BackgroundAll living persons and test subjects have been anonymized for privacy reasons. In this particular study, AncestryDNA performed all DNA tests.
Abstract

A distant relation, JSM, who had been kind enough to lend her DNA matches to Discovering New Ancestors as we develop our CMA and AASK processes, confided that she originally tested her DNA in order to better understand her maternal grandfather's origins, as she had been unable to find any record of him prior to 1918, when he was 29 years of age. Having consulted her close DNA matches on Ancestry and engaged the services of a professional, she was beyond frustrated with her lack of progress on the issue and had essentially given up hope of resolving the question of her granddad's pedigree. We offered to take a look.

The table above lists the most significant DNA matches which appeared to share JSM's maternal grandfather's line. Public member trees and educated inference allowed us to connect each of the individuals listed according to their family relationships:

Members of this group are descended from William Joseph Pearson (1866–1907) and his wife Sarah Knott (1869–1947), who represent the most recent common ancestors (MRCA) of this family group. We've added an additional generation to accommodate hypothetical ways in which JSM might be related to the Pearsons. JSM's connection to this family group would appear to be through her maternal grandfather, Albert James Harris (1889–1963), and so an inheritance chart of her family group is much more straightforward:

As the common ancestors of the Pearson family group predate JSM's grandfather, it's most probable that JSM's family group must connect to the Pearson family, rather than the vice-versa.

Further AnalysisAs such, we are seeking JSM's most probable connection to the Pearson family group — such that JSM's birthdate and those of her maternal ancestors are within the realm of genealogical plausibility and, where multiple scenarios are possible, the scenario with the greatest liklihood that the cM linkage shown in the diagram is maintained. As is often the case, DNApainter.com's What Are The Odds (WATO) Tool is ideally suited to our purposes.
Doing the MathWe'll allow WATO to generate a complement of suggested hypotheses for our Pearson family diagram. The WATO tool uses the probabilities implied by each hypothesis' relationship to the individuals sharing linkage in the Pearson family group to generate genetically — if not genealogically — compatible hypotheses, and then scores these probabilities. (click to enlarge)

Each numbered hypothesis represents a place in the tree where JSM might reside. (Note that the pink and black lines indicate full-sibling relationships, whilst the yellow lines indicate half-siblings.)
However, because we know JSM was born in 1948, her mother in 1922, and her maternal grandfather in 1889, we can immediately remove any hypothesis which violates genealogical plausibility, as shown in the following table:

And so only Hypothesis 15 — coincidentally one of the most probable hypotheses — is also genealogically possible. We'll remove the genealogically impossible hypotheses, but clone Hypothesis 15 to compare it to the liklihood that JSM's grandfather might have been the illigitimate offspring of William Pearson and an unknown woman:

• The resulting calculations show that it's more than 178 times likely that JSM's grandfather is a full sibling to Edward, Lucy, and Arthur Pearson than the heretofore unacknowledged offspring of William Pearson and an unknown woman. William and Sarah Pearson had 6 male children:

• William Pearson (1888–1963)
• Uriah Pearson (1889– )
• Edward Pearson (1897–1959)
• John Pearson (1905– )
• Joseph Pearson (1907– )
• Arthur Ernest Pearson (1912–1973)

Edward and Arthur have been ruled out by our DNA analysis, and John and Joseph would likely be too young to be the father of JSM's mother, born in 1922 — much less to have served in the Great War, as JSM's grandfather reportedly did — which leaves us with William and Uriah Pearson as possible candidates to be JSM's grandfather.
As it happens, William Pearson was already married with children when JSM's mother was born, and likewise appears in the 1939 register living with his family. However, although Uriah married in July, 1911 (and a child may have been born of the union in 1914), there appear to be no records of Uriah after his discharge from the British Army in August of 1917 — approximately one year before the earliest citations of JSM's grandfather Albert James Harris appear, as witness to a wedding. Further, particulars concerning military enlistment and discharge, written on the back of an envelope addressed to JSM's grandfather, appear to reference the same Regimental Number as is shown on Uriah Pearson's British Army World War I Medal Rolls Index Card from 1917 (blue arrow).

Amongst other engagements, Uriah's Gloucestershire Regiment appears to have fought at Gallipoli, so it's anyone's guess as to what havoc the horrors of war wreaked upon the man.
Whatever the circumstances, it would appear that following his discharge from the British Army, Uriah Pearson embarked on a new life as Albert James Harris, taking a new wife and adopting this new identity for the remainder of his days.

ConclusionDNA evidence, in addition to official records, would indicate that JSM's maternal grandfather, heretofore known as Albert James Harris, was born Uriah Pearson, adopting his Harris alias following discharge from the British Army in 1917.

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