Furthermore, in some instances, having details everywhere seems unnatural or fake. Eventually, the final call is yours and you need to make this call not on whether or not you can get details everywhere in the image but, rather, on whether or not you want to show details everywhere in the image.
Given several originals and enough time, patience and skill, it is most likely that you will succeed in most instances in producing an image with details throughout the scene. However, once this is done, you may decide it is not what you want. We are talking here more about your personal style than your technical skills, a subject we will revisit at length later on in this series of articles.
This photograph combines two different scans from two different transparencies. Aware of the extreme contrast range of this scene I decided to make 2 exposures to capture details in the sky and in the cliffs. I combined the two scans in Photoshop with layers using a layer mask to erase areas of the image I did not want to keep. The wall to the right has slight details in it but, just like in Tear Drop Arch Moonrise, I prefer it pure black than dark grey.
A hand held light meter, especially a spot meter, is a great asset if you want to meter a scene carefully. Such a meter can be useful whether you use a camera with a built in meter or not. I use a hand held spotmeter, actually two of them at this time, whenever I photograph a scene which is challenging exposure-wise. Below are several examples of hand held light meters and spotmeters.
I have used this meter since 1980. The spot attachment gives it a measurement angle of 7.5 or 14 degrees. I use the 7.5 degree angle setting nearly all the time. The needle gives you the measurement value by zeroing it in the center of the top dial. Measuring a highlight area, placing it on "2" on the right side of the top dial, and then measuring shadow areas immediately tells you if the film can handle the contrast range in the scene, +2 stops being what most transparency films can handle without burning the highlights. The circular dial below gives you all the f-stop and speed combinations for the measurement as well as zone system placement.
I consider this meter to be the digital version of the Luna Pro. It has the advantage of offering a one degree and 5 degrees measuring angle, flash metering, ambient light metering and more. You can measure several areas of the scene and the meter will calculate the average exposure value. It is very accurate but, in my estimate, it does not offer the immediate visual comparative feedback that the analog meter offers. Digital light meters with similar functions are offered by several other manufacturers such as Seikonic, Minolta, etc.
Introduced over 20 years ago and recently discontinued by Pentax, this meter has become the archetype for spotmeters as well as a favorite of many photographers. Its success is due to its reliability, accuracy, simplicity and ease of use. This meter has a fixed one degree angle of view and gives an EV value in the viewfinder for the measured area. This EV value is dialed on the front settings area (see photo at right above) and the corresponding f-stop/speed combination are immediately available together with zone system placement if one adds a zone system strip to the meter as was done on the photograph above. Providing accurate measurements is only the first step in meter functionality. The second, and very important step is to allow easy contrast evaluation. A good meter allows one to easily calculate where the highlights and shadows will fall.
As I discussed earlier in this article digital cameras not only have a built in light meter they also have a built in histogram display. To my knowledge this function is not available, as of April 2004, on any handheld meter.
I do understand that a histogram is calculated from all the values present in a scene and not from meter readings of specific areas of the scene. In other words, a histogram is created from a photograph, not from meter readings. To get a histogram one needs a photograph and to get a photograph one needs a lens. So far light meters have not been designed to capture photographs.
Certainly you can say that all I need to do to get a histogram is use a low cost digital camera to meter the scene. I do, with the Canon 300D, which I use regularly as a light meter that can also take photographs. But I also carry a light meter because I need the spotmeter function and the comparative metering capalilities provided by a hand held meter such as the different models described above.
A number of PDA's, such as the Sony Clie NZ 90, are equipped with a photographic lens and can capture low-resolution photographs. If such a PDA was able to display a histogram for the photographs they take, and was fitted with a one degree spot meter, we would have a measuring instrument the like of which we have not seen before.
Above is a photograph of the lens on the Sony Clie NZ90.
Merging a digital spotmeter with a PDA fitted with a miniature zoom lens would produce a spotmeter that can take a photograph, zoom in and out to give different compositions and croppings, meter the entire scene just like a digital camera, take spot meter readings of specific areas of this scene and display the resulting histogram on a PDA-like screen. The histogram displayed on this PDA-Spotmeter would be a tri-color histogram to provide reliable information about all three channels of the photograph. The display area would be about 2"x3", much larger than the LCD screen of any currently-available digital camera. Spotmeter readings of specific areas in the scene would be indicated on the photograph and would provide EV and f-stop/speed combination for each area measured. Overexposed and underexposed areas would blink as they do on most digital camera.
The processing and storage capabilities of a PDA would also make it possible to store photographs and their related histograms and spot measurements for later reference. This data could be opened in a separate "Histogram and Light Measurements" software, independent of the browsers, RAW converters or image processing software we currently use. This program would also be able to output a print of the histogram and spot meter readings. We could even combine this with a GPS system and record the GPS coordinate of each photograph at the same time as their exposure information.
Here is a composite image of what such a meter may look like. It would be the size of a PDA, hence much smaller than a digital camera such as the Canon 300D and not much bigger than currently available spot meters.
None of the articles in this series would be complete without specific Photographic Skill Exercises targeted to help you develop your skills in the area covered by each article. I know that many readers conduct these exercises carefully and I therefore designed a set of exercises specifically for this article.
A – Determine the overall contrast range of a specific scene.
This is a simple exercise in which you take a meter reading of the brightest and the darkest area of the scene.
The easiest way to do this is to use a spotmeter. However, if you do not have one you can conduct this exercise using the light meter built in your camera or a hand-held light meter.
If you use your camera you will have to get close to the area you are measuring to make sure you only measure this area. You can also use a long lens to narrow the camera's field of view to just this area. Getting close will work for nearby objects but not for distant ones or for the sky. For these areas you will have to use a long lens. Of course, if your camera has a built in spotmeter function make sure you use it.
Once you have measured the highlight and shadow areas count how many stops there are between these two areas. This is the contrast range of the scene in front of you.
B – Measure the important values of a scene and write each exposure value on a sketch of this scene.
When I started working with my first view camera, in 1983, I made sketches of each scene I photographed using tracing paper placed against the ground glass of the view camera. I made a drawing of each photograph by tracing the outlines of my composition. Then, using a Pentax 1 degree spotmeter, I took readings of all the important value areas in the scene and wrote these values on each drawing. Comparing each print to the corresponding sketch taught me volumes about how light is captured on film and how light is translated into light and dark values onto the final print.
This exercise aims at providing you with the same knowledge. If you use a 35mm camera, or a medium format camera, you will have to do a hand-held sketch as you won't be able to sketch directly on the ground glass. However, the quality of your sketch is not important. What is important is that it shows the rough outlines of the objects in the scene and that you write the exposure information for each of area.
Below you can see one of the sketches I made in 1983 next to the corresponding photograph. The photograph was taken on Polaroid 4×5 Black and White film type 52. The use of Polaroid film enabled me to visually compare the scene in front of me to the photograph of this scene in the field and see how my meter readings were translated into shades of black, white and grey on the print.
The outlines of the shapes in the photograph were traced on the ground glass of my Arca Swiss 4×5 monorail view camera. The exposure values were recorded as EV values because these are the values provided by the Pentax 1 degree spotmeter I was using at the time. EV values are not directly related to a specific speed and f.stop setting. Rather, they are absolute values which express a specific brightness. To use them however, you need a light meter which can give you the f.stop and shutter speed settings for a specific film speed. EV values are used by some camera manufacturers. For example they are provided by Hasselblad light meters and present on each Hasselblad lens.
You do not have to use EV values. You can instead write the f-stop and shutter speed combination on your drawing. Just make sure you keep either the f-stop or the shutter speed constant. Doing so will make it much easier for you to calculate the overall contrast range.
This is one of the sketches I made in 1983 by using tracing paper pressed onto the ground glass of my Arca Swiss 4×5 monorail view camera. Next to it is the correspoding photograph. I drew the outlines of the main shapes in the photograph, took meter readings of the scene with my Pentax 1 degree spotmeter and wrote the corresponding EV values on the drawing. EV values are practical because only one number is needed to find the corresponding f.stop and speed ratio. Plus, they take less space on the drawing. I did similar drawings for over 100 4×5 photographs during my 1983 six-month journey throughout the southwest.
– This is a very useful and informational exercise which will enable you to know exactly the contrast range of your film or digital sensor.
As with many landscape photograph which show both skies and shadow areas, contrast is a problem in this image. The original transparency hold details everywhere but several contrast masks were needed in order to convey a sense of light and brilliance on the final print. I used multiple curve layers in Photoshop to both open the shadows on the hoodoos and reduce the brightness of the sky behind them. The final print conveys how I felt when I photographed this scene.
The goal of learning how to expose is not only being able to determine the "correct" exposure. Most cameras do this automatically and in many instances we can trust what they tell us. Plus, with the histogram function we can visualize how the final image will be exposed. Finally the LCD monitor of digital cameras show us the photograph we just captured.
The goal of learning more about exposure is to gain control over how our photographs are exposed. Our goal is not only to find the proper exposure but rather to find the exposure which will give us the effect we are looking for.
For this we must learn first to calculate the range of contrast present in the scene in front of us. So far, as of April 2004, no camera does this automatically. We also need to determine if objects will be rendered the way we want them to be rendered. As we have seen, very bright and very dark objects, or scenes, will often result in incorrect exposures. Since these are ultimately artistic calls we cannot depend on our cameras alone. Cameras are devices that, at best, can only capture the world the way they were programmed to capture it.
A properly exposed photograph is one we will eventually consider for printing. In this sense such a photograph is what is often referred to as a keeper. But being properly exposed is not enough to make a photograph a keeper. Other things come into play when making this selection. What are those other things? This is what we will look at in my next article in this series: How to decide which photographs are "keepers" and which ones are not.
As with the previous installments, and to use a little bit of French, this series isa suivre. . .


Since the draft statement refers to an "attached presentation", I would prefer to see the presentation (as formatted for distribution) prior to casting a vote.
Paul has delegated the conduct of the EC electronic ballot on the following motion to me. To ensure this liaison letter is sent as soon as practical I'm announcing the possibility of an 'early close' to this ballot (see below).


text:
\section{Introduction}

The method of Maximum (relative) Entropy (ME) \cite%
{ShoreJohnson80,Skilling88,Caticha03} is designed for updating probabilities
when new information is given in the form of a constraint on the family of
allowed posteriors. This is in contrast with the older MaxEnt method \cite%
{Jaynes57} which was designed to assign rather than update probabilities.
The objective of this paper is to strengthen the ME method in two ways.

In \cite{Caticha03} the axioms that define the ME method have been distilled
down to three. In this work the justification of the method is improved by
considerably weakening the axiom that deals with independent subsystems. We
adopt a consistency axiom similar to that proposed by Shore and Johnson \cite%
{ShoreJohnson80}: When two systems are independent it should not matter
whether the inference procedure treats them separately or jointly. The merit
of such a consistency axiom is that it is very compelling. Nevertheless, the
mathematical implementation of the axiom has been criticized by Karbelkar
\cite{Karbelkar86} and by Uffink \cite{Uffink95}. In their view it fails to
single out the usual logarithmic entropy as the unique tool for updating. It
merely restricts the form of the entropy to a one-dimensional continuum
labeled by a parameter $\eta $. The resulting $\eta $-entropies are
equivalent to those proposed by Renyi \cite{Renyi61} and by Tsallis \cite%
{Tsallis88} in the sense that they update probabilities in the same way.

The main result of this paper is to go beyond the insights of Karlbelkar and
Uffink, and show that the consistency axiom selects a unique, universal
value for the parameter $\eta $ and this value ($\eta =0$) corresponds to
the usual logarithmic entropy. The advantage of our approach is that it
shows precisely how it is that $\eta $-entropies with $\eta \neq 0$ are
ruled out as tools for updating.

Our second objective is mostly pedagogical. The preeminent updating method
is based on Bayes' rule and we want to discuss its relation with the ME
method. We start by drawing a distinction between Bayes' \emph{theorem},
which is a straightforward consequence of the product rule for
probabilities, and Bayes' \emph{rule}, which is the actual updating rule. We
show that Bayes' rule can be derived as a special case of the ME method, a
result that was first obtained by Williams \cite{Williams80,Diaconis82} long
before the logical status of the ME method had been sufficiently clarified.
The virtue of our derivation, which hinges on translating information in the
form of data into constraints that can be processed using ME, is that it is
particularly clear. It throws light on Bayes' rule and demonstrates its
complete compatibility with ME updating. A slight generalization of the same
ideas shows that Jeffrey's updating rule is also a special case of the ME
method.

\section{Entropy as a tool for updating probabilities}

Our objective is to devise a general method to update from a prior
distribution $q(x)$ to a posterior distribution $p(x)$ when new information
becomes available. By information, in its most general form, we mean a set
of constraints on the family of acceptable posterior distributions. \emph{%
Information is whatever constrains our beliefs}.

To carry out the update we proceed by ranking the allowed probability
distributions according to increasing \emph{preference}. This immediately
raises two questions: (a) how is the ranking implemented and (b) what makes
one distribution preferable over another? The answer to (a) is that any
useful ranking scheme must be transitive (if $P_{1}$ is better than $P_{2}$,
and $P_{2}$ is better than $P_{3}$, then $P_{1}$ is better than $P_{3}$),
and therefore it can be implemented by assigning a real number $S[P]$ to
each $P$ in such a way that if $P_{1}$ is preferred over $P_{2}$, then $%
S[P_{1}]>S[P_{2}]$. The preferred $P$ is that which maximizes the
\textquotedblleft entropy\textquotedblright\ $S[P]$. This explains why
entropies are real numbers and why they are meant to be maximized.

Question (b), the criterion for preference, is implicitly answered once the
functional form of the entropy $S[P]$ that defines the ranking scheme is
chosen. The basic strategy is inductive. We follow Skilling's method of
induction \cite{Skilling88}: (1) If an entropy $S[P]$ of universal
applicability exists, it must apply to special examples. (2) If in a certain
example the best distribution is known, then this knowledge constrains the
form of $S[P]$. Finally, (3) if enough examples are known, then $S[P]$ will
be completely determined. (Of course, the known examples might turn out to
be incompatible with each other, in which case there is no universal $S[P]$
that accommodates them all.)

It is perhaps worth emphasizing that in this approach entropy is a tool for
reasoning which requires no interpretation in terms of heat, multiplicities,
disorder, uncertainty, or amount of information. \emph{Entropy needs no
interpretation}. We do not need to know what it means, we only need to know
how to use it.

The known special examples, which are called the \textquotedblleft
axioms\textquotedblright\ of ME, reflect the conviction that what was
learned in the past is important and should not be easily ignored. The
chosen posterior distribution should coincide with the prior as closely as
possible and one should only update those aspects of one's beliefs for which
corrective new evidence has been supplied. The first two axioms are listed
below. (The motivation and detailed proofs are found in \cite{Caticha03}.)

\noindent \textbf{Axiom 1: Locality}. \emph{Local information has local
effects.}

\noindent When the new information does not refer to a domain $D$ of the
variable $x$ the conditional probabilities $p(x|D)$ need not be revised. The
consequence of the axiom is that non-overlapping domains of $x$ contribute
additively to the entropy: $S[P]=\int dx\,F(P(x),x)$ where $F$ is some
unknown function.

\noindent \textbf{Axiom 2: Coordinate invariance.} \emph{The ranking should
not depend on the system of coordinates. }

\noindent The coordinates that label the points $x$ are arbitrary; they
carry no information. The consequence of this axiom is that $S[P]=\int
dx\,m(x)\Phi (P(x)/m(x))$ involves coordinate invariants such as $dx\,m(x)$
and $P(x)/m(x)$, where the functions $m(x)$ (which is a density) and $\Phi $
are, at this point, still undetermined.

Next we make a second use of the locality axiom and allow domain $D$ to
extend over the whole space. Axiom 1 then asserts that \emph{when there is
no new information there is no reason to change one's mind. }When there are
no constraints the selected posterior distribution should coincide with the
prior distribution. This eliminates the arbitrariness in the density $m(x)$:
up to normalization $m(x)$ is the prior distribution, $m(x)\propto q(x)$.

In \cite{Caticha03} the remaining unknown function $\Phi $ was determined
using the following axiom:

\noindent \textbf{Old Axiom 3:\ Subsystem independence}. \emph{When a system
is composed of subsystems that are \textbf{believed} to be independent it
should not matter whether the inference procedure treats them separately or
jointly. }

\noindent Let us be very explicit about what this axiom means. Consider a
system composed of two subsystems which our prior evidence has led us to
believe are independent. This belief is reflected in the prior distribution:
if the subsystem priors are $q_{1}(x_{1})$ and $q_{2}(x_{2})$, then the
prior for the whole system is the product $q_{1}(x_{1})q_{2}(x_{2})$.
Further suppose that new information is acquired such that $q_{1}(x_{1})$ is
updated to $p_{1}(x_{1})$ and that $q_{2}(x_{2})$ is updated to $%
p_{2}(x_{2}) $. Nothing in this new information requires us to revise our
previous assessment of independence, therefore there is no need to change
our minds, and the function $\Phi $ must be such that the prior for the
whole system $q_{1}(x_{1})q_{2}(x_{2})$ should be updated to $%
p_{1}(x_{1})p_{2}(x_{2})$.

This idea is implemented as follows: First we treat the two subsystems
separately. Suppose that for subsystem $1$ maximizing
\begin{equation}
S_{1}[P_{1},q_{1}]=\int dx_{1}\,q_{1}(x_{1})\Phi \left( \frac{P_{1}(x_{1})}{%
q_{1}(x_{1})}\right) ,  \label{S1[P1]}
\end{equation}%
subject to constraints $\mathcal{C}_{1}$ on the marginal distribution $%
P_{1}(x_{1})=\int dx_{2}\,P(x_{1},x_{2})$ selects the posterior $%
p_{1}(x_{1}) $. The constraints $\mathcal{C}_{1}$ could, for example,
include normalization, or they could involve the known expected value of a
function $f_{1}(x_{1})$,
\begin{equation}
\int dx_{1}f_{1}(x_{1})P_{1}(x_{1})=\int
dx_{1}dx_{2}\,f_{1}(x_{1})P(x_{1},x_{2})=F_{1}~.  \label{C1}
\end{equation}%
Similarly, suppose that for subsystem $2$ maximizing the corresponding $%
S_{2}[P_{2},q_{2}]$ subject to constraints $\mathcal{C}_{2}$ on $%
P_{2}(x_{2})=\int dx_{1}\,P(x_{1},x_{2})$ selects the posterior $%
p_{2}(x_{2}) $.

Next we treat the subsystems jointly and maximize the joint entropy,
\begin{equation}
S[P,q_{1}q_{2}]=\int dx_{1}dx_{2}\,q_{1}(x_{1})q_{2}(x_{2})\Phi \left( \frac{%
P(x_{1},x_{2})}{q_{1}(x_{1})q_{2}(x_{2})}\right) ,  \label{joint S[P]}
\end{equation}%
subject to the \emph{precisely the same constraints} on the joint
distribution $P$. The function $\Phi $ is determined by the requirement that
the selected posterior be $p_{1}p_{2}$. As shown in \cite{Caticha03} this
leads to the logarithmic form
\begin{equation}
S[P,q]=-\int dx\,P(x)\log \frac{P(x)}{q(x)}~.  \label{S}
\end{equation}

\section{The new independence axiom}

Next we replace our old axiom 3 by an axiom which is more convincing axiom
because it is an explicit requirement of consistency.

\noindent \textbf{New Axiom 3:\ Consistency for independent subsystems}.
\emph{When a system is composed of subsystems that are \textbf{known} to be
independent it should not matter whether the inference procedure treats them
separately or jointly.}

\noindent Again, we have to be very explicit about what this axiom means and
how it differs from the old one. When the subsystems are treated separately
the inference proceeds exactly as described before: for subsystem $1$
maximize the entropy $S_{1}[P_{1},q_{1}]$ subject to the constraints $%
\mathcal{C}_{1}$ to select a posterior $p_{1}$ and similarly for subsystem $%
2 $ to select $p_{2}$. The important difference is introduced when the
subsystems are treated jointly. Since we are only concerned with those
special examples where we \emph{know} that the subsystems are independent,
we are \emph{required} to search for the posterior within the restricted
family of joint distributions that take the form of a product $P=P_{1}P_{2}$%
; this is an \emph{additional} constraint over and above the original $%
\mathcal{C}_{1}$ and $\mathcal{C}_{2}$.

In the previous case we chose $\Phi $ so as to maintain independence because
there was no evidence against it. Here we impose independence by hand as an
additional constraint for the stronger reason that the subsystems are known
to be independent. At first sight it appears that the new axiom does not
place as stringent a restriction on the general form of $\Phi $: it would
seem that $\Phi $ has been relieved of its responsibility of enforcing
independence because it is up to us to impose it explicitly by hand.
However, as we shall see, the fact that we seek an entropy $S$ of \emph{%
general} applicability and that we require consistency for \emph{all possible%
} independent subsystems is sufficiently restrictive.

The new constraint $P=P_{1}P_{2}$ is easily implemented by direct
substitution. Instead of maximizing the joint entropy, $S[P,q_{1}q_{2}]$, we
now maximize
\begin{equation}
S[P_{1}P_{2},q_{1}q_{2}]=\int dx_{1}dx_{2}\,q_{1}(x_{1})q_{2}(x_{2})\Phi
\left( \frac{P_{1}(x_{1})P_{2}(x_{2})}{q_{1}(x_{1})q_{2}(x_{2})}\right) ,
\label{joint S[P1P2]}
\end{equation}%
under independent variations $\delta P_{1}$ and $\delta P_{2}$ subject to
the same constraints $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ and we choose $%
\Phi $ by imposing that the updating leads to the posterior $%
p_{1}(x_{1})p_{2}(x_{2})$.

\subsection{Consistency for identical independent subsystems}

Here we show that applying the axiom to subsystems that happen to be
identical restricts the entropy functional to a member of the one-parameter
family given by
\begin{equation}
S_{\eta }[P,q]=-\int dx\,P(x)\left( \frac{P(x)}{q(x)}\right) ^{\eta }\quad
\text{for}\quad \eta \neq -1,0~.  \label{S eta}
\end{equation}%
Since entropies that differ by additive or multiplicative constants are
equivalent in that they induce the same ranking scheme, we could equally
well have written
\begin{equation}
S_{\eta }[P,q]=\frac{1}{\eta (\eta +1)}\left( 1-\int dx\,P^{\eta +1}q^{-\eta
}\right) ~.  \label{S eta b}
\end{equation}%
This is convenient because the entropies for $\eta =0$ and $\eta =-1$ can be
obtained by taking the appropriate limits. For $\eta \rightarrow 0$ use $%
y^{\eta }=\exp \eta \log y\approx 1+\eta \log y$ to obtain the usual
logarithmic entropy, $S_{0}[P,q]=S[P,q]$ in eq.(\ref{S}). Similarly, for $%
\eta \rightarrow -1$ we get $S_{-1}[P,q]=S[q,P]$.

The proof below is based upon and extends a previous proof by Karbelkar \cite%
{Karbelkar86}. He showed that belonging to the family of $\eta $-entropies
is a sufficient condition to satisfy the consistency axiom for identical
systems and he conjectured but did not prove that this was perhaps also a
necessary condition. Although necessity was not essential to his argument it
is crucial for ours. We show below that for identical subsystems there are
no acceptable entropies outside this family.

\subsubsection*{Proof}

First we treat the subsystems separately. For subsystem $1$ we maximize the
entropy $S_{1}[P_{1},q_{1}]$ subject to normalization and the constraint $%
\mathcal{C}_{1}$ in eq.(\ref{C1}). Introduce Lagrange multipliers $\alpha
_{1}$ and $\lambda _{1}$,
\begin{equation}
\delta \left[ S_{1}[P_{1},q_{1}]-\lambda _{1}\left( \int
dx_{1}f_{1}P_{1}-F_{1}\right) -\alpha _{1}\left( \int dx_{1}\,P_{1}-1\right) %
\right] =0,
\end{equation}%
which gives
\begin{equation}
\Phi ^{\prime }\left( \frac{P_{1}(x_{1})}{q_{1}(x_{1})}\right) =\lambda
_{1}f_{1}(x_{1})+\alpha _{1}~,  \label{phi 1}
\end{equation}%
where the prime indicates a derivative with respect to the argument, $\Phi
^{\prime }(y)=d\Phi (y)/dy$. For subsystem $2$ we need only consider the
extreme situation where the constraints $\mathcal{C}_{2}$ determine the
posterior completely: $P_{2}(x_{2})=$ $p_{2}(x_{2})$.

Next we treat the subsystems jointly. The constraints $\mathcal{C}_{2}$ are
easily implemented by direct substitution and thus, we maximize the entropy $%
S[P_{1}p_{2},q_{1}q_{2}]$ by varying over $P_{1}$ subject to normalization
and the constraint $\mathcal{C}_{1}$ in eq.(\ref{C1}). Introduce Lagrange
multipliers $\alpha $ and $\lambda $,
\begin{equation}
\delta \left[ S[P_{1}p_{2},q_{1}q_{2}]-\lambda \left( \int
dx_{1}f_{1}P_{1}-F_{1}\right) -\alpha \left( \int dx_{1}\,P_{1}-1\right) %
\right] =0,
\end{equation}%
which gives
\begin{equation}
\int dx_{2}\,p_{2}\Phi ^{\prime }\left( \frac{P_{1}p_{2}}{q_{1}q_{2}}\right)
=\lambda \lbrack p_{2}]f_{1}(x_{1})+\alpha \lbrack p_{2}]~,  \label{phi 2}
\end{equation}%
where the multipliers $\lambda $ and $\alpha $ are independent of $x_{1}$
but could in principle be functionals of $p_{2}$.

The consistency condition that constrains the form of $\Phi $ is that if the
solution to eq.(\ref{phi 1}) is $p_{1}(x_{1})$ then the solution to eq.(\ref%
{phi 2}) must also be $p_{1}(x_{1})$, and this must be true irrespective of
the choice of $p_{2}(x_{2})$. Let us then consider a small change $%
p_{2}\rightarrow p_{2}+\delta p_{2}$ that preserves the normalization of $%
p_{2}$. First introduce a Lagrange multiplier $\alpha _{2}$ and rewrite eq.(%
\ref{phi 2}) as
\begin{equation}
\int dx_{2}\,p_{2}\Phi ^{\prime }\left( \frac{p_{1}p_{2}}{q_{1}q_{2}}\right)
-\alpha _{2}\left[ \int dx_{2}\,p_{2}-1\right] =\lambda \lbrack
p_{2}]f_{1}(x_{1})+\alpha \lbrack p_{2}]~,
\end{equation}%
where we have replaced $P_{1}$ by the known solution $p_{1}$ and thereby
effectively transformed eqs.(\ref{phi 1}) and (\ref{phi 2}) into an equation
for $\Phi $. The $\delta p_{2}(x_{2})$ variation gives,
\begin{equation}
\Phi ^{\prime }\left( \frac{p_{1}p_{2}}{q_{1}q_{2}}\right) +\frac{p_{1}p_{2}%
}{q_{1}q_{2}}\Phi ^{\prime \prime }\left( \frac{p_{1}p_{2}}{q_{1}q_{2}}%
\right) =\frac{\delta \lambda }{\delta p_{2}}f_{1}(x_{1})+\frac{\delta
\alpha }{\delta p_{2}}+\alpha _{2}~.
\end{equation}%
Next use eq.(\ref{phi 1}) to eliminate $f_{1}(x_{1})$,
\begin{equation}
\Phi ^{\prime }\left( \frac{p_{1}p_{2}}{q_{1}q_{2}}\right) +\frac{p_{1}p_{2}%
}{q_{1}q_{2}}\Phi ^{\prime \prime }\left( \frac{p_{1}p_{2}}{q_{1}q_{2}}%
\right) =A[\frac{p_{2}}{q_{2}}]\Phi ^{\prime }\left( \frac{p_{1}}{q_{1}}%
\right) +B[\frac{p_{2}}{q_{2}}]~,  \label{phi 3}
\end{equation}%
where
\begin{equation}
A[\frac{p_{2}}{q_{2}}]=\frac{1}{\lambda _{1}}\frac{\delta \lambda }{\delta
p_{2}}\quad \text{and}\quad B[\frac{p_{2}}{q_{2}}]=-\frac{\delta \lambda }{%
\delta p_{2}}\frac{\alpha _{1}}{\lambda _{1}}+\frac{\delta \alpha }{\delta
p_{2}}+\alpha _{2}~,
\end{equation}%
are at this point unknown functionals of $p_{2}/q_{2}$. Differentiating eq.(%
\ref{phi 3}) with respect to $x_{1}$ the $B$ term drops out and we get
\begin{equation}
A[\frac{p_{2}}{q_{2}}]=\left[ \frac{d}{dx_{1}}\Phi ^{\prime }\left( \frac{%
p_{1}}{q_{1}}\right) \right] ^{-1}\frac{d}{dx_{1}}\left[ \Phi ^{\prime
}\left( \frac{p_{1}p_{2}}{q_{1}q_{2}}\right) +\frac{p_{1}p_{2}}{q_{1}q_{2}}%
\Phi ^{\prime \prime }\left( \frac{p_{1}p_{2}}{q_{1}q_{2}}\right) \right] ~,
\end{equation}%
which shows that $A$ is not a functional but a mere function of $p_{2}/q_{2}$%
. Substituting back into eq.(\ref{phi 3}) we see that the same is true for $%
B $. Therefore eq.(\ref{phi 3}) can be written as
\begin{equation}
\Phi ^{\prime }\left( y_{1}y_{2}\right) +y_{1}y_{2}\Phi ^{\prime \prime
}\left( y_{1}y_{2}\right) =A(y_{2})\Phi ^{\prime }\left( y_{1}\right)
+B(y_{2})~,  \label{phi 4}
\end{equation}%
where $y_{1}=p_{1}/q_{1}$, $y_{2}=p_{2}/q_{2}$, and $A(y_{2})$, $B(y_{2})$
are unknown functions of $y_{2}$. If we specialize to identical subsystems
for which we can exchange the labels $1\leftrightarrow 2$, we get
\begin{equation}
A(y_{2})\Phi ^{\prime }\left( y_{1}\right) +B(y_{2})=A(y_{1})\Phi ^{\prime
}\left( y_{2}\right) +B(y_{1})~.  \label{phi 5}
\end{equation}%
To find the unknown functions $A$ and $B$ differentiate with respect to $%
y_{2}$,
\begin{equation}
A^{\prime }(y_{2})\Phi ^{\prime }\left( y_{1}\right) +B^{\prime
}(y_{2})=A(y_{1})\Phi ^{\prime \prime }\left( y_{2}\right)  \label{b}
\end{equation}%
and then with respect to $y_{1}$ to get
\begin{equation}
\frac{A^{\prime }(y_{1})}{\Phi ^{\prime \prime }\left( y_{1}\right) }=\frac{%
A^{\prime }(y_{2})}{\Phi ^{\prime \prime }\left( y_{2}\right) }=a=\func{const%
}~.
\end{equation}%
Integrating,
\begin{equation}
A(y_{1})=a\Phi ^{\prime }\left( y_{1}\right) +b~.
\end{equation}%
Substituting back into eq.(\ref{b}) and integrating gives
\begin{equation}
B^{\prime }(y_{2})=b\Phi ^{\prime \prime }\left( y_{2}\right) \quad \text{and%
}\quad B(y_{2})=b\Phi ^{\prime }\left( y_{2}\right) +c~,
\end{equation}%
where $b$ and $c$ are constants. We can check that $A(y)$ and $B(y)$ are
indeed solutions of eq.(\ref{phi 5}). Substituting into eq.(\ref{phi 4})
gives
\begin{equation}
\Phi ^{\prime }\left( y_{1}y_{2}\right) +y_{1}y_{2}\Phi ^{\prime \prime
}\left( y_{1}y_{2}\right) =a\Phi ^{\prime }\left( y_{1}\right) \Phi ^{\prime
}\left( y_{2}\right) +b\left[ \Phi ^{\prime }\left( y_{1}\right) +\Phi
^{\prime }\left( y_{2}\right) \right] +c~.  \label{phi 6}
\end{equation}%
This is a peculiar differential equation. We can think of it as one
differential equation for $\Phi ^{\prime }\left( y_{1}\right) $ for each
given constant value of $y_{2}$ but there is a complication in that the
various (constant) coefficients $\Phi ^{\prime }\left( y_{2}\right) $ are
themselves unknown. To solve for $\Phi $ choose a fixed value of $y_{2}$,
say $y_{2}=1$,
\begin{equation}
y\Phi ^{\prime \prime }\left( y\right) -\eta \Phi ^{\prime }\left( y\right)
-\kappa =0~,  \label{phi 7}
\end{equation}%
where $\eta =a\Phi ^{\prime }\left( 1\right) +b-1$ and $\kappa =b\Phi
^{\prime }\left( 1\right) +c$. To eliminate the constant $\kappa $
differentiate with respect to $y$,
\begin{equation}
y\Phi ^{\prime \prime \prime }+\left( 1-\eta \right) \Phi ^{\prime \prime
}=0~,  \label{phi 8}
\end{equation}%
which is a linear homogeneous equation and is easy to integrate. For a
generic value of $\eta $ the solution is
\begin{equation}
\Phi ^{\prime \prime }(y)\propto y^{\eta -1}\Rightarrow \Phi ^{\prime
}(y)=\alpha y^{\eta }+\beta ~.
\end{equation}%
The constants $\alpha $ and $\beta $ are chosen so that this is a solution
of eq.(\ref{phi 6}) for all values of $y_{2}$ (and not just for $y_{2}=1$).
Substituting into eq.(\ref{phi 6}) and equating the coefficients of various
powers of $y_{1}y_{2}$, $y_{1}$, and $y_{2}$ gives three conditions on the
two constants $\alpha $ and $\beta $,
\begin{equation}
\alpha (1+\eta )=a\alpha ^{2},\quad 0=a\alpha \beta +b\alpha ,\quad \beta
=a\beta ^{2}+2b\beta +c~.
\end{equation}%
The nontrivial ($\alpha \neq 0$) solutions are $\alpha =(1+\eta )/a$ and $%
\beta =-b/a$, while the third equation gives $c=b(1-b)/4a$. We conclude that
for generic values of $\eta $ the solution of eq.(\ref{phi 6}) is
\begin{equation}
\Phi (y)=\frac{1}{a}y^{\eta +1}-\frac{b}{a}y+C~,  \label{sol a}
\end{equation}%
where $C$ is a new constant. Choosing $a=-\eta (\eta +1)$ and $b=1+Ca$ we
obtain eq.(\ref{S eta b}).

For the special values $\eta =0$ and $\eta =-1$ one can either first take
the limit of the differential eq.(\ref{phi 8}) and then find the relevant
solutions, or one can first solve the differential equation for general $%
\eta $ and then take the limit of the solution eq.(\ref{S eta b}) as
described earlier. Either way one obtains (up to additive and multiplicative
constants which have no effect on the ranking scheme) the entropies $%
S_{0}[P,q]=S[P,q]$ and $S_{-1}[P,q]=S[q,P]$.

\subsection{Consistency for non-identical subsystems}

Let us summarize our results so far. The goal is to update probabilities by
ranking the distributions according to an entropy $S$ that is of general
applicability. The functional form of the entropy $S$ has been constrained
down to a member of the one-dimensional family $S_{\eta }$. One might be
tempted to conclude (see \cite{Karbelkar86, Uffink95}) that there is no $S$
of universal applicability; that inferences about different systems ought to
be carried out with different $\eta $-entropies. But we have not yet
exhausted the full power of our new axiom 3.

To proceed further we ask: What is $\eta $? Is it a property of the
individual carrying out the inference or of the system under investigation?
The former makes no sense; we insist that the updating must be objective in
that different individuals with the same prior and the same information must
make the same inference. Therefore the \textquotedblleft inference
parameter\textquotedblright\ $\eta $ must be a characteristic of the system.

Consider two different systems characterized by $\eta _{1}$ and $\eta _{2}$.
Let us further suppose that these systems are independent (perhaps system $1$
is here on Earth while the other lives in a distant galaxy) so that they
fall under the jurisdiction of the new axiom 3; inferences about system $1$
are carried out with $S_{\eta _{1}}[P_{1},q_{1}]$ while inferences about
system $2$ require $S_{\eta _{2}}[P_{2},q_{2}]$. For the combined system we
are also required to use an $\eta $-entropy $S_{\eta
}[P_{1}P_{2},q_{1}q_{2}] $. The question is what $\eta $ do we choose that
will lead to consistent inferences whether we treat the systems separately
or jointly. The results of the previous section indicate that a joint
inference with $S_{\eta }[P_{1}P_{2},q_{1}q_{2}]$ is equivalent to separate
inferences with $S_{\eta }[P_{1},q_{1}]$ and $S_{\eta }[P_{2},q_{2}]$.
Therefore we must choose $\eta =\eta _{1}$ and also $\eta =\eta _{2}$ which
is possible only when $\eta _{1}=\eta _{2}$. But this is not all: any other
system whether here on Earth or elsewhere that happens to be independent of
the distant system $2$ must also be characterized by the same inference
parameter $\eta =\eta _{2}=\eta _{1}$ even if it is correlated with system $%
1 $. Thus all systems have the same $\eta $ whether they are independent or
not.

The power of a consistency argument resides in its universal applicability:
if a general expression for $S[P,q]$ exists then it must be of the form $%
S_{\eta }[P,q]$ where $\eta $ is a universal constant. The remaining problem
is to determine this universal $\eta $. One possibility is to determine $%
\eta $ experimentally: are there systems for which inferences based on a
known value of $\eta $ have repeatedly led to success? The answer is yes;
they are quite common.

The next step in our argument is provided by the work of Jaynes \cite%
{Jaynes57} who showed that statistical mechanics and thus thermodynamics are
theories of inference based on the value $\eta =0$. His method, called
MaxEnt, can be interpreted as the special case of the ME when one updates
from a uniform prior using the Gibbs-Shannon entropy. Thus, it is an
experimental fact without any known exceptions that inferences about \emph{%
all} physical, chemical and biological systems that are in thermal
equilibrium or close to it can be carried out by assuming that $\eta =0$.
Let us emphasize that this is not an obscure and rare example of purely
academic interest; these systems comprise essentially all of natural
science. (Included is every instance where it is useful to introduce a
notion of temperature.)

In conclusion: consistency for non-identical systems requires that $\eta $
be a universal constant and there is abundant experimental evidence for its
value being $\eta =0$. Other $\eta $-entropies may be useful for other
purposes but the logarithmic entropy $S[P,q]$ in eq.(\ref{S}) provides the
only consistent ranking criterion for updating probabilities that can claim
general applicability.

\section{Bayes updating}

The two preeminent updating methods are the ME method discussed above and
Bayes' rule. The choice between the two methods has traditionally been
dictated by the nature of the information being processed (either
constraints or observed data) but questions about their compatibility are
regularly raised. Our goal here is to show that these two updating
strategies are completely consistent with each other. Let us start by
drawing a distinction between Bayes' theorem and Bayes' rule.

\subsection{Bayes' theorem and Bayes' rule}

The goal here is to update our beliefs about the values of one or several
quantities $\theta \in \Theta $ on the basis of observed values of variables
$x\in \mathcal{X}$ and of the known relation between them represented by a
specific model. The first important point to make is that attention must be
focused on the joint distribution $P_{\text{old}}(x,\theta )$. Indeed, being
a consequence of the product rule, Bayes' theorem requires that $P_{\text{old%
}}(x,\theta )$ be defined and that assertions such as \textquotedblleft $x$
\emph{and} $\theta $\textquotedblright\ be meaningful; the relevant space is
neither $\mathcal{X}$ nor $\Theta $ but the product $\mathcal{X}$ $\times
\,\Theta $. The label \textquotedblleft old\textquotedblright\ is important.
It has been attached to the joint distribution $P_{\text{old}}(x,\theta )$
because this distribution codifies our beliefs about $x$ and about $\theta $
before the information contained in the actual data has been processed. The
standard derivation of Bayes' theorem invokes the product rule,
\begin{equation}
P_{\text{old}}(x,\theta )=P_{\text{old}}(x)P_{\text{old}}(\theta |x)=P_{%
\text{old}}(\theta )P_{\text{old}}(x|\theta )~,
\end{equation}%
so that%
\begin{equation}
P_{\text{old}}(\theta |x)=P_{\text{old}}(\theta )\frac{P_{\text{old}%
}(x|\theta )}{P_{\text{old}}(x)}~.  \tag{Bayes' theorem}
\end{equation}%
It is important to realize that at this point there has been no updating.
Our beliefs have not changed. All we have done is rewrite what we knew all
along in $P_{\text{old}}(x,\theta )$. Bayes' \emph{theorem} is an identity
that follows from requirements on how we should consistently assign degrees
of belief. Whether the justification of the product rule is sought through
Cox's consistency requirement and regraduation or through a Dutch book
betting coherence argument, the theorem is valid irrespective of whatever
data will be or has been collected. Our notation, with the label
\textquotedblleft old\textquotedblright\ throughout, makes this point
explicit.

The real updating from the old prior distribution $P_{\text{old}}(\theta )$
to a new posterior distribution $P_{\text{new}}(\theta )$ occurs when we
take into account the values of $x$ that have actually been observed, which
we will denote with a capital $X$. This requires a new assumption and the
natural choice is that the updated distribution $P_{\text{new}}(\theta )$ be
given by Bayes' \emph{rule},
\begin{equation}
P_{\text{new}}(\theta )=P_{\text{old}}(\theta |X)~.  \tag{Bayes' rule}
\end{equation}%
Combining Bayes' theorem with Bayes' rule leads to the standard equation for
Bayes updating,
\begin{equation}
P_{\text{new}}(\theta )=P_{\text{old}}(\theta )\frac{P_{\text{old}}(X|\theta
)}{P_{\text{old}}(X)}~.
\end{equation}%
The assumption embodied in Bayes' rule is extremely reasonable: we maintain
those old beliefs about $\theta $ that are consistent with data values that
have turned out to be true. Data values that were not observed are discarded
because they are now known to be false.

This argument is indeed so compelling that it may seem unnecessary to seek
any further justification for the Bayes' rule assumption. However, we deal
here with such a basic algorithm for information processing -- it is
fundamental to all experimental science -- that even such a self-evident
assumption should be carefully examined and its compatibility with the ME
method should be verified.

\subsection{Bayes' rule from ME}

Our first concern when using the ME method to update from a prior to a
posterior distribution is to define the space in which the search for the
posterior will be conducted. We argued above that the relevant space is the
product $\mathcal{X}\times \Theta $. Therefore the selected posterior $P_{%
\text{new}}(x,\theta )$ is that which maximizes
\begin{equation}
S[P,P_{\text{old}}]=-\tint dxd\theta ~P(x,\theta )\log \frac{P(x,\theta )}{%
P_{\text{old}}(x,\theta )}~
\end{equation}%
subject to the appropriate constraints.

Next, the information being processed, the observed data $X$, must be
expressed in the form of a constraint on the allowed posteriors. Clearly,
the family of posteriors that reflects the fact that $x$ is now known to be $%
X$ is such that
\begin{equation}
P(x)=\tint d\theta ~P(x,\theta )=\delta (x-X)~.  \label{data constraint}
\end{equation}%
This amounts to an \emph{infinite} number of constraints: there is one
constraint on $P(x,\theta )$ for each value of the variable $x$ and each
constraint will require its own Lagrange multiplier $\lambda (x)$.
Furthermore, we impose the usual normalization constraint,
\begin{equation}
\tint dxd\theta ~P(x,\theta )=1~.
\end{equation}

Maximize $S$ subject to these constraints,
\begin{equation}
\delta \left\{ S+\tint dx\,\lambda (x)\left[ \tint d\theta ~P(x,\theta
)-\delta (x-X)\right] +\alpha \left[ \tint dxd\theta ~P(x,\theta )-1\right]
\right\} =0~,
\end{equation}%
and the selected posterior is
\begin{equation}
P_{\text{new}}(x,\theta )=P_{\text{old}}(x,\theta )\,\frac{e^{\lambda (x)}}{Z%
}~,  \label{solution a}
\end{equation}%
where the normalization $Z$ is
\begin{equation}
Z=\,e^{-\alpha +1}=\tint dxd\theta \,P_{\text{old}}(x,\theta )\,e^{\lambda
(x)}~,  \label{Z}
\end{equation}%
and the multipliers $\lambda (x)$ are determined from eq.(\ref{data
constraint}),
\begin{equation}
\tint d\theta ~P_{\text{old}}(x,\theta )\frac{\,e^{\lambda (x)}}{Z}=P_{\text{%
old}}(x)\frac{\,e^{\lambda (x)}}{Z}=\delta (x-X)~.
\end{equation}%
Therefore, substituting $e^{\lambda (x)}$ back into eq.(\ref{solution a}),
\begin{equation}
P_{\text{new}}(x,\theta )=\frac{P_{\text{old}}(x,\theta )\,\delta (x-X)}{P_{%
\text{old}}(x)}=\delta (x-X)P_{\text{old}}(\theta |x)~.
\end{equation}

The new marginal distribution for $\theta $ is%
\begin{equation}
P_{\text{new}}(\theta )=\tint dxP_{\text{new}}(x,\theta )=P_{\text{old}%
}(\theta |X)~,
\end{equation}%
which is Bayes' rule! Bayes updating is a special case of ME updating.

To summarize: the prior $P_{\text{old}}(x,\theta )=P_{\text{old}}(x)P_{\text{%
old}}(\theta |x)$ is updated to the posterior $P_{\text{new}}(x,\theta )=P_{%
\text{new}}(x)P_{\text{new}}(\theta |x)$ where $P_{\text{new}}(x)=\delta
(x-X)$ is fixed by the observed data while $P_{\text{new}}(\theta |x)=P_{%
\text{old}}(\theta |x)$ remains unchanged. Note that in accordance with the
philosophy that drives the ME method \emph{one only updates those aspects of
one's beliefs for which corrective new evidence has been supplied}.

The generalization to situations where there is some uncertainty about the
actual data is straightforward. In this case the marginal $P(x)$ in eq.(\ref%
{data constraint}) is not a $\delta $ function but a known distribution $%
P_{D}(x)$. The selected posterior $P_{\text{new}}(x,\theta )=\,P_{\text{new}%
}(x)P_{\text{new}}(\theta |x)$ is easily shown to be $P_{\text{new}%
}(x)=P_{D}(x)$ with $P_{\text{new}}(\theta |x)=P_{\text{old}}(\theta |x)$
remaining unchanged. This leads to Jeffrey's conditionalization rule,
\begin{equation}
P_{\text{new}}(\theta )=\tint dx\,P_{\text{new}}(x,\theta )=\tint
dx\,P_{D}(x)P_{\text{old}}(\theta |x)~.
\end{equation}

\section{Conclusions}

We have shown that Skilling's method of induction has led to a unique
general theory of inductive inference, the ME method. The whole approach is
extremely conservative. First, the axioms merely instruct us what not to
update -- do not change your mind except when forced by new information.
Second, the validity of the method does not depend on any particular
interpretation of the notion of entropy -- entropy needs no interpretation.

Our derivation of the consequences of the new axiom show that when applied
to identical subsystems they restrict the entropy to a member of the $\eta $%
-entropy family. Its further application to non-identical systems shows that
consistency requires that $\eta $ be a universal constant which must take
the value $\eta =0$ in order to account for the empirical success of the
inference theory we know as statistical mechanics. Thus, the unique tool for
updating probabilities is the logarithmic relative entropy. Other entropies
with $\eta \neq 0$ or those of Renyi or Tsallis are ruled out; they may be
useful for other purposes but not for inference.

Finally we explored the compatibility of Bayes and ME updating. After
pointing out the distinction between Bayes' theorem and the Bayes' updating
rule, we showed that Bayes' rule is a special case of ME updating by
translating information in the form of data into constraints that can be
processed using ME.

\noindent \textbf{Acknowledgements:} We would like to acknowledge valuable
discussions with N. Caticha, R. Fischer, M. Grendar, K. Knuth, C. Rodr\'{\i}%
guez, and A. Solana-Ortega.

redpajama_set_name: RedPajamaArXiv


The Acropolis, the Acropolis Museum and the National Archaeological Museum in Athens, all located in Greece's capital city are the most visited sites across the nation, according to the latest data released by the Hellenic Statistical Authority for the January – October 2017 period.
Director of the National Archaeological Museum in Athens, Maria Lagogianni-Georgakarakos told ANA the increased number of tourists, diversity of shows, exhibitions and activities, as well as promotional actions is the reason behind the greater numbers compared to previous years.
A total of 2,510,397 people visited the Acropolis in the January and October period and 1,428,037 made their way to the Acropolis Museum, 489,279 visited the National Archaeological Museum, while 612,119 visited Knossos, and 404,660 the Museum in Heraklion, Crete.
"It's much more than just the brand name… The main factor is for the museum to keep track of its visitors, focus on them, try to tap into their expectations and, to the extent that it can, meet their demands," curator at the Acropolis Museum, Dimitrios Pandermalis told ANA.


Three people have died and nine others have been injured in a shooting in the central Dutch city of Utrecht on Monday morning, police said.
The Utrecht police said a square at a tram station outside the city centre had been cordoned off as emergency services were at the scene.
Reports claim a man in a tram on the way to Utrecht CS pulled out a weapon and shot several people.
Counter-terrorism police are at the scene and officers have said they are considering the possibility of a 'terrorist motive'.
Local reports in the Netherlands claim that there may have been shootings in other places.
The shooter is believed to have fled in a red car, however dumped the vehicle a short while later.
A manhunt is now underway.
'Shooting incident… Several injured people reported. Assistance started,' the Utrecht police Twitter account said.
Officers urged the public to stay away to allow first responders 'to do their work'.
Dutch Prime Minister Mark Rutte said he was 'deeply concerned' about the incident.
The Dutch government raised the terrorism threat level to the highest possible this morning.
The National Anti-Terrorism Coordinator said it had raised alert level as the suspected shooter was still at large.
Security was increased at schools, mosques and transportation hubs after the shooting.


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Creativity presents the series of perpetual assumptions where the entire processes involving behavioural ways to understand and think differently on the same set of a series of reasoning in completely different ways. It always asks yourself about the set up of differently understood critical questioning to yourselves in order to understand every point of action and behaviour in completely different ways. Through a series of critical questioning and generating of ideas more and more flow of conclusion towards the understanding of absolute reasoning comes about.
In the series of understanding and its corresponding cases where there had always been the presence of different ideas and understandings in each and every cases still identifying the challenging areas that involved with these entire processes should come handy to understand the flow of ideas generating with each and every process must coincide with a complete understanding of a series of processes attached with it. In these contexts of a series of events, there has always been the tendency of one principal context that works in correlated ways to supplement entire set up of events in completely prudent manners.
Even though you are absolutely certain about the principal context working incoherent and synchronisation manners still you have to explore the set up of differently calculated mechanisms to find out whether all these series of events have some sort of alternative possibilities. Before arriving at such conclusion it is important to follow the imaginative alternative contexts and explore various ideas involved within it and develop the reflective scepticism in order to learn more and more cohesive and alternate methods to reach and challenge existing conclusions.
It does not prove to be you are moving towards the negative zones, instead it goes on to show that with the positive and reflexive thinking parameters you constantly go for complete improvement of the related contexts to find out more pieces of information and intriguing aspects of complete understanding of set up of events that comes forwards at you at various stages of thinkings. While dealing with the complete set up of events that related with problem-solving mechanisms consider moving towards removing of standardised formats that should suit these trends in coping with the aspects of understanding diverse attributes forwarding to movement of standardised formats.
In this manner the most of standardised or common set of problems move towards the form of not moving into it and understanding it as it should be the short cut route for process or movement of ideas and the range of discussions and topics move from wide range of consolidated and divergent range of subjects so that multiple and various perspectives to the same problems could be easily understandable with it. What it makes you to fully understand and update the knowledge base is to find out the series of futuristic oriented understandings of the set of problems and their solutions in the present context.
While dealing with all these aspects to understand the entire process that involves the different contexts one need to sufficiently convenient to approximate about the trend of events that should generate absolute self-confidence in arriving at own judgement that should provide multifaceted ways to understand and find out more pieces of information about detecting and managing the aspects of divergent views. In a way, the series of processes involving with critical thinking includes recognition which underlies the processes attached with considerate assumptions where every conclusion comes to the rescue of understanding the series of contexts and most of these contexts should come from the basic assumptions by removing the series of standardised processes.
Then, comes the series of processes where you should be analysing each and every bit of processing and the arguments that support these processes in making these the single most standalone supportive elements to consider the potential to reach an absolute conclusion. In the set up of entire processes and considerate arguments you only have to seat and discuss within your scope of reasoning and discussing different pros and cons of every argument and in these processes scrutinising each and every aspect in clear and lucid manner so as to find out the single most perfect assumption of your choices.
In this process of brain mapping and thinking and arriving out absolute conclusions you are the sole lawyer, judge and decision maker and for this it is important to understand to recognise the set of problems out there and then consider the process and the argument inherent with it, Then reach some niche conclusion but still judging these ideas goes on for important processes in order to find out the best out of all ideas and contexts that provide inherent understanding and possible marking to point out what should be the best of a lot of all. Then considerate every conclusive context in the form of rationalistic thinking and its related context parameters to reach out to absolute and solid conclusions.
It involves complete comparing out of various stages of events that should enable and provide absolute justification for reaching out and understanding the positives as well as negative approval limits so as to find the most important and generalised ideas related with complete overwhelming of generation of ideas by rejecting bad contexts and accepting good contexts. In this manner, the processes of thinkings become absolutely critical that rejects standardised formats of different problem-solving methods and goes on to see the absolute necessity associated with the processes that involve complete and absolute perspective that involves relating and diversification of views to take multiple perspectives.
In these processes of reaching to an absolute conclusion of the series of contexts of events, one should always use the trial and error methods and find out the most unconventional ways to understand this set of problems to reach out to an absolute conclusion. Ask some questions to yourselves about what could happen this context that does not happen till now and what should be the series of events of attendances that should provide the most vivid and dynamic aspect of accomplishments that should be the prime focus to understand and estimate process that should enable the power of critical thinkings.
In these entire process of examining the contextual elements one aspect to be critically understood is the presence of behavioural implications so that ultimately the forms and the processes and the set of events should be equally understood and examine and to be set up with complete tractable parameters if it succeeds successfully with the entire process associated with the presence of such and such critical elements of human interactions. What it makes the entire set up of contexts and its related mechanisms to behave in such and such manner so as to find out the process optimisation as well as the most vivid and dynamic way to understand what should have been the best choice to think of it and should be the most desirable aspect to move beyond and understand entire set up of events in completely and clearly understandable ways.
In short, when you start thinking of creative ways you ought to think in different ways and go with some of the other mechanisms to find out the single most aspect to move towards the most efficient and most perfected ways to move absolute perfections. It might have been scary ideas to think out of the box and think on some other way but still one needs to understand the aspects of the entire process optimisation in terms of entirely to find out more and more perspective towards efficient maximisations of entire processes. There are many chances of failures but still, they move within such thinking parameters and that is why there had been many respects to creative thinkers worldwide simply for their mission to find out more about the different aspect of thinking by completely moving out from the standardised form of thinking processes.
This entry was posted in LEADERSHIP, MANAGEMENT on February 29, 2016 by Bubu.