How do we know baroque art depicted obese ladies because of a different ideal of beauty?

The idea that, taken across the entire range of media and works, art from any given period represents a society or culture's ideal for how anyone - male or female - should appear has no real basis in fact.

It is vastly more likely - especially in any time frame where we know the names and personalities of the artists being discussed - that any given work represents an example of what the artist considered technically interesting or challenging, or important in some other way by the standards of his craft or in the judgment of his contemporary artists. This was true even in portraiture.

Three centuries of artistic technical fascination with chiaroscuro alone should show us this. Artists chose bodies, poses and themes which allowed them to explore the interaction of light and shadow and the human body; and they did so for technical (and aesthetic) reasons that only rarely had anything to do with depicting ideals of physique.

For most of the history of representational art across all cultures, in fact, there was little attempt to make the figures depicted realistic in any way - "beautiful" or not. Nobody would argue that Greek sculptures from the Geometric or Archaic periods represent the cultural ideal of male and female physical beauty of the time, or that medieval weavers of tapestries or illuminators of manuscripts were attempting to depict the perfect human form.

This aesthetic relativism is also confused by the fact that for most of the history of Western art, women weren't able to model so the artist often just changed male bodies (from his/her model) into female bodies by adding breasts or extra fat or whatever.

Here's a source: https://renresearch.wordpress.com/2011/02/11/men-with-breasts-or-why-are-michelangelos-women-so-muscular-part-1/

Which references Gill Saunders' The Nude: A New Perspective (1989)

Q How do we know baroque art depicted obese ladies because of a different ideal of beauty?

Do we really?
We don't. The anthropological constant to be observed is: "women are considered 'attractive' if: young and healthy" (both more or less relating to fecundity; Whether socio-biological, evolutionary, or just cynical):

Differences in the historical record in the form of paintings of women –– meaning that not L'art-pour-l'art pictures show not that much variation in 'considered beautiful'. Archaeological evidence, that is more material evidence – like statues – does not support major shifts of aesthetic preference or judgement, before the 20th century.

There is the obvious tendency to ascribe a cultural preference for heftier women in baroque times. "Just look at the paintings!" And while it's certainly true that people come in all shapes and sizes, so it is equally true that people come with all kinds of tastes in all times: some preferring slender shapes, some going for the voluptuous.

The "obvious" conclusion is that in times when all kinds of illnesses, harsh winters and not in the least quite frequent real hunger catastrophes were just around every corner, people with a little reserve in body fat are indeed more likely to survive that, and be more fecund in the case of women. They were/are just more robust.

Added to that is the status appeal: you have to be able to afford so much eating.

But with this "tastes"-angle comes a tiny little problem: we have also at least three selection biases for our material to analyse at work here:

1. When Henry VIII was young and coming, he was the sporty guy with not much fat on him. As he aged he became fat, like the tendency to observe in every last one of us today: as we age, the body needs less energy, but habits change slowly and for most average people it gets harder with every year to keep the same weight. But those people who could afford being painted tended to be slightly older as well.

2. With differing tastes there is then this personal preference thing again. Rubens being the prime example. This painter is described as:

   His nudes of various biblical and mythological women are especially well-known. Painted in the Baroque tradition of depicting women as soft-bodied, passive, and highly sexualized beings, his nudes emphasize the concepts of fertility, desire, physical beauty, temptation, and virtue. Skillfully rendered, these paintings of nude women were undoubtedly created to appeal to his largely male audience of patrons. Additionally, Rubens was quite fond of painting full-figured women, giving rise to terms like 'Rubensian' or 'Rubenesque' (sometimes 'Rubensesque'). And while the male gaze features heavily in Rubens's paintings of females generally, he brings multi-layered allegory and symbolism to his portraits.

   "He was indeed a chubby chaser(?)", preferring the looks of some women painted being even above what we think to be the norm in his age. (But also look at Karolien De Clippel: "Defining beauty: Rubens’s female nudes", (PDF) ––/–– Leah Sweet: "Fantasy Bodies, Imagined Pasts: A Critical Analysis of the “Rubenesque” Fat Body in Contemporary Culture", Fat Studies:
   An Interdisciplinary Journal of Body Weight and Society Volume 3, 2014 - Issue 2: Reflective Intersections, (DOI))

   Strong versions of evolutionary psychologists have proposed that men possess perceptual mechanisms that engender a preference for women with low waist-to-hip ratios (WHR), typically 0.70, as this is considered maximally healthy and fertile. This has taken to be culturally and temporally invariant. In the present study, two semi-expert and two non-expert judges made measurements of the WHR of nude females in paintings by Pieter Pauwel Rubens. The results showed that the mean WHR of Rubens’ women was 0.776, significantly higher than the reported preference for WHRs of 0.70. Possible non-adaptive explanations for this result are proposed in conclusion.
   For example, the present study assumes that Rubens portrayed women for their physical beauty, but it may also be possible that he portrayed them to exemplify other traits such as wealth and abundance, that is, things other than beauty and fertility. Finally, although Rubens is perhaps the most well-known painter to depict voluptuous nudes (hence the term ‘Rubenesque’), future studies should examine the stylised depictions of other artists and artistic eras.
   _{Viren Swami et al.: "The Female Nude in Rubens: Disconfirmatory Evidence of the Waist-to-Hip Ratio Hypothesis of Female Physical Attractiveness", Imagination, Cognition and Personality, Vol. 26(1-2) 139-147, 2006-2007.}

3. The current Western 'ideal' – as propagated in most media, and now internalised – is sick!
   While it is quite consensual that too much body fat is unhealthy, it is also quite clear that anorexia, bulimia, orthorexia, surgery, pills, and propaganda are sick as well.

   By observing the art of different eras, as well as the more recent existence of the media, it is obvious that there have been dramatic changes in what is considered a beautiful body. The ideal of female beauty has shifted from a symbol of fertility to one of mathematically calculated proportions. It has taken the form of an image responding to men’s sexual desires. Nowadays there seems to be a tendency towards the destruction of the feminine, as androgynous fashion and appearance dominate our culture. The metamorphosis of the ideal woman follows the shifting role of women in society from mother and mistress to a career-orientated indi- vidual. Her depiction by artists across the centuries reveals this change in role and appearance that should be interpreted within the social and historical context of each era with its own theories of what constituted the ideal female body weight.
   Did medical science contribute to today’s accepted BMI? Obesity has been regarded as a condition that increases the risk of many diseases only in recent decades, when evidence-based medicine took the lead, and scientists discovered saturated and transfats and their relationship with metabolic and cardiovascular diseases. Evidence of the change this may have had on society can be seen in the more prevalent occurrence of eating disorders in the latter half of the 20th century. The advice to lose weight and reach a BMI comprised between 18 and 24 therefore might have accelerated the process of the ideal figure tending towards thin bodies, in particular in professions where there is a strong pressure to control body weight such as athletics and dance. The media brought the relationship between an ideal figure and evidence-based medicine to extreme consequences, presenting exaggeratedly thin figures as symbols of health, while in reality being the opposite as shown by data reporting female athletes with higher rates of eating disorders.
   The metamorphosis of the ideal woman follows the shifting role of women in society from mother and mistress to a career-orientated individual. Her depiction by artists across the centuries reveals this change in role and appearance. Unfortunately, today, beauty (and the ideal body weight) is not exactly in the eye of the beholder, but in the body image presented by the media and sold to a malleable public.
   _{B. A. Bonafini & P. Pozzilli: "Body weight and beauty: the changing face of the ideal female body weight", Obesity Reviews, Volume12, Issue1, 2011, Pages 62-65. (DOI)}

So yes: until just a few decades ago average heterosexual men preferred healthy looking women. Really obese women were always well liked by 'specialists'. But our current understanding of past preferences of beauty seems to be much distorted by what we now commonly call "obesity".

Having a look, I do not see "obese ladies":

(Rubens & his wife!) Portrait of Marchesa Brigida Spinola-Doria

Shoot me, but I believe 'the Aliens' reading it, would tend to agree, that quite recently, the meaning of "obese" seems to encroach on what we once thought of as normal and healthy:

_{(Jake Rosenthal • January 20, 2016 The Pioneer Plaque: Science as a Universal Language)}

Now compare the to:

Aphrodite or 'Crouching Venus' Second century AD Marble | 125 x 53 x 65 cm (whole object) | RCIN 69746: or

Aphrodite Kallipygos "The Venus of the beautiful buttocks"

If the original model for that would be laser-scanned and then 3-D-printed while calculating here insurance-premium based on BMI: then you know that StackExchange policy probably prohibits commenting on it, but I am certain that you have made up your modern mind on that?

To begin with I accepted the conventional, though actually little examined, view that standards of beauty change from age to age. In human affairs much does change - ideologies and institutions, economic and social systems, class structures, the role and status of women; my own study of the evidence compelled me to the conclusion that, relative to these, beauty (in the western world that I am qualified to write about) has changed little. That is why I call it a 'relative constant', a 'relative universal'. To be honest I am not greatly impressed by the oft-repeated accounts of African tribes prizing fatness, South American ones lip plates, Burmese ones necks stretched and ringed like a snake - in these examples the admiration is for symbols of wealth and status, not beauty. Indeed the whole subject is bedevilled by an elementary failure to distinguish between fashion and beauty.
_{Arthur Marwick: "It. A History of Human Beauty", Hambledon and London: London, New York, 2004, p IX;}

A lot of the paintings were commissioned as portraits, why would people pay for themselves to be depicted in an ugly way?

Wealth nowadays is associated with a slim, tanned, and shaped body because those are traits of people who have enough free time, and money to achieve it. In that period, it would be the reverse, being more on the fat side would require wealth, and more refined foods which were more expensive, while majority of commoners would be slim, tanned (for working in the sun) and toned due to hard work.

It makes sense based on those, that it must have been acceptable to be fat back then, even desirable.

Art does not exist in a vacuum, but is rather only one part of the historical record. Just as people comment on our modern standard of beauty today, so does early modern writers on theirs. Fortunately, Baroque art dates from a recent enough period that the historical record is extensive.

For example, a 17th century commentary on a Van Dyck portrait of a heavier women states:

William Sanderson in the treatise Graphice, published in 1658, noted that a beautiful woman was to have "a noble neck, round rising, full and fat . . . brawny arm of good flesh. Such a lady possesses a goodly plump fat."

_{Wind, Barry. A Foul and Pestilent Congregation: Images of Freaks in Baroque Art. Routledge, 2018.}

Beyond artistic contexts, we also find examples of beauty being explicitly attributed to body fat:

The use of fat is . . . It fills up the empty spaces between the Muscles, Vessels, and Skin, and consequently renders the Body smooth, white, soft, fair, and beautiful . . . Persons in a Consumption and decrepit old women are deformed for want of Fat.

_{Bartholin, Thomas, Caspar Bartholin, and Johannes Walaeus. Bartholinus Anatomy: Made from the Precepts of His Father, and from the Observations of All Modern Anatomists. John Streater.}

Historians, on the basis of information like this, therefore argue that people of the period - at least in some countries - considered it desirable for women to have a bit of plump.

As for "rationalisation" - of course, you can definitely make the case that they only considered chubbiness beautiful because of the association with wealth. However, does that necessarily invalidate their standard of beauty? I believe this delves into a philosophical realm over how beauty should be defined, which is beyond the scope of history.